The anisotropic Hooke's law for cancellous bone and wood

Studying trabecular bone samples demonstrates a power law relation between deteriorated structure and mechanical properties - a study combining 3D printing with the finite element method

Introduction

The bone volume fraction (BV/TV) significantly contributes to the mechanical properties of trabecular bone. However, when studies compare normal trabeculae against osteoporotic trabeculae (in terms of BV/TV decrease), only an "average" mechanical result has been determined because of the limitation that no two trabecular structures are the same and that each unique trabecular structure can be mechanically tested only once. The mathematic relation between individual structural deterioration and mechanical properties during aging or the osteoporosis process has yet to be further clarified. Three-dimensional (3D) printing and micro-CT-based finite element method (μFEM) can assist in overcoming this issue.

Methods

In this study, we 3D printed structural-identical but BV/TV value-attenuated trabecular bones (scaled up ×20) from the distal femur of healthy and ovariectomized rats and performed compression mechanical tests. Corresponding μFEM models were also established for simulations. The tissue modulus and strength of 3D printed trabecular bones as well as the effective tissue modulus (denoted as Ez) derived from μFEM models were finally corrected by the side-artifact correction factor.

Results

The results showed that the tissue modulus _corrected, strength _corrected and Ez _corrected exhibited a significant power law function of BV/TV in structural-identical but BV/TV value-attenuated trabecular samples.

Discussion

Using 3D printed bones, this study confirms the long-known relationship measured in trabecular tissue with varying volume fractions. In the future, 3D printing may help us attain better bone strength evaluations and even personal fracture risk assessments for patients who suffer from osteoporosis.

Can DXA image-based deep learning model predict the anisotropic elastic behavior of trabecular bone?

3D image-based finite element (FE) and bone volume fraction (BV/TV)/fabric tensor modeling techniques are currently used to determine the apparent stiffness tensor of trabecular bone for assessing its anisotropic elastic behavior. Inspired by the recent success of deep learning (DL) techniques, we hypothesized that DL modeling techniques could be used to predict the apparent stiffness tensor of trabecular bone directly using dual-energy X-ray absorptiometry (DXA) images. To test the hypothesis, a convolutional neural network (CNN) model was trained and validated to predict the apparent stiffness tensor of trabecular bone cubes using their DXA images. Trabecular bone cubes obtained from human cadaver proximal femurs were used to obtain simulated DXA images as input, and the apparent stiffness tensor of the trabecular cubes determined by using micro-CT based FE simulations was used as output (ground truth) to train the DL model. The prediction accuracy of the DL model was evaluated by comparing it with the micro-CT based FE models, histomorphometric parameter based multiple linear regression models, and BV/TV/fabric tensor based multiple linear regression models. The results showed that DXA image-based DL model achieved high fidelity in predicting the apparent stiffness tensor of trabecular bone cubes (R² = 0.905-0.973), comparable to or better than the histomorphometric parameter based multiple linear regression and BV/TV/fabric tensor based multiple linear regression models, thus supporting the hypothesis of this study. The outcome of this study could be used to help develop DXA image-based DL techniques for clinical assessment of bone fracture risk.

Mapping cortical bone stiffness and mineralization from endosteal to periosteal surfaces of bovine mid-diaphyseal femur

INTRODUCTION

While bone literature abounds with correlations of mechanical stiffness to mineralization, such correlations are reported without relating the findings to specific intracortical locations. This study reports on mapping of stiffness and mineralization distributions in ring-shaped cortical bone samples sliced from mid-diaphyseal bovine femur. Stiffness and mineralization measurements were conducted at points across the intracortical thickness along radial lines emanating from the inner (endosteal) surface to the outer (periosteal) surface. Measurements were taken along approximately 4 mm distance of cortical bone thickness.

MATERIALS AND METHODS

Three experimental techniques were employed: Vickers microhardness (HV), energy-dispersive X-ray (EDX) spectroscopy, and computed tomography (CT). Stiffness values were extracted from the Vickers microhardness tests. Elemental mineralization values (calcium %wt. and phosphorus %wt.) were determined from EDX data. All measurements were repeated on three different femur bones taken from different bovines (collected fresh from butcher).

RESULTS

The study plots stiffness values and elemental mineralization (calcium %wt. and phosphorus %wt.) versus cortical thickness. Both stiffness and Ca %wt. and P %wt. are found to track and to linearly increase when plotted along the radial distance. The stiffness and mineralization trends collected from Vickers and EDX measurements were verified by employing the CT number (Hounsfield units, HU) via CT scans of the same bone samples. Data fitting via statistical methods revealed that all correlations were statistically significant.

CONCLUSION

Starting from endosteal to periosteal surfaces of mid-diaphyseal bovine femur, it was found that stiffness, mineralization, and HU values all exhibit increasing and correlating trends.

