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

Bulk Wave Velocities in Cortical Bone Reflect Porosity and Compression Strength

The goal of this study was to evaluate whether ultrasonic velocities in cortical bone can be considered as a proxy for mechanical quality of cortical bone tissue reflected by porosity and compression strength. Micro-computed tomography, compression mechanical testing and resonant ultrasound spectroscopy were used to assess, respectively, porosity, strength and velocity of bulk waves of both shear and longitudinal polarisations propagating along and perpendicular to osteons, in 92 cortical bone specimens from tibia and femur of elderly human donors. All velocities were significantly associated with strength (r = 0.65-0.83) and porosity (r = -0.64 to -0.77). Roughly, according to linear regression models, a decrease in velocity of 100 m/s corresponded to a loss of 20 MPa in strength (which is approximately 10% of the largest strength value) and to an increase in porosity of 5%. These results provide a rationale for the in vivo measurement of one or several velocities for the diagnosis of bone fragility.

Bone cortical thickness and porosity assessment using ultrasound guided waves: An ex vivo validation study

Several studies showed the ability of the cortex of long bones such as the radius and tibia to guide mechanical waves. Such experimental evidence has given rise to the emergence of a category of quantitative ultrasound techniques, referred to as the axial transmission, specifically developed to measure the propagation of ultrasound guided waves in the cortical shell along the axis of long bones. An ultrasound axial transmission technique, with an automated approach to quantify cortical thickness and porosity is described. The guided modes propagating in the cortex are recorded with a 1-MHz custom made linear transducer array. Measurement of the dispersion curves is achieved using a two-dimensional spatio-temporal Fourier transform combined with singular value decomposition. Automatic parameters identification is obtained through the solution of an inverse problem in which the dispersion curves are predicted with a two-dimensional transverse isotropic free plate model. Thirty-one radii and fifteen tibiae harvested from human cadavers underwent axial transmission measurements. Estimates of cortical thickness and porosity were obtained on 40 samples out of 46. The reproducibility, given by the root mean square error of the standard deviation of estimates, was 0.11 mm for thickness and 1.9% for porosity. To assess accuracy, site-matched micro-computed tomography images of the bone specimens imaged at 9 μm voxel size served as the gold standard. Agreement between micro-computed tomography and axial transmission for quantification of thickness and porosity at the radius and tibia ranged from R²=0.63 for porosity (root mean square error RMSE=1.8%) to 0.89 for thickness (RMSE=0.3 mm). Despite an overall good agreement for porosity, the method performs less well for porosities lower than 10%. The heterogeneity and general complexity of cortical bone structure, which are not fully accounted for by our model, are suspected to weaken the model approximation. This study presents the first validation study for assessing cortical thickness and porosity using the axial transmission technique. The automatic signal processing minimizes operator-dependent errors for parameters determination. Recovering the waveguide characteristics, that is to say cortical thickness and porosity, could provide reliable information about skeletal status and future fracture risk.

Characterization of a new ultrasound device designed for measuring cortical porosity at the human tibia: A phantom study

Quantitative ultrasound (QUS) measurements of trabecular bone are a useful tool for the assessment of osteoporotic fracture risk. However, cortical bone properties (e.g. porosity) have an impact on bone strength as well and thus current research is focused on QUS assessment of cortical bone properties. Simulation studies of ultrasound propagation through cortical bone indicate that anisotropy, calculated from the ratio of the velocities in axial and tangential directions, is correlated with porosity. However, this relationship is affected by error sources, specifically bone surface curvature and variability of probe positioning. With the aim of in vivo estimation of cortical porosity a new ultrasound device was developed, which sequentially measures velocities in 3 different directions (axial=0° and ±37.5°) using the axial transmission method. Measurements on planar porosity phantoms (0-25%) were performed to confirm the results of the afore mentioned simulation studies. Additionally, measurements on cylindrical phantoms without pores (min. radius=34mm for strongest curvature) were performed to estimate the influence of surface curvature on velocity measurements (the tibia bone surface is fairly flat but may show surface curvature in some patients). The velocities in the axial and ±37.5° directions were used to calculate an anisotropy index. The velocities measured on the porosity phantoms showed a decrease by -6.3±0.2m/s and -10.1±0.2m/s per percent increase in porosity in axial and ±37.5° directions, respectively. Surface curvature had an effect on the velocities measured in ±37.5° directions which could be minimized by a correction algorithm resulting in an error of 5m/s. The anisotropy index could be used to estimate porosity with an accuracy error of 1.5%. These results indicate that an estimation of porosity using velocity measurements in different directions might be feasible, even in bones with curved surface. These results obtained on phantom material indicate that the approach tested may be suited for porosity measurements on human tibia bone.

