A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure

A 3D anisotropic micropolar continuum model of vertebral trabecular bone is presently developed accounting for the influence of microstructure-related scale effects on the macroscopic effective properties. Vertebral trabecular bone is modeled as a cellular material with an idealized periodic structure made of open 3D cells. The micromechanical approach relies on the discrete homogenization technique considering lattice microrotations as additional degrees of freedom at the microscale. The effective elastic properties of 3D lattices made of articulated beams taking into account axial, transverse shearing, flexural, and torsional deformations of the cell struts are derived as closed form expressions of the geometrical and mechanical microparameters. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the same microparameters. A finite element model of the local architecture of the trabeculae gives values of the effective moduli that are in satisfactory agreement with the homogenized moduli.     

Stiffness characterization of anisotropic trabecular meshwork

Elevation of intraocular pressure has been correlated to changes in stiffness of trabecular meshwork (TM) in glaucomatous eyes although mechanical properties of the TM remain to be quantitatively determined. Data in the literature suggest that the TM cannot be considered mechanically as a uniform layer of isotropic elastic material, because the value of its Young’s modulus depends on the methods of measurements and can vary up to five orders of magnitude. To this end, we proposed a new theoretical framework for mechanical analysis of the TM, in which the inner wall of Schlemm’s canal and the juxtacanalicular tissue in the TM were treated as a uniform layer of isotropic elastic material, and the rest of the TM, i.e., the uveal and corneoscleral meshworks, were modeled as a uniform layer of transversely isotropic material. Using the model, we demonstrated that the large discrepancy in the apparent Young’s modulus reported in the literature could be caused by the anisotropy of the meshwork that was significantly stiffer in the longitudinal direction than in the transverse direction. The theoretical framework could be used to integrate existing data of the stiffness, investigate anisotropic behaviors of the tissues, and develop new methods to measure mechanical properties of the TM.     

Determination of the geometry of the RVE for cancellous bone by using the effective complex shear modulus

This contribution deals with the application of the inverse homogenization method to the determination of geometrical properties of cancellous bone. The approach represents a combination of an extended version of the Marquardt–Levenberg method with the multiscale finite element method. The former belongs to the group of gradient-based optimization strategies, while the latter is a numerical homogenization method, suitable for the modeling of materials with a highly heterogeneous microstructure. The extension of the Marquardt–Levenberg method is concerned with the selection strategy for distinguishing the global minimum from the plethora of local minima. Within the numerical examples, the bone is modeled as a biphasic viscoelastic medium and three different representative volume elements are taken into consideration. Different models enable the simulation of the bone either as a purely isotropic or as a transversally anisotropic medium. Main geometrical properties of trabeculae are determined from data on effective shear modulus but alternative schemes are also possible.     

An estimate of anisotropic poroelastic constants of an osteon

The anisotropic poroelastic constants of an osteon are estimated by micromechanical analysis. Two extreme cases are examined, the drained and the undrained elastic constants. The drained elastic constants are the porous medium’s effective elastic constants when the fluid in the pores easily escapes and the pore fluid can sustain no pore pressure. The undrained elastic constants are the porous medium’s effective elastic constants when the medium is fully saturated with pore fluid and the fluid cannot escape. The drained and undrained elastic constants at the lacunar and canalicular porosity tissue levels are estimated by using an effective moduli model consisting of the periodic distribution of ellipsoidal cavities. These estimated anisotropic poroelastic constants provide a database for the development of an accurate anisotropic poroelastic model of an osteon.     

Dehomogenized Elastic Properties of Heterogeneous Layered Materials in AFM Indentation Experiments

Atomic force microscopy (AFM) is used to study mechanical properties of biological materials at submicron length scales. However, such samples are often structurally heterogeneous even at the local level, with different regions having distinct mechanical properties. Physical or chemical disruption can isolate individual structural elements but may alter the properties being measured. Therefore, to determine the micromechanical properties of intact heterogeneous multilayered samples indented by AFM, we propose the Hybrid Eshelby Decomposition (HED) analysis, which combines a modified homogenization theory and finite element modeling to extract layer-specific elastic moduli of composite structures from single indentations, utilizing knowledge of the component distribution to achieve solution uniqueness. Using finite element model-simulated indentation of layered samples with micron-scale thickness dimensions, biologically relevant elastic properties for incompressible soft tissues, and layer-specific heterogeneity of an order of magnitude or less, HED analysis recovered the prescribed modulus values typically within 10% error. Experimental validation using bilayer spin-coated polydimethylsiloxane samples also yielded self-consistent layer-specific modulus values whether arranged as stiff layer on soft substrate or soft layer on stiff substrate. We further examined a biophysical application by characterizing layer-specific microelastic properties of full-thickness mouse aortic wall tissue, demonstrating that the HED-extracted modulus of the tunica media was more than fivefold stiffer than the intima and not significantly different from direct indentation of exposed media tissue. Our results show that the elastic properties of surface and subsurface layers of microscale synthetic and biological samples can be simultaneously extracted from the composite material response to AFM indentation. HED analysis offers a robust approach to studying regional micromechanics of heterogeneous multilayered samples without destructively separating individual components before testing.     

