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

X-ray 3D imaging-based microunderstanding of granular mixtures: Stiffness enhancement by adding small fractions of soft particles

This research focuses on performing ultrasound propagation measurements and micro-X-ray computed tomography (µXRCT) imaging on prestressed granular packings prepared with biphasic mixtures of monodisperse glass and rubber particles at different compositions/fractions. Ultrasound experiments employing piezoelectric transducers, mounted in an oedometric cell (complementing earlier triaxial cell experiments), are used to excite and detect longitudinal ultrasound waves through randomly prepared mixtures of monodisperse stiff/soft particles. While the fraction of the soft particles is increasing linearly from zero, the effective macroscopic stiffness of the granular packings transits nonlinearly and nonmonotonically toward the soft limit, remarkably via an interesting stiffer regime for small rubber fractions between 0.1 ≲ ν ≲ 0.2. The contact network of dense packings, as accessed from µXRCT, plays a key role in understanding this phenomenon, considering the structure of the network, the chain length, the grain contacts, and the particle coordination. While the maximum stiffness is due to surprisingly shortened chains, the sudden drop in elastic stiffness of the mixture packings, at ν ≈ 0.4, is associated with chains of particles that include both glass and rubber particles (soft chains); for ν ≲ 0.3, the dominant chains include only glass particles (hard chains). At the drop, ν ≈ 0.4, the coordination number of glass and rubber networks is approximately four and three, respectively, i.e., neither of the networks are jammed, and the chains need to include particles from another species to propagate information.

Biomechanical properties and clinical significance of cancellous bone in proximal femur: A review

Trabecular bone plays an important role in the load-bearing capacity of the femur. Understanding the structural characteristics, biomechanics, and mechanical conduction of the trabecular bone is of great value in studying the mechanism of fractures and formulating surgical plans. The past decade has witnessed unprecedented progress in imaging, biomechanics and finite element analysis techniques, translating into a better understanding of trabecular bone. This article reviews the research progress achieved over the years regarding femoral trabecular bone, especially on factors influencing the strength of the proximal femoral cancellous bone and cancellous bone microfractures and provides a comprehensive overview of the latest findings on proximal femoral trabecular bone and their clinical significance.

Deformation of Gels with Spherical Auxetic Inclusions

Auxetic metamaterials possess unnatural properties, such as a negative Poisson’s ratio, which offers interesting features when combined with traditional materials. This paper describes the deformation behavior of a gel consisting of spherical auxetic inclusions when embedded in a conventional matrix. The auxetic inclusions and conventional matrix were modeled as spherical objects with a controlled pore shape. The auxetic particle had a reentrant honeycomb, and the conventional phase contained honeycomb-shaped pores. The deformation behavior was simulated using various existing models based on continuum mechanics. For the continuum mechanics models—the simplest of which are the Mori–Tanaka theory and self-consistent field mechanics models—the auxetic particle was homogenized as a solid element with Young’s modulus and Poisson’s ratio and compared with the common composite gel filled with rigid spheres. The finite element analysis simulations using these models were performed for two cases: (1) a detailed model of one particle and its surroundings in which the structure included the design of both the reentrant and conventional honeycombs; and (2) a multiparticle face-centered cubic lattice where both the classic matrix and auxetic particle were homogenized. Our results suggest that auxetic inclusion-filled gels provide an unsurpassed balance of low density and enhanced stiffness.

A Nonparametric Approach for Estimating Three-Dimensional Fiber Orientation Distribution Functions (ODFs) in Fibrous Materials

Many biological tissues contain an underlying fibrous microstructure that is optimized to suit a physiological function. The fiber architecture dictates physical characteristics such as stiffness, diffusivity, and electrical conduction. Abnormal deviations of fiber architecture are often associated with disease. Thus, it is useful to characterize fiber network organization from image data in order to better understand pathological mechanisms. We devised a method to quantify distributions of fiber orientations based on the Fourier transform and the Qball algorithm from diffusion MRI. The Fourier transform was used to decompose images into directional components, while the Qball algorithm efficiently converted the directional data from the frequency domain to the orientation domain. The representation in the orientation domain does not require any particular functional representation, and thus the method is nonparametric. The algorithm was verified to demonstrate its reliability and used on datasets from microscopy to show its applicability. This method increases the ability to extract information of microstructural fiber organization from experimental data that will enhance our understanding of structure-function relationships and enable accurate representation of material anisotropy in biological tissues.

Patient-specific bone modeling and analysis: the role of integration and automation in clinical adoption

Patient-specific analysis of bones is considered an important tool for diagnosis and treatment of skeletal diseases and for clinical research aimed at understanding the etiology of skeletal diseases and the effects of different types of treatment on their progress. In this article, we discuss how integration of several important components enables accurate and cost-effective patient-specific bone analysis, focusing primarily on patient-specific finite element (FE) modeling of bones. First, the different components are briefly reviewed. Then, two important aspects of patient-specific FE modeling, namely integration of modeling components and automation of modeling approaches, are discussed. We conclude with a section on validation of patient-specific modeling results, possible applications of patient-specific modeling procedures, current limitations of the modeling approaches, and possible areas for future research.