Mechanism and microstructure based concept to predict skull fracture using a hybrid-experimental-modeling-computational approach

Cellular and tissue-scale indent/impact thresholds for different mechanisms of functional impairments to the brain would be the preferred method to predict head injuries, but a comprehensive understanding of the dominant possible injury mechanisms under multiaxial stress-states and rates is currently not available. Until then, skull fracture could serve as an indication of head injury. Therefore the ability to predict the initiation of skull fracture through finite element simulation can serve as an in silico tool for assessing the effectiveness of various head protection scenarios. For this objective, skull fracture initiation was represented with a microstructurally-inspired, mechanism-based (MIMB) failure surface assuming three different dominant mechanisms of skull failure: each element, with deformation and failure properties selected based on its microstructure, was allowed to fail either in tension, compression, or shear, corresponding to clinical linear, depressed or penetrating shear-plug failure (fracture), respectively. Microstructure-inspired a priori values for the initiation threshold of each mechanism, obtained previously from uniaxial and simple-shear experiments, were iterated and optimized for the predicted load-displacement to represent that of the corresponding indentation experiment. Element-level failure enabled the visualization of the evolution of fracture by different mechanisms. The final crack pattern at the time of macroscopic (clinically-identifiable) injury was compared between the simulation and experiment obtained through 3D tomography. Even though the timing was slightly different, the simulated prediction represented remarkably well the experimental crack pattern before the appearance of the catastrophic unstable fast crack in the experiment, thus validating the implemented hybrid-experimental-modeling-computational (HEMC) concept as a tool to predict skull fracture initiation.

The influence of musculoskeletal forces on the growth of the prenatal cortex in the ilium: a finite element study

Remodelling and adaptation of bone within the pelvis is believed to be influenced by the mechanical strains generated during locomotion. Variation in the cortical bone thickness observed in the prenatal ilium has been linked to the musculoskeletal loading associated with in utero movements; for example the development of a thicker gluteal cortex is a possible response to contractions of the gluteal muscles. This study examines if the strains generated in the prenatal iliac cortex due to musculoskeletal loading in utero are capable of initiating bone remodelling to either maintain homeostasis or form new bone. Computational modelling techniques were used firstly to predict the muscle forces and resultant joint reaction force acting on the pelvis during a range of in utero movements. Finite element analyses were subsequently performed to calculate the von Mises strains induced in the prenatal ilium. The results demonstrated that strains generated in the iliac cortex were above the thresholds suggested to regulate bone remodelling to either maintain homeostasis or form new bone. Further simulations are required to investigate the extent to which the heterogeneous cortex forms in response to these strains (i.e., remodelling) or if developmental bone modelling plays a more pivotal role.

Multiscale design of artificial bones with biomimetic elastic microstructures

Cancellous bone is a highly porous, heterogeneous, and anisotropic material which can be found at the epiphyses of long bones and in the vertebral bodies. The hierarchical architecture makes cancellous bone a prime example of a lightweight natural material that combines strength with toughness. Better understanding the mechanics of cancellous bone is of interest for the diagnosis of bone diseases, the evaluation of the risk of fracture, and for the design of artificial bones and bone scaffolds for tissue engineering. A multiscale optimization method to maximize the stiffness of artificial bones using biomimetic cellular microstructures described by a finite set of geometrical micro-parameters is presented here. The most outstanding characteristics of its implementation are the use of: an interior point optimization algorithm, a precalculated response surface methodology for the evaluation of the elastic tensor of the microstructure as an analytical function of the micro-parameters, and the adjoint method for the computation of the sensitivity of the macroscopic mechanical response to the variation of the micro-parameters. The performance and effectiveness of the tool are evaluated by solving a problem that consists in finding the optimal distribution of the microstructures for a proximal end of a femur subjected to physiological loads. Two strategies for the specification of the solid volume fraction constraints are assessed. The results are compared with data of a computed tomography study of an actual human bone. The model successfully predicts the main features of the spatial arrangement of the trabecular and cortical microstructures of the natural bone.

Finite Element Analysis and Biomechanical Testing to Analyze Fracture Displacement of Alveolar Ridge Splitting

The alveolar ridge splitting technique enables reconstruction of atrophied alveolar ridges prior implantation. However, in cases of severe atrophy, there is an unpredictable risk of fracturing the buccal lamella during the expansion. Currently, there is no preoperative assessment to predict the maximum distraction of the lamella. The aim of this study was to develop a biomechanical model to mimic the alveolar ridge splitting and a finite element (FE) model to predict the experimental results. The biomechanical testing was conducted on porcine mandibles. To build the FE model high resolution peripheral quantitative computer tomography scans of one specimen was performed after the osteotomy outline, but before the lamella displacement. A servo-electric testing machine was used for the axial tension test to split the lamellae. Results showed, in line with clinical observations, that the lamellae broke primarily at the base of the splits with a median displacement of 1.27 mm. The FE model could predict fracture force and fracture displacement. Fracture force showed a nonlinear correlation with the height of the bone lamella. In conclusion, good correspondence between mechanical testing and virtual FE analysis showed a clinically relevant approach that may help to predict maximum lamella displacement to prevent fractures in the future.

The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone

Cancellous bone is a highly porous material, and two types of waves, fast and slow, are observed when ultrasound is used for detecting bone diseases. There are several possible stimuli for bone remodelling processes, including bone fluid flow, streaming potential, and piezoelectricity. Poroelasticity has been widely used for elucidating the bone fluid flow phenomenon, but the combination of poroelasticity with charge density has not been introduced. Theoretically, general poroelasticity with a varying charge density is employed for determining the relationship between wave velocity and attenuation with charge density. Fast wave velocity and attenuation are affected by porosity as well as charge density; however, for a slow wave, both slow wave velocity and attenuation are not as sensitive to the effect of charge density as they are for a fast wave. Thus, employing human femoral data, we conclude that charged ions gather on trabecular struts, and the fast wave, which moves along the trabecular struts, is significantly affected by charge density.