Nonlinear hierarchical multiscale modeling of cortical bone considering its nanoscale microstructure

We have used a hierarchical multiscale modeling scheme for the analysis of cortical bone considering it as a nanocomposite. This scheme consists of definition of two boundary value problems, one for macroscale, and another for microscale. The coupling between these scales is done by using the homogenization technique. At every material point in which the constitutive model is needed, a microscale boundary value problem is defined using a macroscopic kinematical quantity and solved. Using the described scheme, we have studied elastic properties of cortical bone considering its nanoscale microstructural constituents with various mineral volume fractions. Since the microstructure of bone consists of mineral platelet with nanometer size embedded in a protein matrix, it is similar to the microstructure of soft matrix nanocomposites reinforced with hard nanostructures. Considering a representative volume element (RVE) of the microstructure of bone as the microscale problem in our hierarchical multiscale modeling scheme, the global behavior of bone is obtained under various macroscopic loading conditions. This scheme may be suitable for modeling arbitrary bone geometries subjected to a variety of loading conditions. Using the presented method, mechanical properties of cortical bone including elastic moduli and Poisson's ratios in two major directions and shear modulus is obtained for different mineral volume fractions.     

Influence of anisotropic bone properties on the biomechanical behavior of the acetabular cup implant: a multiscale finite element study

Although the biomechanical behavior of the acetabular cup (AC) implant is determinant for the surgical success, it remains difficult to be assessed due to the multiscale and anisotropic nature of bone tissue. The aim of the present study was to investigate the influence of the anisotropic properties of peri-implant trabecular bone tissue on the biomechanical behavior of the AC implant at the macroscopic scale. Thirteen bovine trabecular bone samples were imaged using micro-computed tomography (μCT) with a resolution of 18 μm. The anisotropic biomechanical properties of each sample were determined at the scale of the centimeter based on a dedicated method using asymptotic homogenization. The material properties obtained with this multiscale approach were used as input data in a 3D finite element model to simulate the macroscopic mechanical behavior of the AC implant under different loading conditions. The largest stress and strain magnitudes were found around the equatorial rim and in the polar area of the AC implant. All macroscopic stiffness quantities were significantly correlated (R² > 0.85, p < 6.5 e-6) with BV/TV (bone volume/total volume). Moreover, the maximum value of the von Mises stress field was significantly correlated with BV/TV (R² > 0.61, p < 1.6 e-3) and was always found at the bone-implant interface. However, the mean value of the microscopic stress (at the scale of the trabeculae) decrease as a function of BV/TV for vertical and torsional loading and do not depend on BV/TV for horizontal loading. These results highlight the importance of the anisotropic properties of bone tissue.     

Anisotropic elastic network modeling of entire microtubules

Microtubules are supramolecular structures that make up the cytoskeleton and strongly affect the mechanical properties of the cell. Within the cytoskeleton filaments, the microtubule (MT) exhibits by far the highest bending stiffness. Bending stiffness depends on the mechanical properties and intermolecular interactions of the tubulin dimers (the MT building blocks). Computational molecular modeling has the potential for obtaining quantitative insights into this area. However, to our knowledge, standard molecular modeling techniques, such as molecular dynamics (MD) and normal mode analysis (NMA), are not yet able to simulate large molecular structures like the MTs; in fact, their possibilities are normally limited to much smaller protein complexes. In this work, we developed a multiscale approach by merging the modeling contribution from MD and NMA. In particular, MD simulations were used to refine the molecular conformation and arrangement of the tubulin dimers inside the MT lattice. Subsequently, NMA was used to investigate the vibrational properties of MTs modeled as an elastic network. The coarse-grain model here developed can describe systems of hundreds of interacting tubulin monomers (corresponding to up to 1,000,000 atoms). In particular, we were able to simulate coarse-grain models of entire MTs, with lengths up to 350 nm. A quantitative mechanical investigation was performed; from the bending and stretching modes, we estimated MT macroscopic properties such as bending stiffness, Young modulus, and persistence length, thus allowing a direct comparison with experimental data.     