Predicting bone strength with ultrasonic guided waves

Recent bone quantitative ultrasound approaches exploit the multimode waveguide response of long bones for assessing properties such as cortical thickness and stiffness. Clinical applications remain, however, challenging, as the impact of soft tissue on guided waves characteristics is not fully understood yet. In particular, it must be clarified whether soft tissue must be incorporated in waveguide models needed to infer reliable cortical bone properties. We hypothesize that an inverse procedure using a free plate model can be applied to retrieve the thickness and stiffness of cortical bone from experimental data. This approach is first validated on a series of laboratory-controlled measurements performed on assemblies of bone- and soft tissue mimicking phantoms and then on in vivo measurements. The accuracy of the estimates is evaluated by comparison with reference values. To further support our hypothesis, these estimates are subsequently inserted into a bilayer model to test its accuracy. Our results show that the free plate model allows retrieving reliable waveguide properties, despite the presence of soft tissue. They also suggest that the more sophisticated bilayer model, although it is more precise to predict experimental data in the forward problem, could turn out to be hardly manageable for solving the inverse problem.

Effect of porosity, tissue density, and mechanical properties on radial sound speed in human cortical bone

PURPOSE

The purpose of this study was to investigate the effect of simultaneous changes in cortical porosity, tissue mineral density, and elastic properties on radial speed of sound (SOS) in cortical bone. The authors applied quantitative pulse-echo (PE) ultrasound techniques that hold much potential especially for screening of osteoporosis at primary healthcare facilities. Currently, most PE measurements of cortical thickness, a well-known indicator of fracture risk, use a predefined estimate for SOS in bone to calculate thickness. Due to variation of cortical bone porosity, the use of a constant SOS value propagates to an unknown error in cortical thickness assessment by PE ultrasound.

METHODS

The authors conducted 2.25 and 5.00 MHz focused PE ultrasound time of flight measurements on femoral diaphyses of 18 cadavers in vitro. Cortical porosities of the samples were determined using microcomputed tomography and related to SOS in the samples. Additionally, the effect of cortical bone porosity and mechanical properties of the calcified matrix on SOS was investigated using numerical finite difference time domain simulations.

RESULTS

Both experimental measurements and simulations demonstrated significant negative correlation between radial SOS and cortical porosity (R(2) ≥ 0.493, p < 0.01 and R(2) ≥ 0.989, p < 0.01, respectively). When a constant SOS was assumed for cortical bone, the error due to variation of cortical bone porosity (4.9%-16.4%) was about 6% in the cortical thickness assessment in vitro.

CONCLUSIONS

Use of a predefined, constant value for radial SOS in cortical bone, i.e., neglecting the effect of measured variation in cortical porosity, propagated to an error of 6% in cortical thickness. This error can be critical as characteristic cortical thinning of 1.10% ± 1.06% per yr decreases bending strength of the distal radius and results in increased fragility in postmenopausal women. Provided that the cortical porosity can be estimated in vivo, the relationship between radial SOS and cortical porosity can be utilized and a porosity based radial SOS estimate could be implemented to determine cortical thickness. This would constitute a step toward individualized quantitative ultrasound diagnostics of osteoporosis.