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.

Alterations in trabecular bone microarchitecture in the ovine spine and distal femur following ovariectomy

Osteoporosis is a bone disease resulting in increased fracture risk as a result of alterations in both quantity and quality of bone. Bone quality is a combination of metabolic and microarchitectural properties of bone that can help to explain the increased susceptibility to fracture. Translational animal models are essential to understanding the pathology and for evaluating potential treatments of this disease. Large animals, such as the ovariectomized sheep, have been used as models for post-menopausal osteoporosis. However, long-term studies have not been carried out to observe the effects of ovariectomy after more than one year. This study employed micro-computed tomography to quantify changes in microarchitectural and mechanical parameters in femoral condyles and vertebral bodies of sheep that were sacrificed one or two years following ovariectomy. In the vertebral body, microarchitectural characteristics were significantly degraded following one year of ovariectomy in comparison to controls. The mechanical anisotropy, determined from micro-scale finite element models, was also greater in the ovariectomized groups, although the fabric tensor anisotropy was similar. There was no greater architectural degradation following two years of ovariectomy compared to one. Ovariectomy had minimal effects on the trabecular architecture of the distal femur even after two years. These results indicate that the vertebral body is the preferred anatomic site for studying bone from the ovariectomized sheep model, and that architectural changes stabilize after the first year.

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.

Predicting trabecular bone elastic properties from measures of bone volume fraction and fabric on the basis of micromagnetic resonance images

The relationship between fabric (a measure of structural anisotropy) and elastic properties of trabecular bone was examined by invoking morphology and homogenization theory on the basis of micromagnetic resonance images from the distal tibia in specimens (N = 30) and human subjects (N = 16) acquired at a 160 × 160 × 160 μm(3) voxel size. The fabric tensor was mapped in 7.5 × 7.5 × 7.5 mm(3) cubic subvolumes by a three-dimensional mean-intercept-length method. Elastic constants (three Young's and three shear moduli) were derived from linear microfinite element simulations of three-dimensional grayscale bone volume fraction-mapped images. In the specimen data, moduli fit power laws of bone volume fraction (bone volume/total volume) for all three test directions and subvolumes (R(2) = 0.92-0.98) with exponents ranging from 1.3 to 1.8. Weaker linear relationships were found for the in vivo data because of a narrower range in bone volume/total volume. When pooling the data for all test directions and subvolumes, bone volume/total volume predicted elastic moduli less well in the specimens (mean R(2) = 0.74) and not at all in vivo. A model of bone volume/total volume and fabric was highly predictive of microfinite element-derived Young's moduli: mean R(2) s of 0.98 and 0.82 (in vivo). The results show that fabric, an important predictor of bone mechanical properties, can be assessed in the limited resolution and signal-to-noise ratio regime of micromagnetic resonance images.

Mixture theory-based poroelasticity as a model of interstitial tissue growth

This contribution presents an alternative approach to mixture theory-based poroelasticity by transferring some poroelastic concepts developed by Maurice Biot to mixture theory. These concepts are a larger RVE and the subRVE-RVE velocity average tensor, which Biot called the micro-macro velocity average tensor. This velocity average tensor is assumed here to depend upon the pore structure fabric. The formulation of mixture theory presented is directed toward the modeling of interstitial growth, that is to say changing mass and changing density of an organism. Traditional mixture theory considers constituents to be open systems, but the entire mixture is a closed system. In this development the mixture is also considered to be an open system as an alternative method of modeling growth. Growth is slow and accelerations are neglected in the applications. The velocity of a solid constituent is employed as the main reference velocity in preference to the mean velocity concept from the original formulation of mixture theory. The standard development of statements of the conservation principles and entropy inequality employed in mixture theory are modified to account for these kinematic changes and to allow for supplies of mass, momentum and energy to each constituent and to the mixture as a whole. The objective is to establish a basis for the development of constitutive equations for growth of tissues.

Role of structural anisotropy of biological tissues in poroelastic wave propagation

Ultrasound waves have a broad range of clinical applications as a non-destructive testing approach in imaging and in the diagnoses of medical conditions. Generally, biological tissues are modeled as an homogenized equivalent medium with an apparent density through which a single wave propagates. Only the first wave arriving at the ultrasound probe is used for the measurement of the speed of sound. However, the existence of a second wave in tissues such as cancellous bone has been reported and its existence is an unequivocal signature of Biot type poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as density, a fabric-dependent anisotropic poroelastic ultrasound (PEU) propagation theory was recently developed. Key to this development was the inclusion of the fabric tensor - a quantitative stereological measure of the degree of structural anisotropy of bone - into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of waves in several soft and hard tissues. It was found that collagen fibers in soft tissues and the mineralized matrix in hard tissues are responsible for the anisotropy of the solid tissue constituent through the fabric tensor in the model.