INVESTIGATION OF BONE DRILLING FOR SECURE IMPLANT FIXATION IN HUMAN FEMURS: TAGUCHI OPTIMIZATION AND PREDICTIVE FORCE MODELS WITH EXPERIMENTAL VALIDATION

Drilling procedures are important to optimize and ensure the strongest fixation in bone fracture treatment and reconstruction surgery. The mechanistic force models currently available for bovine bones, human spines and human mandibles are not relevant to perform drilling through human femurs. The present study addresses this lack of information and aims to develop the predictive force models for drilling human femurs at different regions and directions. In this study, 10 freshly harvested cadaveric human femurs were included, and a surgical drill bit of 3.2[Formula: see text]mm diameter was used to make 4[Formula: see text]mm deep holes. Different spindle speeds (500, 1000 and 1500[Formula: see text]rpm), feed rates (40, 60 and 80[Formula: see text]mm/min), and apparent density between 0.98 and 1.98[Formula: see text]g/cm3were considered. The optimal parameters [Formula: see text], [Formula: see text], and [Formula: see text] respectively obtained for longitudinal, radial, and circumferential direction could minimize the thrust forces in bone drilling by up to 7.70, 10.50, and 16.20 N, respectively. Validation study demonstrated that the force model developed could predict the thrust force from computed tomography data sets of the patient, only with 5.05%, 6.74%, and 4.91% as a maximum error in longitudinal, radial, and circumferential direction. This important tool can assist to perform complicated surgical operations.

Bone Mechanical Properties in Healthy and Diseased States

The mechanical properties of bone are fundamental to the ability of our skeletons to support movement and to provide protection to our vital organs. As such, deterioration in mechanical behavior with aging and/or diseases such as osteoporosis and diabetes can have profound consequences for individuals' quality of life. This article reviews current knowledge of the basic mechanical behavior of bone at length scales ranging from hundreds of nanometers to tens of centimeters. We present the basic tenets of bone mechanics and connect them to some of the arcs of research that have brought the field to recent advances. We also discuss cortical bone, trabecular bone, and whole bones, as well as multiple aspects of material behavior, including elasticity, yield, fracture, fatigue, and damage. We describe the roles of bone quantity (e.g., density, porosity) and bone quality (e.g., cross-linking, protein composition), along with several avenues of future research.

DEVELOPMENT OF IN-SITU TEMPERATURE PREDICTION MODELS FROM CADAVERIC HUMAN FEMUR FOR BONE DRILLING

Thermal osteonecrosis of bone in drilling procedure is caused by improper parameters which can lead to poor bone-implant integration and loss of fixation. In this study, Taguchi technique for parameter optimization and multiple regression models for temperature prediction were employed. The main aim of the study was to determine the optimal parameters of bone drilling to control the temperature rise below the thermal osteonecrosis threshold (47[Formula: see text]C) in respect of the bone density variations at different drilling directions. A 32 full factorial design with nine sets of parameters was used in the study. Drilling operations were performed along the longitudinal, radial and circumferential directions at the proximal-diaphysis, mid-diaphysis and distal-diaphysis regions of the 10 adult cadaveric femurs with different feed rates (40, 60 and 80[Formula: see text]mm/min) and spindle speeds (500, 1000 and 1500[Formula: see text]rpm) using 3.2[Formula: see text]mm diameter surgical drill bit. The in-situ drilling temperatures were measured with T-type thermocouple. The optimum drilling parameters for each drilling direction were determined from signal to noise ratios and the effect of each parameter was determined using analysis of variance. By using computed tomography scan data of patients, the proposed method is able to predict the temperature rise at the bone-drilling sites, optimal parameters and possibility for the occurrence of thermal osteonecrosis. This important tool could assist in reducing localized temperature induced from surgical drilling by up to 32% and 18[Formula: see text]C and as such significantly reduce associated osteonecrosis and improve patient outcome and quality of life.

The potential role of variations in juvenile hip geometry on the development of Legg-Calvé-Perthes disease: a biomechanical investigation

Legg-Calvé-Perthes disease (LCP) is one of the most poorly understood diseases in paediatric orthopaedics. One common trait of LCP is the marked morphological difference between healthy and pathological hips, early deviations of which (i.e. prior to disease onset) have been suggested to lead to the overload and collapse of the epiphysis. Here, the impact of common variations in geometry is investigated with a finite element model of a juvenile femur under single leg standing and landing. Here, the impact of typical variations in geometry is investigated with a finite element model of a juvenile femur under single leg standing and landing. The variations appear to have only a limited effect on the stress distribution in the femoral epiphysis even during high impact activities. This suggests that, for this individual at least, they would be unlikely to cause epiphyseal overload and collapse, even in the presence of a skeletally immature epiphysis.

New insights into the biomechanics of Legg-Calvé-Perthes' disease: The Role of Epiphyseal Skeletal Immaturity in Vascular Obstruction

Objectives

Legg–Calvé–Perthes’ disease (LCP) is an idiopathic osteonecrosis of the femoral head that is most common in children between four and eight years old. The factors that lead to the onset of LCP are still unclear; however, it is believed that interruption of the blood supply to the developing epiphysis is an important factor in the development of the condition.

Methods

Finite element analysis modelling of the blood supply to the juvenile epiphysis was investigated to understand under which circumstances the blood vessels supplying the femoral epiphysis could become obstructed. The identification of these conditions is likely to be important in understanding the biomechanics of LCP.

Results

The results support the hypothesis that vascular obstruction to the epiphysis may arise when there is delayed ossification and when articular cartilage has reduced stiffness under compression.

Conclusion

The findings support the theory of vascular occlusion as being important in the pathophysiology of Perthes disease.