In Vitro Fracture Testing of Submicron Diameter Collagen Fibril Specimens

Mechanical testing of collagenous tissues at different length scales will provide improved understanding of the mechanical behavior of structures such as skin, tendon, and bone, and also guide the development of multiscale mechanical models. Using a microelectromechanical-systems (MEMS) platform, stress-strain response curves up to failure of type I collagen fibril specimens isolated from the dermis of sea cucumbers were obtained in vitro. A majority of the fibril specimens showed brittle fracture. Some displayed linear behavior up to failure, while others displayed some nonlinearity. The fibril specimens showed an elastic modulus of 470 ± 410 MPa, a fracture strength of 230 ± 160 MPa, and a fracture strain of 80% ± 44%. The fibril specimens displayed significantly lower elastic modulus in vitro than previously measured in air. Fracture strength/strain obtained in vitro and in air are both significantly larger than those obtained in vacuo, indicating that the difference arises from the lack of intrafibrillar water molecules produced by vacuum drying. Furthermore, fracture strength/strain of fibril specimens were different from those reported for collagenous tissues of higher hierarchical levels, indicating the importance of obtaining these properties at the fibrillar level for multiscale modeling.     

Biomechanical properties across trabeculae from the proximal femur of normal and ovariectomised sheep

The elastic behaviour of trabecular bone is a function not only of bone volume and architecture, but also of tissue material properties. Variation in tissue modulus can have a substantial effect on the biomechanical properties of trabecular bone. However, the nature of tissue property variation within a single trabecula is poorly understood. This study uses nanoindentation to determine the mechanical properties of bone tissue in individual trabeculae. Using an ovariectomised ovine model, the modulus and hardness distribution across trabeculae were measured. In both normal and ovariectomised bone, the modulus and hardness were found to increase towards the core of the trabeculae. Across the width of the trabeculae, the modulus was significantly less in the ovariectomised bone than in the control bone. However, in contrast to this hardness was found not to differ significantly between the two groups. This study provides valuable information on the variation of mechanical material properties in healthy and diseased trabecular bone tissue. The results of the current study will be useful in finite element modelling where more accurate values of trabecular bone modulus will enable the prediction of the macroscale behaviour of trabecular bone.     

Finite element evaluation of the AIA shear specimen for bone

An elastic-plastic finite element analysis is performed on the AIA shear specimen to evaluate its effectiveness to yield ultimate shear strength values. The effect of geometry, material properties, and yield criteria are discussed in the light of applications to human femoral cortical bone. Specimen dimensions are noted as follows: W, width, D, hole diameter and H, distance between holes. As the H/D ratio increases the stress distribution tends more toward pure shear at the same time the overshoot in the shear distribution increases. An H/D ratio equal to 1.2-1.5 is optimal. The H/W parameter does not affect the overshoot noticeably but it does slightly affect the purity of shear. The material parameters do affect the performance of the shear specimen. However, the effect of the material parameters are far more pronounced in the anisotropic case than it is in the isotropic case. In the isotropic case, the Young modulus does not affect the overshoot. The increase in Poisson's ratio does slightly decrease the overshoot. For the anisotropic case, the increase in the ratio of shear modulus to Young modulus in the transverse direction (G/E2) results in an increase in the overshoot (in the shear distribution). The increase in the ratio of the Young modulus in the transverse direction to that of the axial direction (E2/E1) also results in an increase in the overshoot. Creating a notch at the top of the hole is shown to have the effect of decreasing the overshoot. Its effect on the purity of the shear is rather slight. It is found that plasticity is initiated at the sides of the two holes where the tensile normal stresses are maximum. The plastic region first expands around the perimeter of the hole then radially outward; and finally, it expands into the significant region. If the W/H parameter is less than 5, a sizable portion of the width of the specimen around the hole can go plastic with the significant region still being in the elastic state. Such a situation can cause tearing of the specimen across the width. A W/H ratio of 6 or more can prevent that danger. It is also found that the onset of plasticity brings about higher overshoot and higher purity of shear. The notched shear specimen performs better in actual tests and is more reliable in producing shear failures. The shear strength results obtained from AIA shear tests tend to confirm those shear strength results obtained from torsion tests.     

