Tensile behavior of cortical bone: dependence of organic matrix material properties on bone mineral content

A porous composite model is developed to analyze the tensile mechanical properties of cortical bone. The effects of microporosity (volksman's canals, osteocyte lacunae) on the mechanical properties of bone tissue are taken into account. A simple shear lag theory, wherein tensile loads are transferred between overlapped mineral platelets by shearing of the organic matrix, is used to model the reinforcement provided by mineral platelets. It is assumed that the organic matrix is elastic in tension and elastic-perfectly plastic in shear until it fails. When organic matrix shear stresses at the ends of mineral platelets reach their yield values, the stress-strain curve of bone tissue starts to deviate from linear behavior. This is referred as the microscopic yield point. At the point where the stress-strain behavior of bone shows a sharp curvature, the organic phase reaches its shear yield stress value over the entire platelet. This is referred as the macroscopic yield point. It is assumed that after macroscopic yield, mineral platelets cannot contribute to the load bearing capacity of bone and that the mechanical behavior of cortical bone tissue is determined by the organic phase only. Bone fails when the principal stress of the organic matrix is reached. By assuming that mechanical properties of the organic matrix are dependent on bone mineral content below the macroscopic yield point, the model is used to predict the entire tensile mechanical behavior of cortical bone for different mineral contents. It is found that decreased shear yield stresses and organic matrix elastic moduli are required to explain the mechanical behavior of bones with lowered mineral contents. Under these conditions, the predicted values (elastic modulus, 0.002 yield stress and strain, and ultimate stress and strain) are within 15% of experimental data.     

Effect of bone mineral content on the tensile properties of cortical bone: experiments and theory

The effect of mineral volume fraction on the tensile mechanical properties of cortical bone tissue is investigated by theoretical and experimental means. The mineral content of plexiform, bovine bone was lowered by 18% and 29% by immersion in fluoride solutions for 3 days and 12 days, respectively. The elastic modulus, yield strength and ultimate strength of bone tissue decreased, while the ultimate strain increased with a decrease in mineral content. The mechanical behavior of bone tissue was modeled by using a micromechanical shear lag theory consisting of overlapped mineral platelets reinforcing the organic matrix. The decrease in yield stress, by the 0.002 offset method, of the fluoride treated bones were matched in the theoretical curves by lowering the shear yield stress of the organic matrix. The failure criterion used was based on failure stresses determined from a failure envelope (Mohr's circle), which was constructed using experimental data. It was found that the model predictions of elastic modulus got worse with a decrease in mineral content (being 7.9%, 17.2% and 33.0% higher for the control, 3-day and 12-day fluoride-treated bones). As a result, the developed theory could not fully predict the yield strain of bones with lowered mineral content, being 12.9% and 21.7% lower than the experimental values. The predicted ultimate stresses of the bone tissues with lower mineral contents were within +/- 10% of the experimental values while the ultimate strains were 12.7% and 26.3% lower than the experimental values. Although the model developed in this study did not take into account the presence of hierarchical structures, voids, orientation of collagen molecules and micro cracks, it still indicated that the mechanical properties of the organic matrix depend on bone mineral content.     

Interfacial mechanical behaviour of protein–mineral nanocomposites: A molecular dynamics investigation

Biological composite materials, such as bone, tooth and nacre, are comprised of a mixture of nano-sized hard components (e.g. mineral platelets) and soft components (e.g. protein molecules). Their mechanical behaviour greatly depends on the protein–mineral interfaces. This paper investigates the effects of mineral surface nanostructures on the interfacial interaction and mechanical behaviour of protein–mineral nanocomposites. Interfacial shear between osteopontin (OPN) and hydroxyapatite (HA) mineral layers with surface nanostructures is investigated using the atomistic molecular dynamics (MD) simulations. The results indicate that the OPN residues can be attached to HA surfaces but the surface nanostructures greatly affect the interfacial interaction and mechanical behaviour. The HA layers with a higher number of nano-sized grooves (defects) increase the surface roughness but reduce the pulling force and energy dissipation.     

Structure-strength relations in mammalian tendon.

The stress-strain relations in mammalian tendon are analyzed in terms of the structure and mechanics of its constituents. The model considers the tensile and bending strength of the collagen fibers, the tensile strength of the elastin fibers, and the interaction between the matrix and the collagen fibers. The stress-strain relations are solved through variational considerations by assuming that the fibermaxtrix interactions can be modeled as beam on elastic foundation. The tissue thus modeled is a hyperelastic material. It is further shown that on the basis of the model, the dominant parameters to the tendon's behavior can be evaluated from simple tensile tests.     

Failure of mineralized collagen fibrils: modeling the role of collagen cross-linking

Experimental evidence demonstrates that collagen cross-linking in bone tissue significantly influences its deformation and failure behavior yet difficulties exist in determining the independent biomechanical effects of collagen cross-linking using in vitro and in vivo experiments. The aim of this study is to use a nano-scale composite material model of mineral and collagen to determine the independent roles of enzymatic and non-enzymatic cross-linking on the mechanical behavior of a mineralized collagen fibril. Stress–strain curves were obtained under tensile loading conditions without any collagen cross-links, with only enzymatic cross-links (modeled by cross-linking the end terminal position of each collagen domain), or with only non-enzymatic cross-links (modeled by random placement of cross-links within the collagen–collagen interfaces). Our results show enzymatic collagen cross-links have minimal effect on the predicted stress–strain curve and produce a ductile material that fails through debonding of the mineral–collagen interface. Conversely, non-enzymatic cross-links significantly alter the predicted stress–strain response by inhibiting collagen sliding. This inhibition leads to greater load transfer to the mineral, which minimally affects the predicted stress, increases modulus and decreases post-yield strain and toughness. As a consequence the toughness of bone that has more non-enzymatically mediated collagen cross-links will be drastically reduced.     