Cite this article: M. Pinheiro, C. A. Dobson, D. Perry, M. J. Fagan. New insights into the biomechanics of Legg-Calvé-Perthes’ disease: The Role of Epiphyseal Skeletal Immaturity in Vascular Obstruction. Bone Joint Res 2018;7:148–156. DOI: 10.1302/2046-3758.72.BJR-2017-0191.R1.

Calculation of cancellous bone elastic properties with the polarization-based FFT iterative scheme

The Fast Fourier Transform-based method, originally introduced by Moulinec and Suquet in 1994 has gained popularity for computing homogenized properties of composites. In this work, the method is used for the computational homogenization of the elastic properties of cancellous bone. To the authors' knowledge, this is the first study where the Fast Fourier Transform scheme is applied to bone mechanics. The performance of the method is analyzed for artificial and natural bone samples of 2 species: bovine femoral heads and implanted femurs of Hokkaido rats. Model geometries are constructed using data from X-ray tomographies, and the bone tissue elastic properties are measured using microindentation and nanoindentation tests. Computed results are in excellent agreement with those available in the literature. The study shows the suitability of the method to accurately estimate the fully anisotropic elastic response of cancellous bone. Guidelines are provided for the construction of the models and the setting of the algorithm.

Mimetization of the elastic properties of cancellous bone via a parameterized cellular material

Bone tissue mechanical properties and trabecular microarchitecture are the main factors that determine the biomechanical properties of cancellous bone. Artificial cancellous microstructures, typically described by a reduced number of geometrical parameters, can be designed to obtain a mechanical behavior mimicking that of natural bone. In this work, we assess the ability of the parameterized microstructure introduced by Kowalczyk (Comput Methods Biomech Biomed Eng 9:135-147, 2006. doi: 10.1080/10255840600751473 ) to mimic the elastic response of cancellous bone. Artificial microstructures are compared with actual bone samples in terms of elasticity matrices and their symmetry classes. The capability of the parameterized microstructure to combine the dominant isotropic, hexagonal, tetragonal and orthorhombic symmetry classes in the proportions present in the cancellous bone is shown. Based on this finding, two optimization approaches are devised to find the geometrical parameters of the artificial microstructure that better mimics the elastic response of a target natural bone specimen: a Sequential Quadratic Programming algorithm that minimizes the norm of the difference between the elasticity matrices, and a Pattern Search algorithm that minimizes the difference between the symmetry class decompositions. The pattern search approach is found to produce the best results. The performance of the method is demonstrated via analyses for 146 bone samples.

A new multiscale micromechanical model of vertebral trabecular bones

A new three-dimensional (3D) multiscale micromechanical model has been suggested as adept at predicting the overall linear anisotropic mechanical properties of a vertebral trabecular bone (VTB) highly porous microstructure. A nested 3D modeling analysis framework spanning the multiscale nature of the VTB is presented herein. This hierarchical analysis framework employs the following micromechanical methods: the 3D parametric high-fidelity generalized method of cells (HFGMC) as well as the 3D sublaminate model. At the nanoscale level, the 3D HFGMC method is applied to obtain the effective elastic properties of a representative unit cell (RUC) representing the mineral collagen fibrils composite. Next at the submicron scale level, the 3D sublaminate model is used to generate the effective elastic properties of a repeated stack of multilayered lamellae demonstrating the nature of the trabeculae (bone-wall). Thirdly, at the micron scale level, the 3D HFGMC method is used again on a RUC of the highly porous VTB microstructure. The VTB-RUC geometries are taken from microcomputed tomography scans of VTB samples harvested from different vertebrae of human cadavers [Formula: see text]. The predicted anisotropic overall elastic properties for native VTBs are, then, examined as a function of age and sex. The predicted results of the VTBs longitudinal Young's modulus are compared to reported values found in the literature. The proposed 3D nested modeling analysis framework provides a good agreement with reported values of Young's modulus of single trabeculae as well as for VTB-RUC in the literature.

ULTRASONIC WAVE IS DETERMINED BY FABRIC TENSOR: AN APPLICATION TO CALCANEUS

The fabric tensor is a good measure for determining the mechanical properties of cancellous bone. Ultrasound is one method used to measure these mechanical properties. Ultrasound-generated speed of sound (SOS) measures the mechanical properties of cancellous bone. Thus, in this paper, we started with the fact that the fast wave in poroelastic theory is identical to the bulk wave velocity. We then formulate the equation for the fast wave in terms of fabric tensor for the calcaneus. The formulation in this paper is simpler than previously published results and will be easy to use in future experiments.

Reasons why dynamic compression plates are inferior to locking plates in osteoporotic bone: a finite element explanation

While locking plate fixation is becoming increasingly popular for complex and osteoporotic fractures, for many indications compression plating remains the standard choice. This study compares the mechanical behaviour of the more recent locking compression plate (LCP) device, with the traditional dynamic compression plates (DCPs) in bone of varying quality using finite element modelling. The bone properties considered include orthotropy, inhomogeneity, cortical thinning and periosteal apposition associated with osteoporosis. The effect of preloads induced by compression plating was included in the models. Two different fracture scenarios were modelled: one with complete reduction and one with a fracture gap. The results show that the preload arising in DCPs results in large principal strains in the bone all around the perimeter of the screw hole, whereas for LCPs large principal strains occur primarily on the side of the screw proximal to the load. The strains within the bone produced by the two screw types are similar in healthy bone with a reduced fracture gap; however, the DCP produces much larger strains in osteoporotic bone. In the presence of a fracture gap, the DCP results in a considerably larger region with high tensile strains and a slightly smaller region with high compressive strains. These findings provide a biomechanical basis for the reported improved performance of locking plates in poorer bone quality.

