A review of morphology–elasticity relationships in human trabecular bone: theories and experiments

New suggestions for the mechanical control of bone remodeling

Bone is constantly renewed over our lifetime through the process of bone (re)modeling. This process is important for bone to allow it to adapt to its mechanical environment and to repair damage from everyday life. Adaptation is thought to occur through the mechanosensitive response controlling the bone-forming and -resorbing cells. This report shows a way to extract quantitative information about the way remodeling is controlled using computer simulations. Bone resorption and deposition are described as two separate stochastic processes, during which a discrete bone packet is removed or deposited from the bone surface. The responses of the bone-forming and -resorbing cells to local mechanical stimuli are described by phenomenological remodeling rules. Our strategy was to test different remodeling rules and to evaluate the time evolution of the trabecular architecture in comparison to what is known from micro-CT measurements of real bone. In particular, we tested the reaction of virtual bone to standard therapeutic strategies for the prevention of bone deterioration, i.e., physical activity and medications to reduce bone resorption. Insensitivity of the bone volume fraction to reductions in bone resorption was observed in the simulations only for a remodeling rule including an activation barrier for the mechanical stimulus above which bone deposition is switched on. This is in disagreement with the commonly used rules having a so-called lazy zone.

A comparison of enhanced continuum FE with micro FE models of human vertebral bodies

Continuum finite element (FE) models are standard tools for determination of biomechanical properties of bones and bone-implant systems. This study investigates the accuracy of an enhanced continuum FE model by taking muFE as the gold standard. The enhanced continuum models account for trabecular bone morphology (density and fabric) as well as for an anatomically correct cortical shell. Vertebral body slice models are extracted from high-resolution CT images using an algorithm proposed in [Pahr and Zysset, 2008b. From high-resolution CT data to FE models: development of an integrated modular framework. Computer Methods in Biomechanics and Biomedical Engineering, in press.]. Three different models are generated: the proposed enhanced density-fabric-based model with a subject-specific cortex and two classical isotropic density-only models, with and without explicit modeling of the cortical shell. The material property errors of the used morphology-elasticity relationship are minimized by using elasticity tensors from 60 cubical muFE models which are cropped from the trabecular centrums of the investigated vertebral bodies. Two different boundary conditions-kinematic [Van Rietbergen et al., 1995. A new method to determine trabecular bone elastic properties and loading using micromechanical FE models. Journal of Biomechanics 28 (1), 69-81] and mixed [Pahr, D.H., Zysset, P.K., 2008a. Influence of boundary conditions on computed apparent elastic properties of cancellous bone. Biomechanics and Modeling in Mechanobiology 7, 463-476.]-are used in these FE models. After removal of the endplates, compressive and antero-posterior shear loading is applied on the investigated vertebral bodies. Individual error sources are studied in more detail by loading also the trabecular centrum (removed shell) and the cortical shell alone. It is found that the cortex-only models need a correction of the shell thickness when transforming from a voxel to a smooth description. The trabecular centrum alone gives too stiff and too soft a response using material calibration with kinematic and mixed boundary conditions, respectively. A comparison of the whole vertebral body stiffnesses shows that an orthotropic cancellous bone material calibrated with kinematic boundary conditions corresponds best with muFE. Taken together, the proposed enhanced homogenized surface-based FE model is structurally more accurate than density-only models.

Elastic Anisotropy of Human Cortical Bone Secondary Osteons Measured by Nanoindentation