Inter-individual changes in cortical bone three-dimensional microstructure and elastic coefficient have opposite effects on radial sound speed

Knowledge about simultaneous contributions of tissue microstructure and elastic properties on ultrasound speed in cortical bone is limited. In a previous study, porosities and elastic coefficients of cortical bone in human femurs were shown to change with age. In the present study, influences of inter-individual and site-dependent variation in cortical bone microstructure and elastic properties on radial speed of sound (SOS; at 4, 6, and 8 MHz) were investigated using three-dimensional (3D) finite difference time domain modeling. Models with fixed (nominal model) and sample-specific (sample-specific model) values of radial elastic coefficients were compared. Elastic coefficients and microstructure for samples (n = 24) of human femoral shafts (n = 6) were derived using scanning acoustic microscopy and micro-computed tomography images, respectively. Porosity-related SOS varied more extensively in nominal models than in sample-specific models. Linear correlation between pore separation and SOS was similar (R = 0.8, p < 0.01, for 4 MHz) for both models. The determination coefficient (R(2)= 0.75, p < 0.05) between porosity and radial SOS, especially at 4 MHz, was highest in the posterior quadrant. The determination coefficient was lower for models with sample-specific values of radial elastic coefficient implemented (R(2) < 0.33, p < 0.05), than for nominal models (0.48 < R(2)< 0.63, p < 0.05). This information could be useful in in vivo pulse-echo cortical thickness measurements applying constant SOS.

To what extent can cortical bone millimeter-scale elasticity be predicted by a two-phase composite model with variable porosity?

An evidence gap exists in fully understanding and reliably modeling the variations in elastic anisotropy that are observed at the millimeter scale in human cortical bone. The porosity (pore volume fraction) is known to account for a large part, but not all, of the elasticity variations. This effect may be modeled by a two-phase micromechanical model consisting of a homogeneous matrix pervaded by cylindrical pores. Although this model has been widely used, it lacks experimental validation. The aim of the present work is to revisit experimental data (elastic coefficients, porosity) previously obtained from 21 cortical bone specimens from the femoral mid-diaphysis of 10 donors and test the validity of the model by proposing a detailed discussion of its hypotheses. This includes investigating to what extent the experimental uncertainties, pore network modeling, and matrix elastic properties influence the model's predictions. The results support the validity of the two-phase model of cortical bone which assumes that the essential source of variations of elastic properties at the millimeter-scale is the volume fraction of vascular porosity. We propose that the bulk of the remaining discrepancies between predicted stiffness coefficients and experimental data (RMSE between 6% and 9%) is in part due to experimental errors and part due to small variations of the extravascular matrix properties. More significantly, although most of the models that have been proposed for cortical bone were based on several homogenization steps and a large number of variable parameters, we show that a model with a single parameter, namely the volume fraction of vascular porosity, is a suitable representation for cortical bone. The results could provide a guide to build specimen-specific cortical bone models. This will be of interest to analyze the structure-function relationship in bone and to design bone-mimicking materials.

Combined estimation of thickness and velocities using ultrasound guided waves: a pioneering study on in vitro cortical bone samples

This paper reports for the first time on inverse estimation of several bone properties from guided-wave measurements in human bone samples. Previously, related approaches have focused on ultrasonic estimation of a single bone property at a time. The method is based on two steps: the multi-Lamb mode response is analyzed using the singular value decomposition signal processing method recently introduced in the field, then an identification procedure is run to find thickness and anisotropic elastic properties of the considered specimen. Prior to the measurements on bone, the method is validated on cortical bone-mimicking phantoms. The repeatability and the trueness of the estimated parameters on bone-mimicking phantoms were found around a few percent. Estimation of cortical thickness on bone samples was in good agreement with cortical thickness derived from high-resolution peripheral quantitative computed tomography data analysis of the samples.