Changes of elastic constants and anisotropy patterns in trabecular bone during disuse-induced bone loss assessed by poroelastic ultrasound

Currently, the approach most widely used to examine bone loss is the measurement of bone mineral density (BMD) using dual X-ray absorptiometry (DXA). However, bone loss due to immobilization creates changes in bone microarchitecture, which in turn are related to changes in bone mechanical function and competence to resist fracture.Unfortunately, the relationship between microarchitecture and mechanical function within the framework of immobilization and antiresorptive therapy has not being fully investigated. The goal of the present study was to investigate the structure–function relationship in trabecular bone in the real-world situations of a rapidly evolving osteoporosis(disuse), both with and without antiresorptive treatment. We evaluated the structure–function relationship in trabecular bone after bone loss (disuse-induced osteoporosis)and bisphosphonate treatment (antiresorptive therapy using risedronate) in canine trabecular bone using lCT and ultrasound wave propagation. Microstructure values determined from lCT images were used into the anisotropic poroelastic model of wave propagation in order to compute the apparent elastic constants (EC) and elastic anisotropy pattern of bone. Immobilization resulted in a significant reduction in trabecular thickness (Tb.Th) and bone volume fraction (BV/TV), while risedronate treatment combined with immobilization exhibited a lesser reduction in Tb.Th and BV/TV, suggesting that risedronate treatment decelerates bone loss, but it was unable to fully stop it. Risedronate treatment also increased the tissue mineral density (TMD), which when combined with the decrease in Tb.Th and BV/TV may explain the lack of significant differences invBMD in both immobilization and risedronate treated groups. Interestingly, changes inapparent EC were much stronger in the superior–inferior (SI) direction than in the medial–lateral (ML) and anterior–posterior (AP) anatomical directions, producing changes in elastic anisotropy patterns. When data were pooled together, vBMD was able to explain 58% of ultrasound measurements variability, a poroelastic wave propagation analytical model (i.e., BMD modulated by fabric directionality) was able to predict 81%of experimental wave velocity variability, and also explained 91% of apparent EC and changes in elastic anisotropy patterns. Overall, measurements of vBMD were unable to distinguish changes in apparent EC due to immobilization or risedronate treatment.However, anisotropic poroelastic ultrasound (PEUS) wave propagation was able to distinguish functional changes in apparent EC and elastic anisotropy patterns due to immobilization and antiresorptive therapy, providing an enhanced discrimination of anisotropic bone loss and the structure–function relationship in immobilized and risedronate-treated bone, beyond vBMD.

Prediction of failure load using micro-finite element analysis models: Toward in vivo strength assessment

Finite element analysis (FEA) is the method of choice to nondestructively quantify stresses and strains in bones. Moderate to good estimates of bone strength can be obtained from continuum-level FEA. Improved predictive capacity is expected from microstructural FE models that represent the trabecular architecture in detail. With the advent of recently developed high-resolution in vivo bone imaging systems and the steady increase in computational power, such microstructural FE analyses are now becoming available to estimate bone strength in humans in a clinical setting. The procedure can help improve predictions of fracture risk, clarify the pathophysiology of skeletal diseases, and monitor the response to therapy.:

Quantification of trabecular spatial orientation from low-resolution images

No accepted methodology exists to assess trabecular bone orientation from clinical CT scans. The aim of this study was to test the hypothesis that the distribution of grey values in clinical CT images is related to the underlying trabecular architecture and that this distribution can be used to identify the principal directions and local anisotropy of trabecular bone. Fourteen trabecular bone samples were extracted from high-resolution (30 μm) micro-CT scans of seven human femoral heads. Trabecular orientations and local anisotropy were calculated using grey-level deviation (GLD), a novel method providing a measure of the three-dimensional distribution of image grey values. This was repeated for different image resolutions down to 300 μm and for volumes of interest (VOIs) ranging from 1 to 7 mm. Outcomes were compared with the principal mechanical directions and with mean intercept length (MIL) as calculated for the segmented 30-μm images. For the 30-μm images, GLD predicted the mechanical principal directions equally well as MIL. For the 300-μm images, which are resolutions that can be obtained in vivo using clinical CT, only a small increase (3°-6°) in the deviation from the mechanical orientations was found. VOIs of 5 mm resulted in a robust quantification of the orientation. We conclude that GLD can quantify structural bone parameters from low-resolution CT images.

Biaxial shear of confined colloidal hard spheres: the structure and rheology of the vorticity-aligned string phase

Using a novel biaxial confocal rheoscope, we investigate the flow of the shear induced vorticity aligned string phase [X. Cheng et al., Proc. Natl. Acad. Sci. U. S. A., 2011, 109, 63], which has a highly anisotropic microstructure. Using biaxial shear protocols we show that we have excellent control of the string phase anisotropic morphology. We choose a shear protocol that drives the system into the string phase. Subsequently, a biaxial force measurement device is used to determine the suspension rheology along both the flow and vorticity directions. We find no measurable dependence of the suspension stress response along the shear and vorticity directions due to the hydrodynamically induced string morphology. In particular, we find that the suspension's high frequency stress response is nearly identical along the two orthogonal directions. While we do observe an anisotropic stress response at lower shear frequencies associated with shear thinning, we show that this anisotropy is independent of the shear induced string structure. These results suggest that for the range of flows explored, Brownian and hydrodynamic contributions to the stress arising from the anisotropic suspension microstructure are sufficiently weak that they do not significantly contribute to the rheology. Collectively, this study presents a general and powerful approach for using biaxial confocal rheometry to elucidate the relationship between microstructure and rheology in complex fluids driven far-from-equilibrium.