The identification of anisotropic elastic properties of lamellar bone based on nanoindentation data is an open problem. Therefore, the purpose of this study was to develop a method to estimate the orthotropic elastic constants of human cortical bone secondary osteons using nanoindentation in two orthogonal directions. Since the indentation modulus depends on all elastic constants and, for anisotropic materials, also on the indentation direction, a theoretical model quantifying the indentation modulus from the stiffness tensor of a given material was implemented numerically (Swadener and Pharr, 2001, “Indentation of Elastically Anisotropic Half-Spaces by Cones and Parabolae of Revolution,” Philos. Mag. A, 81(2), pp. 447–466). Nanoindentation was performed on 22 osteons of the distal femoral shaft: A new holding system was designed in order to indent the same osteon in two orthogonal directions. To interpret the experimental results and identify orthotropic elastic constants, an inverse procedure was developed by using a fabric-based elastic model for lamellar bone. The experimental indentation moduli were found to vary with the indentation direction and showed a marked anisotropy. The estimated elastic constants showed different degrees of anisotropy among secondary osteons of the same bone and these degrees of anisotropy were also found to be different than the one of cortical bone at the macroscopic level. Using the log-Euclidean norm, the relative distance between the compliance tensors of the estimated mean osteon and of cortical bone at the macroscopic level was 9.69%: Secondary osteons appeared stiffer in their axial and circumferential material directions, and with a greater bulk modulus than cortical bone, which is attributed to the absence of vascular porosity in osteonal properties. The proposed method is suitable for identification of elastic constants from nanoindentation experiments and could be adapted to other (bio)materials, for which it is possible to describe elastic properties using a fabric-based model.

In vivo trabecular bone morphologic and mechanical relationship using high-resolution 3-T MRI

OBJECTIVE

The purpose of this study was to investigate the in vivo morphologic and elastic parameters of trabecular bone with high-resolution 3-T MRI in a healthy reference population.

SUBJECTS AND METHODS

A series of wrist MR images were acquired with high-spatial-resolution (180 mum) isotropic voxels from 40 subjects without reported bone disease. After image postprocessing, the bone volume-to-total volume ratio, trabecular thickness, trabecular separation, and trabecular number were calculated in the morphologic analysis. Trabecular bone was mechanically simulated using the finite-element method to calculate the apparent elastic modulus parameter. The relationship between morphologic and mechanical parameters was studied. The influence of the analyzed bone volume was also investigated.

RESULTS

Statistically significant sex influences were found on the bone volume-to-total volume ratio (p = 0.003), trabecular thickness (p = 0.02), and apparent elastic modulus (p = 0.01); these parameters were lower in women. However, trends were found only on trabecular separation (p = 0.06) and trabecular number (p = 0.07). Age had no statistically significant influence in any morphologic (bone volume-to-total volume ratio, r = -0.24, p = 0.13; trabecular thickness, r = -0.03, p = 0.88; trabecular separation, r = 0.12, p = 0.47; and trabecular number, r = -0.23, p = 0.16) or elastic (apparent elastic modulus, r = -0.18, p = 0.26) parameter. A statistically significant relationship between apparent elastic modulus and the square of bone volume-to-total volume ratio was found (r = 0.968, p < 0.001). This association was not seen (r = 0.185, p = 0.25) and apparent elastic modulus results were considerably different (p < 0.001) if the volume of analyzed bone was reduced.

CONCLUSION

We found that bone volume-to-total volume ratio, trabecular thickness, and apparent elastic modulus are parameters significantly influenced by sex. Apparent elastic modulus results show a relationship with bone volume-to-total volume ratio. Trabecular bone volume should be maximized for an appropriate mechanical analysis.

Sensitivity of qus parameters to controlled variations of bone strength assessed with a cellular model

The physical principles underlying quantitative ultrasound (QUS) measurements are not fully understood yet. Therefore, the translation of QUS results into bone strength remains elusive. In the present study, we derive the sensitivity of broadband ultrasonic attenuation (BUA) and speed of sound (SOS) to variations of bone strength. For this purpose, a mechanical cellular model is combined to a multiple regression resulting from the analysis of finite-difference time domain (FDTD) simulations. Specifically, we investigate how QUS variables respond to a variation in strength of 10%, realized either by a change in material properties or a change in bone volume fraction (BV/TV). The results show that except when BV/TV is high, the variations of BUA in response to a variation in strength realized by a pure change of BV/TV exceeds the technique imprecision and thus can be detected. When the variation of strength is realized by changes of compressive or shear stiffness, the response in QUS properties is dominated by the variation in C(11), whereas changes in C(44), remaining below the precision error, cannot be detected. The interpretation of these data, however, is not straightforward due to sparse description of elastic properties at the tissue level. To overcome the limitation of the cellular model, more realistic computational models such as micro- finite element analysis have to be considered.