Ultrasound to assess bone quality

Bone quality is determined by a variety of compositional, micro- and ultrastructural properties of the mineralized tissue matrix. In contrast to X-ray-based methods, the interaction of acoustic waves with bone tissue carries information about elastic and structural properties of the tissue. Quantitative ultrasound (QUS) methods represent powerful alternatives to ionizing x-ray based assessment of fracture risk. New in vivo applicable methods permit measurements of fracture-relevant properties, [eg, cortical thickness and stiffness at fragile anatomic regions (eg, the distal radius and the proximal femur)]. Experimentally, resonance ultrasound spectroscopy and acoustic microscopy can be used to assess the mesoscale stiffness tensor and elastic maps of the tissue matrix at microscale resolution, respectively. QUS methods, thus, currently represent the most promising approach for noninvasive assessment of components of fragility beyond bone mass and bone microstructure providing prospects for improved assessment of fracture risk.

Using the gradient of human cortical bone properties to determine age-related bone changes via ultrasonic guided waves

Bone fragility depends not only on bone mass but also on bone quality (structure and material). To accurately evaluate fracture risk or propose therapeutic treatment, clinicians need a criterion, which reflects the determinants of bone strength: geometry, structure and material. In human long bone, the changes due to aging, accentuated by osteoporosis are often revealed through the trabecularization of cortical bone, i.e., increased porosity of endosteal bone inducing a thinning of the cortex. Consequently, the intracortical porosity gradient corresponding to the spatial variation in porosity across the cortical thickness is representative of loss of mass, changes in geometry (thinning) and variations in structure (porosity). This article examines the gradient of material properties and its age-related evolution as a relevant parameter to assess bone geometry, structure and material. By applying a homogenization process, cortical bone can be considered as an anisotropic functionally graded material with variations in material properties. A semi-analytical method based on the sextic Stroh formalism is proposed to solve the wave equation in an anisotropic functionally graded waveguide for two geometries, a plate and a tube, without using a multilayered model to represent the structure. This method provides an analytical solution called the matricant and explicitly expressed under the Peano series expansion form. Our findings indicate that ultrasonic guided waves are sensitive to the age-related evolution of realistic gradients in human bone properties across the cortical thickness and have their place in a multimodal clinical protocol.

Relative contributions of porosity and mineralized matrix properties to the bulk axial ultrasonic wave velocity in human cortical bone

Velocity of ultrasound waves has proved to be a useful indicator of bone biomechanical competence. A detailed understanding of the dependence of ultrasound parameters such as velocity on bone characteristics is a key to the development of bone quantitative ultrasound (QUS). The objective of this study is to investigate the relative contributions of porosity and mineralized matrix properties to the bulk compressional wave velocity (BCV) along the long bone axis. Cross-sectional slabs from the diaphysis of four human femurs were included in the study. Seven regions of interest (ROIs) were selected in each slab. BCV was measured in through-transmission at 5 MHz. Impedance of the mineralized matrix (Z(m)) and porosity (Por) were obtained from 50 MHz scanning acoustic microscopy. Por and Z(m) had comparable effects on BCV along the bone axis (R=-0.57 and R=0.72, respectively).

Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women

At the mesoscale (i.e. over a few millimeters), cortical bone can be described as two-phase composite material consisting of pores and a dense mineralized matrix. The cortical porosity is known to influence the mesoscopic elasticity. Our objective was to determine whether the variations of porosity are sufficient to predict the variations of bone mesoscopic anisotropic elasticity or if change in bone matrix elasticity is an important factor to consider. We measured 21 cortical bone specimens prepared from the mid-diaphysis of 10 women donors (aged from 66 to 98 years). A 50-MHz scanning acoustic microscope (SAM) was used to evaluate the bone matrix elasticity (reflected in impedance values) and porosity. Porosity evaluation with SAM was validated against Synchrotron Radiation μCT measurements. A standard contact ultrasonic method was applied to determine the mesoscopic elastic coefficients. Only matrix impedance in the direction of the bone axis correlated to mesoscale elasticity (adjusted R(2)=[0.16-0.25], p<0.05). The mesoscopic elasticity was found to be highly correlated to the cortical porosity (adj-R(2)=[0.72-0.84], p<10(-5)). Multivariate analysis including both matrix impedance and porosity did not provide a better statistical model of mesoscopic elasticity variations. Our results indicate that, for the elderly population, the elastic properties of the mineralized matrix do not undergo large variations among different samples, as reflected in the low coefficients of variation of matrix impedance (less than 6%). This work suggests that change in the intracortical porosity accounts for most of the variations of mesoscopic elasticity, at least when the analyzed porosity range is large (3-27% in this study). The trend in the variation of mesoscale elasticity with porosity is consistent with the predictions of a micromechanical model consisting of an anisotropic matrix pervaded by cylindrical pores.