PMMA-hydroxyapatite composite material retards fatigue failure of augmented bone compared to augmentation with plain PMMA: in vivo study using a sheep model

Polymethylmethacrylate (PMMA) is the most commonly used void filler for augmentation of osteoporotic vertebral fracture, but the differing mechanical features of PMMA and osteoporotic bone result in overload and failure of adjacent bone. The aim of this study was to compare fatigue failure of bone after augmentation with PMMA-nanocrystalline hydroxyapatite (HA) composite material or with plain PMMA in a sheep model. After characterization of the mechanical properties of a composite material consisting of PMMA and defined amounts (10, 20, and 30% volume fraction) of HA, the composite material with 30% volume fraction HA was implanted in one distal femur of sheep; plain PMMA was implanted in the other femur. Native non-augmented bone served as control. Three and 6 months after implantation, the augmented bone samples were exposed to cyclic loading and the evolution of damage was investigated. The fatigue life was highest for the ovine native bone and lowest for bone-PMMA specimens. Bone-composite specimens showed significantly higher fatigue life than the respective bone-PMMA specimens in both 3- and 6-month follow-up groups. These results suggest that modification of mechanical properties of PMMA by addition of HA to approximate those of cancellous bone retards fatigue failure of the surrounding bone compared to augmented bone with plain PMMA.

A symmetry invariant formulation of the relationship between the elasticity tensor and the fabric tensor

The fabric tensor is employed as a quantitative stereological measure of the structural anisotropy in the pore architecture of a porous medium. Earlier work showed that the fabric tensor can be used additionally to the porosity to describe the anisotropy in the elastic constants of the porous medium. This contribution presents a reformulation of the relationship between fabric tensor and anisotropic elastic constants that is approximation free and symmetry-invariant. From specific data on the elastic constants and the fabric, the parameters in the reformulated relationship can be evaluated individually and efficiently using a simplified method that works independent of the material symmetry. The well-behavedness of the parameters and the accuracy of the method was analyzed using the Mori-Tanaka model for aligned ellipsoidal inclusions and using Buckminster Fuller's octet-truss lattice. Application of the method to a cancellous bone data set revealed that employing the fabric tensor allowed explaining 75-90% of the total variance. An implementation of the proposed methods was made publicly available.

Analytical basis for the determination of the lacunar-canalicular permeability of bone using cyclic loading

An analytical model for the determination of the permeability in the lacunar-canalicular porosity of bone using cyclic loading is described in this contribution. The objective of the analysis presented is to relate the lacunar-canalicular permeability to a particular phase angle that is measurable when the bone is subjected to infinitesimal cyclic strain. The phase angle of interest is the lag angle between the applied strain and the resultant stress. Cyclic strain causes the interstitial fluid to move. This movement is essential for the viability of osteocytes and is believed to play a major role in the bone mechanotransduction mechanism. However, certain bone fluid flow properties, notably the permeability of the lacunar-canalicular porosity, are still not accurately determined. In this paper, formulas for the phase angle as a function of permeability for infinitesimal cyclic strain are presented and mathematical expressions for the storage modulus, loss modulus, and loss tangent are obtained. An accurate determination of the PLC permeability will improve our ability to understand mechanotransduction and mechanosensory mechanisms, which are fundamental to the understanding of how to treat osteoporosis, how to cope with microgravity in long-term manned space flights, and how to increase the longevity of prostheses that are implanted in bone tissue.

Relating age and micro-architecture with apparent-level elastic constants: a micro-finite element study of female cortical bone from the anterior femoral midshaft

Homogenized elastic properties are often assumed for macro-finite element (FE) models used in orthopaedic biomechanics. The accuracy of material property assignments may have a strong effect on the ability of these models to make accurate predictions. For cortical bone, most macro-scale FE models assume isotropic elastic material behaviour and do not include variation of material properties due to bone micro-architecture. The first aim of the present study was to evaluate the variation of apparent-level (homogenized) orthotropic elastic constants of cortical bone with age and indices of bone micro-architecture. Considerable age-dependent differences in porosity were noted across the cortical thickness in previous research. The second aim of the study was to quantify the resulting differences in elastic constants between the periosteum and endosteum. Specimens were taken from the anterior femoral midshaft of 27 female donors (age 53.4 ± 23.6 years) and micro-FE (µFE) analysis was used to derive orthotropic elastic constants. The variation of orthotropic elastic constants (Young’s moduli, shear moduli, and Poisson’s ratios) with various cortical bone micro-architectural indices was investigated. The ratio of canal volume to tissue volume, Ca.V/TV, analogous to porosity, was found to be the strongest predictor ( r ave2 = 0.958) of the elastic constants. Age was less predictive ( r ave2 = 0.385) than Ca.V/TV. Elastic anisotropy increased with increasing Ca.V/TV, leading to lower elastic moduli in the transverse, typically less frequently loaded, directions. Increased Ca.V/TV led to a more substantial reduction in elastic constants at the endosteal aspect than at the periosteal aspect. The results are expected to be most applicable in similar midshaft locations of long bones; specific analysis of other sites would be necessary to evaluate elastic properties elsewhere. It was concluded that Ca.V/TV was the most predictive of cortical bone elastic constants and that considerable periosteal–endosteal variations in these constants can develop with bone loss.