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.

Effects of loading orientation on the morphology of the predicted yielded regions in trabecular bone

While the effects of bone mineral density and architecture in osteoporotic bone have been studied extensively, the micromechanics of yielding and failure have received less attention. However, understanding architectural features associated with failure should provide insight into assessing bone quality. In this study, microstructural finite element models were used to compute regions of tissue level yielding in ten bovine tibial trabecular bone samples. The morphology, number, and mean volume of the yielded regions were quantified for four apparent strains under two loading conditions. For on-axis loading, the mean aspect ratio of the tissue that yielded due to compressive strain increased with increasing apparent strain, expanding along the principal trabecular orientation. This suggests that tissue level yielding progresses along vertical trabeculae when a specimen is loaded on-axis. The number, but not the volume, of the regions yielded due to tensile strain increased with increasing applied load, consistent with relaxation and redistribution of stresses around the yielded regions. When the specimens were compressed perpendicular to the principal axis, the aspect ratio of the yielded regions was close to one, while the number, mean volume, and mean thickness of the yielded regions increased. This indicates that localized high strains consistent with bending rather than axial deformation of struts occur at the tissue level. Overall, the results provide new insight into trabecular bone failure, which is relevant to assessing diagnostic tests for fracture risk or evaluating osteoporosis treatments.

A non-invasive in vitro technique for the three-dimensional quantification of microdamage in trabecular bone

An accurate analysis and quantification of microdamage is critical to understand how microdamage affects the mechanics and biology of bone fragility. In this study we demonstrate the development and validation of a novel in vitro micro-computed tomography (microCT) method that employs lead-uranyl acetate as a radio-opaque contrast agent for automated quantification of microdamage in trabecular bone. Human trabecular bone cores were extracted from the femoral neck, scanned via microCT, loaded in unconfined compression to a range of apparent strains (0.5% to 2.25%), stained in lead-uranyl acetate, and subsequently re-scanned via microCT. An investigation of the regions containing microdamage using the backscatter mode of a scanning electron microscope (BSEM) showed that the lead-uranyl sulfide complex was an effective contrast agent for microdamage in bone. Damaged volume fraction (DV/BV), as determined by microCT, increased exponentially with respect to applied strains and proportionately to mechanically determined modulus reduction (p<0.001). Furthermore, the formation of microdamage was observed to occur before any apparent stiffness loss, suggesting that the localized tissue yielding occurs prior to the structural yielding of trabecular bone. This non-invasive in vitro technique for the detection of microdamage using microCT may serve as a valuable complement to existing morphometric analyses of bone.

Comparative assessment of bone mass and structure using texture-based and histomorphometric analyses

The purpose of this study was to develop a methodology for quantitatively assessing bone quantity and anisotropy based on texture analysis using Gabor wavelets. The wavelet approach has the capability to simultaneously examine the images at low and high resolutions to gain information on both global and detailed local features of the bone image. The program that implemented the texture analysis gave measures of density (M(Density)) and anisotropy (M(Anisotropy)). It also allowed us to examine the texture energy at four orientations (0 degrees , 45 degrees , 90 degrees , 135 degrees) to gain insight about the details of the anisotropy. Analysis of templates of four simulated patterns, which had same number of dots but with differing orientations, demonstrated how the texture-based analysis differentiated between these templates. The measures of M(Anisotropy) discriminated between the four simulated patterns. The M(Density) measures were similar across all patterns. These outcomes matched the design intent of the simulated patterns. We also compared the trabecular bone images obtained from a previous study, in which the right forelimbs of normal female retired breeder beagle dogs (5-7 years old) were cast for 12 months to induce bone loss, using both histomorphometry and texture analysis. Both histomorphometry and the texture analysis detected significant differences in the trabecular bone of the distal metatarsal between the control and disuse groups. Percent trabecular bone (Tb.Ar/T.Ar) and the textural density parameter (M(Density)) were highly correlated (r=0.962). M(Anisotropy) was decreased (3.9%) after the 12-month disuse protocol, but was not significantly different from normal. However, the texture energy values at all orientations (0 degrees , 45 degrees , 90 degrees and 135 degrees) were significantly decreased in the disuse group. Therefore, texture analysis was able to assess anisotropy, which could not be extracted from histomorphometric parameters. We conclude that texture analysis is an effective tool for assessing 2D bone images that yields information regarding the quantity of bone as well as the orientation of the trabecular structure that can augment our ability to discriminate between normal and pathological bone tissue.