Biological thermal detection: micromechanical and microthermal properties of biological infrared receptors

Bioinspired design of biomimetic sensors relies upon the complete understanding of properties and functioning of biological analogues in conjunction with an understanding of their microstructural organization at various length scales. In the spirit of this approach, the microscopic properties of infrared (IR) receptors of snakes with "infrared vision" were studied with scanning thermal microscopy and micromechanical analysis. Low surface thermal conductivity of 0.11 W/(m K) was measured for the IR receptor surfaces as compared to the nonspecific skin areas. This difference in surface thermal conductivity should result in a significant local temperature gradient around the receptor areas. Micromechanical analysis showed that pit organs were more compliant than surrounding skin areas with an elastic modulus close to 40 MPa. In addition, the maximum elastic modulus was detected for the outermost layer with gradually reduced elastic resistance for the interior. The porous microstructure of the underlying tissue combined with the highly branched microfibrillar network (Biomacromolecules 2001, 2, 757) is thought to be responsible for such a combination of biomaterial properties. Considering these biomaterials features, we postulated a possible design of an artificial photothermal detector inspired by the microstructure of natural receptors. This bioinspired design would include a microfabricated cavity filled with an ordered lattice of microspheres with a gradient periodicity from the surface to the interior. Such a "photonic cavity" could provide an opportunity for multiple scattering at wavelength tuned to 8-12 microm as a range of highest sensitivity.     

Mechanical model of the nucleosome and chromatin

A theoretical framework for evaluating the approximate energy and dynamic properties associated with the folding of DNA into nucleosomes and chromatin is presented. Experimentally determined elastic constants of linear DNA and a simple fold geometry are assumed in order to derive elastic constants for extended and condensed chromatin. The model predicts the Young s modulus of extended and condensed chromatin to within an order of magnitude of experimentally determined values. Thus we demonstrate that the elastic properties of DNA are a primary determinant of the elastic properties of the higher order folded states. The derived elastic constants are used to predict the speed of propagation of small amplitude waves that excite an extension(sound), twist, bend or shear motion in each folded state. Taken together the results demonstrate that folding creates a hierarchy of time, length and energy scales.     

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

The role of mineral content in determining the micromechanical properties of discrete trabecular bone remodeling packets

In trabecular bone, each remodeling event results in the resorption and/or formation of discrete structural units called ‘packets’. These remodeling packets represent a fundamental level of bone’s structural hierarchy at which to investigate composition and mechanical behaviors. The objective of this study was to apply the complementary techniques of quantitative backscattered electron microscopy (qBSEM) and nanoindentation to investigate inter-relationships between packet mineralization, elastic modulus, contact hardness and plastic deformation resistance. Indentation arrays were performed across nine trabecular spicules from 3 human donors; these spicules were then imaged using qBSEM, and discretized into their composite remodeling packets (127 in total). Packets were classified spatially as peripheral or central, and mean contact hardness, plastic deformation resistance, elastic modulus and calcium content calculated for each. Inter-relationships between measured parameters were analysed using linear regression analyses, and dependence on location assessed using Student’s t-tests. Significant positive correlations were found between all mechanical parameters and calcium content. Elastic modulus and contact hardness were significantly correlated, however elastic modulus and plastic deformation resistance were not. Calcium content, contact hardness and elastic modulus were all significantly higher for central packets than for peripheral, confirming that packet mineral content contributes to micromechanical heterogeneity within individual trabecular spicules. Plastic deformation resistance, however, showed no such regional dependence, indicating that the plastic deformation properties in particular, are determined not only by mineral content, but also by the organic matrix and interactions between these two components.     

Effects of age on elastic moduli of human lungs.

The model of the lung as an elastic continuum undergoing small distortions from a uniformly inflated state has been used to describe many lung deformation problems. Lung stress-strain material properties needed for this model are described by two elastic moduli: the bulk modulus, which describes a uniform inflation, and the shear modulus, which describes an isovolume deformation. In this study we measured the bulk modulus and shear modulus of human lungs obtained at autopsy at several fixed transpulmonary pressures (Ptp). The bulk modulus was obtained from small pressure-volume perturbations on different points of the deflation pressure-volume curve. The shear modulus was obtained from indentation tests on the lung surface. The results indicated that, at a constant Ptp, both bulk and shear moduli increased with age, and the increase was greater at higher Ptp values. The micromechanical basis for these changes remains to be elucidated.     