A model for prediction of bone stiffness using a mechanical approach of composite materials

A model to predict the bone stiffness is presented in this paper. The objective is to obtain a description of bone stiffness of a representative elementary volume (REV) based on a small set of physical parameters. The main idea is to use measurable information related to the orientation and the density of a basic elementary submicrostructure (ESMS). This ESMS is the first arrangement of the basic components. A simple rule-of-mixtures approach is used to provide the elastic properties for the ESMS. The basic properties are dependent on the volume fraction of the mineralized phase. The orientation and the density of the ESMS is described by a tensor and a scalar, respectively. The model is used to obtain the elastic properties of both the cortical and trabecular bones. Data from femoral bone are used to verify this approach.

Spatial autocorrelation and mean intercept length analysis of trabecular bone anisotropy applied to in vivo magnetic resonance imaging

Osteoporosis is characterized by bone loss and deterioration of the trabecular bone (TB) architecture that leads to impaired overall mechanical strength of the bone. Bone mineral density (BMD) measured by dual-energy x-ray absorptiometry is currently the standard clinical metric assessing bone integrity but it fails to capture the structural changes in the TB. Recent research suggests that structure contributes to bone strength in a manner complementary to BMD. Besides parameters of scale such as the mean TB thickness and mean bone volume fraction, parameters describing the anisotropy of the trabecular architecture play an important role in the characterization of TB since trabeculae are preferentially oriented along the direction of local loading. Therefore, the degree of structural anisotropy is of pivotal importance to the bone's mechanical competence. The most common method for measuring structural anisotropy of TB is the mean-intercept length (MIL). In this work we present a method, based on the three-dimensional spatial autocorrelation function (ACF), for mapping of the full structural anisotropy ellipsoid of both TB thickness and spacing and we examine its performance as compared to that of MIL. Not only is the ACF method faster by several orders of magnitude, it is also considerably more robust to noise. Further, it is applicable at lower spatial resolution and is relatively insensitive to image shading. The chief reason for ACF's superior performance is that it does not require binarization, which is difficult to achieve in the limited spatial regime of in vivo magnetic resonance imaging. MIL and ACF have been applied to high-resolution magnetic resonances images of the tibia in a group of ten healthy postmenopausal women by comparing the structural anisotropy and principal direction of the computed fabric tensor for each method. While there is fair agreement between the two methods, ACF analysis yielded greater anisotropy than MIL for both TB thickness and spacing. There was good agreement between the two techniques as far as the eigenvectors of the fabric ellipsoids were concerned, which parallel the bone's macroscopic axis.

Quantifying anisotropy of trabecular bone from gray-level images

In this study, a gray-level image-based approach to quantifying structural anisotropy is described. The secant modulus was estimated for thirty L(3) vertebral bodies using nondestructive testing. The vertebral bodies were imaged with a clinical CT scanner. QCT measurements of BMD were also performed for trabecular regions. Structural anisotropy in trabecular regions was quantified from binarized images using the mean intercept length (MIL) method and from gray-level images using the gray-level structure tensor (GST) method. BMD alone explained 28% of the variation of the secant modulus. Multivariable regression models combining BMD and measures of anisotropy, as proposed by the relations formulated by Cowin, revealed significant improvement in the prediction of the secant modulus. Combining a principal value of the fabric tensor, as computed by either MIL or GST methods, with BMD resulted in increased correlation with the secant modulus. The highest correlation (R(2)=0.81) was achieved with a combination of BMD and the third principal value of the GST. Adding a term proportional to the minimal cross-sectional area of the vertebral body explained 86% of the variation of the secant modulus.