Propagation of elastic waves in an anisotropic functionally graded hollow cylinder in vacuum

As a non-destructive, non-invasive and non-ionizing evaluation technique for heterogeneous media, the ultrasonic method is of major interest in industrial applications but especially in biomedical fields. Among the unidirectionally heterogeneous media, the continuously varying media are a particular but widespread case in natural materials. The first studies on laterally varying media were carried out by geophysicists on the Ocean, the atmosphere or the Earth, but the teeth, the bone, the shells and the insects wings are also functionally graded media. Some of them can be modeled as planar structures but a lot of them are curved media and need to be modeled as cylinders instead of plates. The present paper investigates the influence of the tubular geometry of a waveguide on the propagation of elastic waves. In this paper, the studied structure is an anisotropic hollow cylinder with elastic properties (stiffness coefficients c(ij) and mass density ρ) functionally varying in the radial direction. An original method is proposed to find the eigenmodes of this waveguide without using a multilayered model for the cylinder. This method is based on the sextic Stroh's formalism and an analytical solution, the matricant, explicitly expressed under the Peano series expansion form. This approach has already been validated for the study of an anisotropic laterally-graded plate (Baron et al., 2007; Baron and Naili, 2010) [6,5]. The dispersion curves obtained for the radially-graded cylinder are compared to the dispersion curves of a corresponding laterally-graded plate to evaluate the influence of the curvature. Preliminary results are presented for a tube of bone in vacuum modelling the in vitro conditions of bone strength evaluation.

Determination of the heterogeneous anisotropic elastic properties of human femoral bone: from nanoscopic to organ scale

Cortical bone is a multiscale composite material. Its elastic properties are anisotropic and heterogeneous across its cross-section, due to endosteal bone resorption which might affect bone strength. The aim of this paper was to describe a homogenization method leading to the estimation of the variation of the elastic coefficients across the bone cross-section and along the bone longitudinal axis. The method uses the spatial variations of bone porosity and of the degree of mineralization of the bone matrix (DMB) obtained from the analysis of 3-D synchrotron micro-computed tomography images. For all three scales considered (the foam (100 nm), the ultrastructure (5 microm) and the mesoscale (500 microm)), the elastic coefficients were determined using the Eshelby's inclusion problem. DMB values were used at the scale of the foam. Collagen was introduced at the scale of the ultrastructure and bone porosity was introduced at the mesoscale. The pores were considered as parallel cylinders oriented along the bone axis. Each elastic coefficient was computed for different regions of interest, allowing an estimation of its variations across the bone cross-section and along the bone longitudinal axis. The method was applied to a human femoral neck bone specimen, which is a site of osteoporotic fracture. The computed elastic coefficients for cortical bone were in good agreement with experimental results, but some discrepancies were obtained in the endosteal part (trabecular bone). These results highlight the importance of accounting for the heterogeneity of cortical bone properties across bone cross-section and along bone longitudinal axis.