3D finite element analysis of porous Ti-based alloy prostheses

In this paper, novel designs of porous acetabular cups are created and tested with 3D finite element analysis (FEA). The aim is to develop a porous acetabular cup with low effective radial stiffness of the structure, which will be near to the architectural and mechanical behavior of the natural bone. For the realization of this research, a 3D-scanner technology was used for obtaining a 3D-CAD model of the pelvis bone, a 3D-CAD software for creating a porous acetabular cup, and a 3D-FEA software for virtual testing of a novel design of the porous acetabular cup. The results obtained from this research reveal that a porous acetabular cup from Ti-based alloys with 60 ± 5% porosity has the mechanical behavior and effective radial stiffness (Young’s modulus in radial direction) that meet and exceed the required properties of the natural bone. The virtual testing with 3D-FEA of a novel design with porous structure during the very early stage of the design and the development of orthopedic implants, enables obtaining a new or improved biomedical implant for a relatively short time and reduced price.     

Mechanical behavior of a new biphasic calcium phosphate bone graft

The aim of this study was to create a new porous calcium phosphate implant for use as a synthetic bone graft substitute. Porous bioceramic was fabricated using a foam-casting method. By using polyurethane foam and a slurry containing hydroxyapatite-dicalcium phosphate powder, water, and additives, a highly porous structure (66 ± 5%) was created. The porous specimens possess an elastic modulus of 330 ± 32 MPa and a compressive strength of 10.3 ± 1.7 MPa. The X-ray diffraction patterns show hydroxyapatite and beta-pyrophosphate phases after sintering. A rabbit model was developed to evaluate the compressive strength and elastic modulus of cancellous bone defects treated with these porous synthetic implants. The compressive mechanical properties became weaker until the second month post implantation. After the second month, these properties increased slightly and remained higher than control values. New bone formed on the outside surface and on the macropore walls of the specimens, as osteoids and osteoclasts were evident two months postoperatively. Considering these properties, these synthetic porous calcium phosphate implants could be applicable as cancellous bone substitutes.     

Mechanical properties of cancellous bone in the human mandibular condyle are anisotropic

The objective of the present study was (1) to test the hypothesis that the elastic and failure properties of the cancellous bone of the mandibular condyle depend on the loading direction, and (2) to relate these properties to bone density parameters. Uniaxial compression tests were performed on cylindrical specimens (n=47) obtained from the condyles of 24 embalmed cadavers. Two loading directions were examined, i.e., a direction coinciding with the predominant orientation of the plate-like trabeculae (axial loading) and a direction perpendicular to the plate-like trabeculae (transverse loading). Archimedes' principle was applied to determine bone density parameters. The cancellous bone was in axial loading 3.4 times stiffer and 2.8 times stronger upon failure than in transverse loading. High coefficients of correlation were found among the various mechanical properties and between them and the apparent density and volume fraction. The anisotropic mechanical properties can possibly be considered as a mechanical adaptation to the loading of the condyle in vivo.     

Nanostructure and elastic modulus of single trabecula in bovine cancellous bone

We aimed to investigate the elastic modulus of trabeculae using tensile tests and assess the effects of nanostructure at the hydroxyapatite (HAp) crystal scale on the elastic modulus. In the experiments, 18 trabeculae that were at least 3 mm in length in the proximal epiphysis of three adult bovine femurs were used. Tensile tests were conducted using a small tensile testing device coupled with microscopy under air-dried condition. The c-axis orientation of HAp crystals and the degree of orientation were measured by X-ray diffraction. To observe the deformation behavior of HAp crystals under tensile loading, the same tensile tests were conducted in X-ray diffraction measurements. The mineral content of specimens was evaluated using energy dispersive X-ray spectrometry. The elastic modulus of a single trabecula varied from 4.5 to 23.6 GPa, and the average was 11.5±5.0 GPa. The c-axis of HAp crystals was aligned with the trabecular axis and the crystals were lineally deformed under tensile loading. The ratio of the HAp crystal strain to the tissue strain (strain ratio) had a significant correlation with the elastic modulus (r=0.79; P<0.001). However, the mineral content and the degree of orientation did not vary widely and did not correlate with the elastic modulus in this study. It suggests that the strain ratio may represent the nanostructure of a single trabecula and would determine the elastic modulus as well as mineral content and orientation.