Fabric dependence of wave propagation in anisotropic porous media

Current diagnosis of bone loss and osteoporosis is based on the measurement of the bone mineral density (BMD) or the apparent mass density. Unfortunately, in most clinical ultrasound densitometers: 1) measurements are often performed in a single anatomical direction, 2) only the first wave arriving to the ultrasound probe is characterized, and 3) the analysis of bone status is based on empirical relationships between measurable quantities such as speed of sound (SOS) and broadband ultrasound attenuation (BUA) and the density of the porous medium. However, the existence of a second wave in cancellous bone has been reported, which is an unequivocal signature of poroelastic media, as predicted by Biot's poroelastic wave propagation theory. In this paper, the governing equations for wave motion in the linear theory of anisotropic poroelastic materials are developed and extended to include the dependence of the constitutive relations upon fabric-a quantitative stereological measure of the degree of structural anisotropy in the pore architecture of a porous medium. This fabric-dependent anisotropic poroelastic approach is a theoretical framework to describe the microarchitectural-dependent relationship between measurable wave properties and the elastic constants of trabecular bone, and thus represents an alternative for bone quality assessment beyond BMD alone.

Scale and boundary conditions effects on the apparent elastic moduli of trabecular bone modeled as a periodic cellular solid

We study apparent elastic moduli of trabecular bone, which is represented, for simplicity, by a two- or three-dimensional periodic cellular network. The term "apparent" refers to the case when the region used in calculations (or specimen size) is smaller than a representative volume element and the moduli depend on the size of that region and boundary conditions. Both the bone tissue forming the network and the pores (represented by a very soft material) are assumed, for simplicity, as homogeneous, linear elastic, and isotropic. In order to investigate the effects of scale and boundary conditions on the moduli of these networks we vary the specimen size and apply four different boundary conditions: displacement, traction, mixed, and periodic. The analysis using periodic boundary conditions gives the effective moduli, while the displacement, traction, and mixed boundary conditions give apparent moduli. The apparent moduli calculated using displacement and traction boundary conditions bound the effective moduli from above and below, respectively. The larger is the size of the region used in our calculations, the closer are the bounds. Our choice of mixed boundary conditions gives results that are very close to those obtained using periodic boundary conditions. We conduct this analysis computationally using a finite element method. We also investigate the effect of mismatch in elastic moduli of bone tissue and soft fill, trabecular bone structure geometry, and bone tissue volume fraction on the apparent elastic moduli of idealized periodic models of trabecular bone. This study gives guidance on how the size of the specimen and boundary conditions (used in experiments or simulations) influence elastic moduli of cellular materials. This approach is applicable to heterogeneous materials in general.

Specimen diameter and "side artifacts" in cancellous bone evaluated using end-constrained elastic tension

In cancellous bone testing of cored samples, side artifacts are the underestimation of the true (i.e. in situ) mechanical properties due to the severing of the trabecular network during specimen preparation. Although other researchers have suggested correction factors derived from finite element method (FEM) models, it is proposed that side effects can be minimized by increasing the specimen diameter. Six different diameter specimens (3.1-10.6 mm), from two different anatomic sites (bovine femoral condyle and bovine lumbar vertebrae), were mechanically tested in elastic tension using an epoxy endcap protocol to eliminate end artifacts. Elastic modulus was found to be significantly affected by diameter in both sites. For example, the 5.1 mm samples underestimated the elastic modulus of the 10.6 mm samples by an average of roughly 20%. Yet no statistical difference was detected between the 8.3 and 10.6 mm samples in either anatomic site, suggesting that 8.3 mm diameter specimens were sufficiently large to avoid side artifacts. FEM models created from micro-CT images reveal that modulus approaches an asymptotic value with increasing diameter, and demonstrate an architecture-dependent drop in modulus at decreasing diameters. These results confirm, both experimentally and numerically, that side effects can be ignored given a suitably large specimen diameter and that this minimum diameter will be dependent on the cancellous architecture. An important implication of the latter result is that specimen diameters must be chosen appropriately when comparing test groups with different architectures (e.g. normal versus osteoporotic) to ensure that the magnitude of side artifacts does not confound the true differences between the groups.

Hierarchical microimaging of bone structure and function

With recent advances in molecular medicine and disease treatment in osteoporosis, quantitative image processing of three-dimensional bone structures is critical in the context of bone quality assessment. Biomedical imaging technology such as MRI or CT is readily available, but few attempts have been made to expand the capabilities of these systems by integrating quantitative analysis tools and by exploring structure-function relationships in a hierarchical fashion. Nevertheless, such quantitative end points are an important factor for success in basic research and in the development of novel therapeutic strategies. CT is key to these developments, as it images and quantifies bone in three dimensions and provides multiscale biological imaging capabilities with isotropic resolutions of a few millimeters (clinical CT), a few tens of micrometers (microCT) and even as high as 100 nanometers (nanoCT). The technology enables the assessment of the relationship between microstructural and ultrastructural measures of bone quality and certain diseases or therapies. This Review focuses on presenting strategies for three-dimensional approaches to hierarchical biomechanical imaging in the study of microstructural and ultrastructural bone failure. From this Review, it can be concluded that biomechanical imaging is extremely valuable for the study of bone failure mechanisms at different hierarchical levels.