The role of fabric in the quasi-static compressive mechanical properties of human trabecular bone from various anatomical locations

Osteoporosis leads to an increased risk of bone fracture. While bone density and architecture can be assessed in vivo with increasing accuracy using CT and MRI, their relationship with the critical mechanical properties at various anatomical sites remain unclear. The objective of this study was to quantify the quasi-static compressive mechanical properties of human trabecular bone among different skeletal sites and compare their relationships with bone volume fraction and a measure of microstructural anisotropy called fabric. Over 600 trabecular bone samples from six skeletal sites were assessed by microCT and tested in uniaxial compression. Bone volume fraction correlated positively with elastic modulus, yield stress, ultimate stress, and the relationships depended strongly on skeletal site. The account of fabric improved these correlations substantially, especially when the data of all sites were pooled together, but the fabric-mechanical property relationships remained somewhat distinct among the anatomical sites. The study confirms that, beyond volume fraction, fabric plays an important role in determining the mechanical properties of trabecular bone and should be exploited in mechanical analysis of clinically relevant sites of the human skeleton.

Predicting fracture of the femoral neck

A prediction of the probability of safe loading of the femoral neck, based on queueing theory, is presented. The following methods have been applied: (I) criterion of bone fracture was formulated, taking into consideration the complex state of stress-strain in the porosity zones of the bone; (II) tensile stresses around pores in the stretched zone of the bone were evaluated; (III) the influence of random events of the critical regimes of loading was modeled. The evaluation of the probability of safe loading of bones was obtained based on the levels of the tensile stresses, Young's moduli and ultimate tensile stresses which are affected by the increase in bone porosity and the distribution of the pores. Examples of analysis involving typical mechanical properties of bone in areas of vascular and lacunar-canalicular porosity are demonstrated. The ranges of initial average values of effective Young's moduli and ultimate tensile strengths were taken as 15.8-17.5GPa and 83-95MPa, respectively. The present analysis discovers the existence of three levels of safe loading: (1) a relatively safe level of the nominal tensile stresses (smaller than (2.8-3.2)MPa) where the probability of safe loading is of the order of 0.95 for the bone porosity which is less than 0.15; (2) an intermediate level of safety where the nominal tensile stresses are below (4.2-4.8)MPa and the probability of safe loading is 0.89 or higher, for the same level of bone porosity; (3) a critical level of safe loading where the nominal tensile stresses are about (8.3-9.5)MPa; they lead to sharp drop of probabilities of safe loading to 0.85-0.8 if the porosity is about 0.10 and to probabilities of 0.41-0.4 if the porosity is about 0.15.

Interrelationships between electrical properties and microstructure of human trabecular bone

Microstructural changes, such as reduction of trabecular thickness and number, are characteristic signs of osteoporosis leading to diminished bone strength. Electrical and dielectric parameters might provide diagnostically valuable information on trabecular bone microstructure not extractable from bone mineral density measurements. In this study, structural properties of human trabecular bone samples (n=26) harvested from the distal femur and proximal tibia were investigated using the computed microtomography (microCT) technique. Quantitative parameters, e.g. structural model index (SMI) or trabecular bone volume fraction (BV/TV), were calculated. In addition, the samples were examined electrically over a wide frequency range (50 Hz-5 MHz) using a two-electrode impedance spectroscopy set-up. Relative permittivity, loss factor, conductivity, phase angle, specific impedance and dissipation factor were determined. Significant linear correlations were obtained between the dissipation factor and BV/TV or SMI (|r| 0.70, p<0.01, n=26). Principal component analyses, conducted on electrical and structural parameters, revealed that the high frequency principal component of the dissipation factor was significantly related to SMI (r=0.72, p<0.01, n=26). The linear combination of high and low frequency relative permittivity predicted 73% of the variation in BV/TV. To conclude, electrical and dielectric parameters of trabecular bone, especially relative permittivity and dissipation factor, were significantly and specifically related to a trabecular microstructure as characterized with microCT. The data gathered in this study constitute a useful basis for theoretical and experimental work towards the development of impedance spectroscopy techniques for detection of bone quality in vitro or in special cases of open surgery.

A sensitive method for measuring spatial orientation in bone structures

OBJECTIVES

This article introduces the newly developed line frequency deviation (LFD) method for measuring the orientation of the trabecular structure and shows that it is more sensitive than the mean intercept length (MIL) method that is commonly used.