Influence of viscoelastic and viscous absorption on ultrasonic wave propagation in cortical bone: Application to axial transmission

Cortical bone and the surrounding soft tissues are attenuating and heterogeneous media, which might affect the signals measured with axial transmission devices. This work aims at evaluating the effect of the heterogeneous acoustic absorption in bone and in soft tissues on the bone ultrasonic response. Therefore, a two-dimensional finite element time-domain method is derived to model transient wave propagation in a three-layer medium composed of an inhomogeneous transverse isotropic viscoelastic solid layer, sandwiched between two viscous fluid layers. The model couples viscous acoustic propagation in both fluid media with the anisotropic viscoelastic response of the solid. A constant spatial gradient of material properties is considered for two values of bone thicknesses (0.6 and 4 mm). In the studied configuration, absorption in the surrounding fluid tissues does not affect the results, whereas bone viscoelastic properties have a significant effect on the first arriving signal (FAS) velocity. For a thin bone, the FAS velocity is governed by the spatially averaged bone properties. For a thick bone, the FAS velocity may be predicted using a one-dimensional model.

Propagation of elastic waves in a fluid-loaded anisotropic functionally graded waveguide: application to ultrasound characterization

Non-destructive evaluation of heterogeneous materials is of major interest not only in industrial but also in biomedical fields. In this work, the studied structure is a three-layered one: A laterally heterogeneous anisotropic solid layer is sandwiched between two acoustic fluids. An original method is proposed to solve the wave equation in such a structure without using a multilayered model for the plate. This method is based on an analytical solution, the matricant, explicitly expressed under the Peano series expansion form. This approach is validated for the study of a fluid-loaded anisotropic and homogeneous plane waveguide with two different fluids on each side. Then, original results are given on the propagation of elastic waves in an asymmetrically fluid-loaded waveguide with laterally varying properties. This configuration notably corresponds to the axial transmission technique to the ultrasound characterization of cortical bone in vivo.

Influence of a gradient of material properties on ultrasonic wave propagation in cortical bone: application to axial transmission

The aim of this work is to evaluate the effect of a spatial gradient of material properties (mass density and stiffness coefficients) of cortical bone on its ultrasonic response obtained with an axial transmission device. Therefore, a two-dimensional finite element time-domain method is derived to model transient wave propagation in a three-layer medium composed of an inhomogeneous transverse isotropic solid layer sandwiched between two acoustic fluid layers and excited by an acoustic linear source located in one fluid layer, delivering broadband ultrasonic pulses. The model couples the acoustic propagation in both fluid media with the elastodynamic response of the solid layer. A constant spatial gradient of material properties is considered for two values of bone thicknesses corresponding to relatively thick and thin bone widths. For a thin bone (0.6 mm) compared to wavelength (around 4 mm at 1 MHz), the results are in good agreement with a S(0) Lamb wave assuming a homogeneous material with spatially averaged material properties. For a thick bone (4 mm), the results are in agreement with the propagation of a lateral wave and allow the derivation of an equivalent contributing depth in the case of a transverse isotropic inhomogeneous solid layer.

The influence of mesoscale porosity on cortical bone anisotropy. Investigations via asymptotic homogenization

Recently, the mesoscale of cortical bone has been given particular attention in association with novel experimental techniques such as nanoindentation, micro-computed X-ray tomography and quantitative scanning acoustic microscopy (SAM). A need has emerged for reliable mathematical models to interpret the related microscopic and mesoscopic data in terms of effective elastic properties. In this work, a new model of cortical bone elasticity is developed and used to assess the influence of mesoscale porosity on the induced anisotropy of the material. Only the largest pores (Haversian canals and resorption cavities), characteristic of the mesoscale, are considered. The input parameters of the model are derived from typical mesoscale experimental data (e.g. SAM data). We use the method of asymptotic homogenization to determine the local effective elastic properties by modelling the propagation of low-frequency elastic waves through an idealized material that models the local mesostructure. We use a novel solution of the cell problem developed by Parnell & Abrahams. This solution is stable for the physiological range of variation of mesoscopic porosity and elasticity found in bone. Results are computed efficiently (in seconds) and the solutions can be implemented easily by other workers. Parametric studies are performed in order to assess the influence of mesoscopic porosity, the assumptions regarding the material inside the mesoscale pores (drained or undrained bone) and the shape of pores. Results are shown to be in good qualitative agreement with existing schemes and we describe the potential of the scheme for future use in modelling more complex microstructures for cortical bone. In particular, the scheme is shown to be a useful tool with which to predict the qualitative changes in anisotropy due to variations in the structure at the mesoscale.