Correlation between local stress and strain and lamina cribrosa connective tissue volume fraction in normal monkey eyes

PURPOSE

To investigate the biomechanical response to IOP elevation of normal monkey eyes using eye-specific, three-dimensional (3-D) finite element (FE) models of the ONH that incorporate lamina cribrosa (LC) microarchitectural information.

METHODS

A serial sectioning and episcopic imaging technique was used to reconstruct the ONH and peripapillary sclera of four pairs of eyes fixed at 10 mm Hg. FE models were generated with local LC material properties representing the connective tissue volume fraction (CTVF) and predominant LC beam orientation and used to simulate an increase in IOP from 10 to 45 mm Hg. An LC material stiffness constant was varied to assess its influence on biomechanical response.

RESULTS

Strains and stresses within contralateral eyes were remarkably similar in both magnitude and distribution. Strain correlated inversely, and nonlinearly, with CTVF (median, r (2) = 0.73), with tensile strains largest in the temporal region. Stress correlated linearly with CTVF (median r(2) = 0.63), with the central and superior regions bearing the highest stresses. Net average LC displacement was either posterior or anterior, depending on whether the laminar material properties were compliant or stiff.

CONCLUSIONS

The results show that contralateral eyes exhibit similar mechanical behavior and suggest that local mechanical stress and strain within the LC are correlate highly with local laminar CTVF. These simulations emphasize the importance of developing both high-resolution imaging of the LC microarchitecture and next-generation, deep-scanning OCT techniques to clarify the relationships between IOP-related LC displacement and CTVF-related stress and strain in the LC. Such imaging may predict sites of IOP-related damage in glaucoma.

Multi-level patient-specific modelling of the proximal femur. A promising tool to quantify the effect of osteoporosis treatment

Preventing femoral fractures is an important goal in osteoporosis research. In order to evaluate a person's fracture risk and to quantify response to treatment, bone competence is best assessed by bone strength. Finite-element (FE) modelling based on medical imaging is considered a very promising technique for the assessment of in vivo femoral bone strength. Over the past decades, a number of different FE models have been presented focusing on the effect of several methodological aspects, such as mesh type, material properties and loading conditions, on the precision and accuracy of these models. In this paper, a review of this work is presented. We conclude that moderate to good predictions can be made, especially when the models are tuned to specific loading scenarios. However, there is room for improvement when multiple loading conditions need to be evaluated. We hypothesize that including anisotropic material properties is the first target. As a proof of the concept, we demonstrate that the main orientation of the femoral bone structure can be calculated from clinical computed tomography scans. We hypothesize that this structural information can be used to estimate the anisotropic bone material properties, and that in the future this could potentially lead to a greater predictive value of FE models for femoral bone strength.

Non-invasive bone competence analysis by high-resolution pQCT: an in vitro reproducibility study on structural and mechanical properties at the human radius

Osteoporosis is defined as a skeletal disorder characterized by compromised bone strength. Bone strength depends, among others, on bone density, bone geometry and its internal architecture. With the recent introduction of a new generation high-resolution 3D peripheral quantitative computed tomography (HR-pQCT) system, direct quantification of structural bone parameters has become feasible. Furthermore, it has recently been demonstrated that bone mechanical competence can be derived from HR-pQCT based micro-finite element modeling (microFE). However, reproducibility data for HR-pQCT-derived mechanical indices is not well-known. Therefore, the aim of this study was to quantify reproducibility of HR-pQCT-derived indices. We measured 14 distal formalin-fixed cadaveric forearms three times and analyzed three different regions for each measurement. For each region cortical and trabecular parameters were determined. Reproducibility was assessed with respect to precision error (PE) and intraclass correlation coefficient (ICC). Reproducibility values were found to be best in all three regions for the full bone compartment with an average PE of 0.79%, followed by the cortical compartment (PE=1.19%) and the trabecular compartment with an average PE of 2.31%. The mechanical parameters showed similar reproducibility (PE=0.48%-2.93% for bone strength and stiffness, respectively). ICC showed a very high reproducibility of subject-specific measurements, ranging from 0.982 to 1.000, allowing secure identification of individual donors ranging from healthy to severely osteoporotic subjects. From these in vitro results we conclude that HR-pQCT derived morphometric and mechanical parameters are highly reproducible such that differences in bone structure and strength can be detected with a reproducibility error smaller than 3%; hence, the technique has a high potential to become a tool for detecting bone quality and bone competence of individual subjects.

Quantification of bone structural parameters and mechanical competence at the distal radius

For the clinician, predicting the fracture risk for individual patients is mainly restricted to the quantitative analysis of bone density. Several studies have shown that bone strength, an indicator for bone fracture risk, is only predicted moderately by bone density, indicating that there are other factors influencing bone competence. However, the relative importance of "bone quantity" and "bone quality" remains poorly understood. The objectives of this article are to describe some of the techniques used to measure the microarchitectural aspects of bone quality, how they can be quantified, and how these quantitative endpoints can be used in the assessment of bone competence. Special focus will be on the distal radius, a site with a high fracture incidence. With the introduction of high-resolution in vivo bone imaging systems, a new generation of imaging instruments has entered the arena allowing the reconstruction of the 3-dimensional microarchitecture of the bones at the wrist, thereby giving researchers and clinicians a powerful tool for the quantitative assessment of bone microstructure. In combination with large-scale finite element modeling, these methodologies have reached a level that it is now becoming possible to assess bone stiffness and strength in humans in a clinical setting. The procedure can help improve predictions of fracture risk, clarify the pathophysiology of skeletal diseases, and monitor the response to therapy.