METHODS

The LFD method, which has been developed to measure the orientation of bone on two-dimensional X-ray images, was expanded to handle three-dimensional shapes. For the purpose of comparison, both the LFD and the MIL methods were applied to micro CT scans of 24 trabecular bone samples as well as to 24 simple synthetic samples. LFD and MIL values were calculated in various directions and collected in polar plots. Next, the anisotropy was quantified by calculating the coefficient of variation as well as by fitting ellipsoids through the plots.

RESULTS

The MIL method yielded smooth rather spherical ellipsoidal polar plots with almost no sensitivity for changes in structure. The LFD method yielded more slender polar plots and more sensitivity for geometrical changes. The LFD method yielded significantly more anistropy and larger variation in anisotropy.

CONCLUSIONS

The LFD method is a more sensitive descriptor of spatial orientation of bone structures than the MIL method.

Micro-Computed Tomography Evaluation of the Glenoid Fossa and Mandibular Condyle Bone After Bilateral Vertical Ramus Mandibular Distraction in a Canine Model

The aim of this study was to quantify bone microarchitecture within the glenoid fossa and mandibular condyle following mandibular distraction osteogenesis. Eight 6- to 9-month-old male beagle dogs underwent bilateral vertical mandibular distraction with semiburied distractors (12 days of distraction at 1 mm per day). One unoperated animal served as control. After distraction the animals were divided into two groups (N = 4) and killed after 1 or 2 months of consolidation. Three-dimensional trabecular architecture was analyzed by micro-computed tomography (microCT). At both sites the overall trends were similar. In the glenoid fossa, there was decreased bone volume, trabecular number, and connectivity density and increased trabecular separation at 1 month and decreased trabecular thickness and increased structure model index compared with the control (P < 0.05). In the mandibular condyle, there was decreased bone volume, trabecular number, and connectivity density at both 1 and 2 months, with decreased trabecular thickness and increased structure model index at 2 months only compared with the control (P < 0.05). The bone became less dense and more rodlike. These bone changes are similar to those seen by the effects of aging or impaired normal function. Thus, in the short term, changes occur in the bone microstructure of the glenoid fossa and mandibular condyle after vertical mandibular ramus distraction in the canine model.

Micro-computed tomography evaluation of the glenoid fossa and mandibular condyle bone after bilateral vertical ramus mandibular distraction in a canine model

The aim of this study was to quantify bone microarchitecture within the glenoid fossa and mandibular condyle following mandibular distraction osteogenesis. Eight 6- to 9-month-old male beagle dogs underwent bilateral vertical mandibular distraction with semiburied distractors (12 days of distraction at 1 mm per day). One unoperated animal served as control. After distraction the animals were divided into two groups (N = 4) and killed after 1 or 2 months of consolidation. Three-dimensional trabecular architecture was analyzed by micro-computed tomography (muCT). At both sites the overall trends were similar. In the glenoid fossa, there was decreased bone volume, trabecular number, and connectivity density and increased trabecular separation at 1 month and decreased trabecular thickness and increased structure model index compared with the control (P < 0.05). In the mandibular condyle, there was decreased bone volume, trabecular number, and connectivity density at both 1 and 2 months, with decreased trabecular thickness and increased structure model index at 2 months only compared with the control (P < 0.05). The bone became less dense and more rodlike. These bone changes are similar to those seen by the effects of aging or impaired normal function. Thus, in the short term, changes occur in the bone microstructure of the glenoid fossa and mandibular condyle after vertical mandibular ramus distraction in the canine model.

Importance of individual rods and plates in the assessment of bone quality and their contribution to bone stiffness

Local morphometry based on the assessment of individual rods and plates was applied to 42 human vertebral trabecular bone samples. Results showed that multiple linear regression models based on local morphometry as a measure for bone microstructure helped improving our understanding of the role of local structural changes in the determination of bone stiffness as assessed from direct and computational biomechanics.

INTRODUCTION

In a recent study, we proposed a method for local morphometry of trabecular bone, i.e., morphometry as applied to individual rods and plates. In this study, we used this method to study the relative importance of local morphometry in the assessment of bone architecture and its relative contribution to the stiffness of human vertebral bone.