Spatial distribution of anisotropic acoustic impedance assessed by time-resolved 50-MHz scanning acoustic microscopy and its relation to porosity in human cortical bone

We used quantitative scanning acoustic microscopy (SAM) to assess tissue acoustic impedance and microstructure of cortical bone of human radii with the aim to provide data on regional distribution of acoustic impedance along the circumferential and across the radial directions in the entire cross-section of the radius diaphysis as well as to determine the range of impedance values in transverse (perpendicular to bone axis) and longitudinal (parallel to bone axis) cross-sections. Several microstructural features related to cortical porosity were analyzed in order to determine whether these features differ in different parts of the cortex and to assess the relationship between the microstructure and tissue acoustic impedance. Fifteen fresh bone specimens (human radius) were investigated using a SAM (center frequency of 50 MHz and -6 dB lateral resolution of approximately 23 microm). The sample acoustic impedance was obtained by means of a calibration curve correlating the reflected signal amplitude of reference materials with their corresponding well-known acoustic impedance. Tissue acoustic impedance and microstructural features were derived from the morphometric analysis of the segmented impedance images. A higher porosity was found in the inner cortical layer (mean+/-SD=8.9+/-2.3%) compared to the peripheral layer (2.7+/-1.5%) (paired t-test, p<10(-5)). ANOVA showed that most of the variance can be explained by the regional effect across the radial direction with a minor contribution due to between-sample variability. Similar to porosity, the number and diameter of pores were greater in the inner layer. In contrast to porosity, ANOVA showed that impedance variability can mostly be explained by between-specimen variability. Two-way ANOVA revealed that after compensation for the between-sample variability the variation in acoustic impedance across the radial direction was much larger than that along the circumferential direction. In addition to the significant difference between the inner cortical layer (8.25+/-0.4 Mrayl) and peripheral layer (8.0+/-0.5 Mrayl) (unilateral paired t-test, p<10(-4)), the values in the anterior region (8.2+/-0.5 Mrayl) were found to be significantly higher than those of the posterior region (7.9+/-0.6 Mrayl). Impedance mean value of longitudinal sections was lower than mean value measured in transverse cross-sections, resulting in an impedance acoustic anisotropy ratio of 1.17+/-0.03 in the inner cortical layer and 1.19+/-0.02 in the peripheral layer. SAM is a valuable tool to provide data on the spatial distribution of microstructural and microelastic bone properties that is useful to improve our understanding of the impact of bone microstructure on tissue material properties.

Ultrasound simulation in bone

The manner in which ultrasound interacts with bone is of key interest in therapy and diagnosis alike. These may include applications directly to bone, as, for example, in treatment to accelerate the healing of bone fractures and in assessment of bone density in osteoporosis, or indirectly in diagnostic imaging of soft tissue with interest in assessing exposure levels to nearby bone. Because of the lack of analytic solutions to virtually every "practical problem" encountered clinically, ultrasound simulation has become a widely used technique for evaluating ultrasound interactions in bone. This paper provides an overview of the use of ultrasound simulation in bone. A brief description of the mathematical model used to characterize ultrasound propagation in bone is first provided. A number of simulation examples are then presented that explain how simulation may be utilized in a variety of practical configurations. The focus of this paper in terms of examples presented is on diagnostic applications in bone, and, in particular, for assessment of osteoporosis. However, the use of simulation in other areas of interest can easily be extrapolated from the examples presented. In conclusion, this paper describes the use of ultrasound simulation in bone and demonstrates the power of computational methods for ultrasound research in general and tissue and bone applications in particular.