MATERIALS AND METHODS

We extracted 42 human trabecular bone autopsies from nine intact spinal columns. The cylindrical samples were imaged with muCT to assess bone microstructure. From these images, global and local morphometric indices were derived and related to Young's modulus as assessed by experimental uniaxial compression testing (Emeas) and computational finite element analysis (EFE).

RESULTS

We found the best single predictor for Young's modulus to be apparent bone volume density (BV/TV), which explained 89% of the variance in EFE when fitted with a power law. A multiple linear regression model combining mean trabecular spacing (Tb.Sp), mean slenderness of the rods (<Ro.Sl>), and the relative amount of rod volume to total bone volume (Ro.BV/BV) was able to explain 90% of the variance in EFE. This model could not be improved by adding BV/TV as an independent variable. Furthermore, we found that mean trabecular thickness of the rods was significantly related to EFE (r2 = 0.42), whereas mean trabecular thickness of plates had no correlation to Young's modulus. Because the globally determined trabecular thickness does not discriminate between rods and plates, this index had only a poor predictive power for EFE (r2 = 0.09), showing the importance of local analysis of individual rods and plates.

CONCLUSIONS

From these results, we conclude that models based on local morphometry help improving our understanding of the relative importance of local structural changes in the determination of the stiffness of bone. Separate analysis of individual rods and plates may help to better predict age and disease-related fractures as well as to shed new light on the effect of pharmaceutical intervention in the prevention of such fractures beyond BMD.

Volumetric spatial decomposition of trabecular bone into rods and plates--a new method for local bone morphometry

Bone microarchitecture is believed to play a key role in determining bone quality. We therefore present a new method for the volumetric spatial decomposition of trabecular bone samples into its basic elements (rods and plates). This new method is a framework for the element based description of bone microarchitecture. First, the newly developed algorithm was validated on computer-generated models. Then, it was applied to 328 human trabecular bone samples harvested from 70 donors at five different anatomical sites (calcaneus, femoral head, iliac crest, lumbar spine 2 and 4), which were previously scanned by microcomputed tomography. Standard three-dimensional morphometric algorithms were used to analyze the trabeculae on an individual basis with respect to their volume, surface, and thickness. The results were statistically compared for the five sites. In this study, it was possible for the first time to spatially decompose trabecular bone structures in its volumetric elements; rods and plates. The size of the largest element in the structures showed significant differences for the five compared sites. In samples from femoral head, we found that basically one "major element" was spanning through the whole structure whereas in lumbar spine and calcaneus, smaller elements dominate. From this, we suggest that the strength of strong, dense plate-like structures is determined by the major elements whereas in looser rod-like structures the strength is given by the arrangement, quality, and shape of a whole set of elements. Furthermore, we found that globally determined structural indices such as the mean curvature of the bone surface (<H>) or related to this the structure model index (SMI) are almost exclusively explained by the arrangement of the plates. This also suggests that rods hold independent information characterizing trabecular bone quality, especially in the spine. These findings may improve the understanding of the site-specific role of bone microarchitecture in determining bone quality and in future studies the competence of bone.

Compressive fatigue behavior of human vertebral trabecular bone

Damage accumulation under compressive fatigue loading is believed to contribute significantly to non-traumatic, age-related vertebral fractures in the human spine. Only few studies have explored trabecular bone fatigue behavior under compressive loading and none examined the influence of trabecular architecture on fatigue life. In this study, trabecular bone samples of human lumbar and thoracic vertebrae (4 donors from age 29 to 86, n=29) were scanned with a microCT system prior to compressive fatigue testing to determine morphology-mechanical relationships for this relevant loading mode. Inspired from previous fabric-based relationships for elastic properties and quasi-static strength of trabecular bone, a simple power relationship between volume fraction, fabric eigenvalue, applied stress and the number of cycles to failure is proposed. The experimental results demonstrate a high correlation for this relationship (R2=0.95) and detect a significant contribution of the degree of anisotropy towards prediction of fatigue life. Step-wise regression for total and residual strains at failure suggested a weak, but significant correlation with volume fraction. From the obtained results, we conclude that the applied stress normalized by volume fraction and axial fabric eigenvalue can estimate fatigue life of human vertebral trabecular bone in axial compressive loading.

Prediction of density and mechanical properties of human trabecular bone in vitro by using ultrasound transmission and backscattering measurements at 0.2-6.7 MHz frequency range

The ultrasound (US) backscattering method has been introduced as an alternative for the through-transmission measurement of sound attenuation and speed in diagnosis of osteoporosis. Both attenuation and backscattering depend strongly on the US frequency. In this study, 20 human trabecular bone samples were measured in transmission and pulse-echo geometry in vitro. The aim of the study was to find the most sensitive frequency range for the quantitative ultrasound (QUS) analyses. Normalized broadband US attenuation (nBUA), speed of sound (SOS), broadband US backscatter (BUB) and integrated reflection coefficient (IRC) were determined for each sample. The samples were spatially scanned with five pairs of US transducers covering a frequency range of 0.2-6.7 MHz. Furthermore, mechanical properties and density of the same samples were determined. At all frequencies, SOS, BUB and IRC showed statistically significant linear correlations with the mechanical properties or density of human trabecular bone (0.51 < r < 0.82, 0.54 < r < 0.81 and 0.70 < r < 0.85, respectively). In contrast to SOS, IRC and BUB, nBUA showed statistically significant correlations with mechanical parameters or density at the centre frequency of 1 MHz only. Our results suggest that frequencies up to 5 MHz can be useful in QUS analyses for the prediction of bone mechanical properties and density. Since the use of higher frequencies provides better axial and spatial resolution, improved structural analyses may be possible. While extensive attenuation of high frequencies in trabecular bone limits the clinically feasible frequency range, selection of optimal frequency range for in vivo QUS application should be carefully considered.

Modelling of anisotropic growth in biological tissues. A new approach and computational aspects

In this contribution, we develop a theoretical and computational framework for anisotropic growth phenomena. As a key idea of the proposed phenomenological approach, a fibre or rather structural tensor is introduced, which allows the description of transversely isotropic material behaviour. Based on this additional argument, anisotropic growth is modelled via appropriate evolution equations for the fibre while volumetric remodelling is realised by an evolution of the referential density. Both the strength of the fibre as well as the density follow Wolff-type laws. We however elaborate on two different approaches for the evolution of the fibre direction, namely an alignment with respect to strain or with respect to stress. One of the main benefits of the developed framework is therefore the opportunity to address the evolutions of the fibre strength and the fibre direction separately. It is then straightforward to set up appropriate integration algorithms such that the developed framework fits nicely into common, finite element schemes. Finally, several numerical examples underline the applicability of the proposed formulation.

Ability of ultrasound backscattering to predict mechanical properties of bovine trabecular bone

Ultrasound (US) backscatter measurements have been proposed for the quantitative evaluation of bone quality. In this study, we explored the ability of broadband US backscatter (BUB) and integrated reflection coefficient (IRC) to predict density and mechanical properties of trabecular bone, as compared to normalized broadband US attenuation (nBUA) and speed of sound (SOS). These acoustic parameters were measured in 41 in vitro samples of bovine trabecular bone and correlated with a number of mechanical parameters and with volumetric bone mineral density (BMDvol). BUB correlated statistically significantly with the volumetric bone mineral density (r = 0.61, p < 0.01), Young's modulus (r = 0.40, p < 0.01) and ultimate strength (r = 0.40, p < 0.01). IRC was even more strongly correlated with BMD(vol) (r = 0.92, p < 0.01) and most of the mechanical parameters (0.81 < r < 0.85). Strong correlations were also found between mechanical parameters and SOS (0.87 < r < 0.90). No significant correlation was found between attenuation (nBUA) and either BMD(vol) or mechanical parameters. Reproducibilities (standardized CV%) of BUB (3.5%) and IRC (1.5%) were comparable to those of nBUA (2.3%) and SOS (0.5%). To conclude, BUB and IRC are promising parameters for the evaluation of density and mechanical properties of trabecular bone. Advantageously, BUB and IRC can be determined with a single transducer, hypothetically enabling measurements at many clinically relevant fracture sites.