Mechanical interaction between cells and fluid for bone tissue engineering scaffold: Modulation of the interfacial shear stress

An analytical model of the fluid/cell mechanical interaction was developed. The interfacial shear stress, due to the coupling between the fluid and the cell deformation, was characterized by a new dimensionless number N_fs. For N_fs above a critical value, the fluid/cell interaction had a damping effect on the interfacial shear stress. Conversely, for N_fs below this critical value, interfacial shear stress was amplified. As illustration, the role of the dynamic fluid/cell mechanical coupling was studied in a specific biological situation involving cells seeded in a bone scaffold. For the particular bone scaffold chosen, the dimensionless number N_fs was higher than the critical value. In this case, the dynamic shear stress at the fluid/cell interface is damped for increasing excitation frequency. Interestingly, this damping effect is correlated to the pore diameter of the scaffold, furnishing thus target values in the design of the scaffold. Correspondingly, an efficient cell stimulation might be achieved with a scaffold of pore size larger than 300 μm as no dynamic damping effect is likely to take place. The analytical model proposed in this study, while being a simplification of a fluid/cell mechanical interaction, brings complementary insights to numerical studies by analyzing the effect of different physical parameters.     

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.     

Predicting fracture of the femoral neck

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

The role of viscoelasticity of collagen fibers in articular cartilage: theory and numerical formulation

The relative importance of fluid-dependent and fluid-independent transient mechanical behavior in articular cartilage was examined for tensile and unconfined compression testing using a fibril reinforced model. The collagen matrix of articular cartilage was modeled as viscoelastic using a quasi-linear viscoelastic formulation with strain-dependent elastic modulus, while the proteoglycan matrix was considered as linearly elastic. The collagen viscoelastic properties were obtained by fitting experimental data from a tensile test. These properties were used to investigate unconfined compression testing, and the sensitivity of the properties was also explored. It was predicted that the stress relaxation observed in tensile tests was not caused by fluid pressurization at the macroscopic level. A multi-step tensile stress relaxation test could be approximated using a hereditary integral in which the elastic fibrillar modulus was taken to be a linear function of the fibrillar strain. Applying the same formulation to the radial fibers in unconfined compression, stress relaxation could not be simulated if fluid pressurization were absent. Collagen viscoelasticity was found to slightly weaken fluid pressurization in unconfined compression, and this effect was relatively more significant at moderate strain rates. Therefore, collagen viscoelasticity appears to play an import role in articular cartilage in tensile testing, while fluid pressurization dominates the transient mechanical behavior in compression. Collagen viscoelasticity plays a minor role in the mechanical response of cartilage in unconfined compression if significant fluid flow is present.     

Deformation behaviour and damage accumulation of cortical bone specimens from the equine tibia under cyclic loading

Despite its clinical importance, the fatigue behaviour of cortical bone has not been examined as widely as its static behaviour. In the present study, specimens from the tibiae of horses have been subjected to load-controlled single step tests. The cyclic deformation behaviour was described by the development of stress-strain hysteresis parameters over the lifetime. The fatigue behaviour of bone is characterised by cyclic softening which is most distinctive towards the end of the lifetime. The microstructural damage accumulated during cyclic loading results in a loss of stiffness, asymmetrical deformation of the bone in tension and compression in cyclic creep. As shown by light and scanning electron microscopy, microcrack formation and growth is the main damage mechanism. The crack growth behaviour is strongly influenced by the microstructure, the stress components and the absolute value of the local stresses. Lower local stresses and/or compressive mean stresses lead to a dominant influence of the shear stress components with shear failure at inner interfaces. With increasing crack length, that is, higher local stress amplitudes, or tensile mean stresses, the microstructure is more and more ignored and failure occurs primarily under the influence of the normal stress components. This can be clearly seen on the fracture and specimen surfaces.     

Tensile damage and its effects on cortical bone

Plexiform bovine bone samples are repeatedly loaded in tension along their longitudinal axis. In order to induce damage in the bone tissue, bone samples are loaded past their yield point. Half of the bone samples from the damaged group were stored in saline to allow for viscoelastic recovery while the others were decalcified. Tensile tests were conducted on these samples to characterize the effects of damage on the mechanical behavior of the organic matrix (decalcified samples) as well as on bone tissue (stored in saline). The ultimate strain of the damaged decalcified bone is 29% higher compared to that of non-damaged decalcified (control) bone. The ultimate stresses as well as the elastic moduli are similar in both decalcified groups. This phenomenon is also observed in other collagenous tissue (tendon and ligament). This may suggest that damage in bone is caused by shear failure of the organic matrix; transverse separation of the collagen molecules or microfibrils from each other. In contrast, there is a trend towards lowered ultimate strains in damaged bone, which is soaked in saline, with respect to control bone samples (not damaged). The damaged bone tissue exhibits a bi-linear behavior in contrast to the mechanical behavior of non-damaged bone. The initial elastic modulus (below 55 MPa) and ultimate strength of damaged bone are similar to that in non-damaged bone.     

A comparison of biomechanical properties between human and porcine cornea

Due to the difficulty in obtaining human corneas, pig corneas are often substituted as models for cornea research. The purpose of this study is to find the similarities and differences in the biomechanical properties between human and porcine corneas. Uniaxial tests were conducted using an Instron apparatus to determine their tensile strength, stress-strain relationship, and stress-relaxation properties. The tensile strength and stress-strain relation were very similar but significant differences between the two tissues were observed in the stress-relaxation relationship. Under the same stretch ratio lambda=1.5, porcine cornea relaxed much more than human cornea. If tensile strength and the stress-strain relation are the only mechanical factors to be investigated, porcine cornea can be used as a substitute model for human cornea research. However, when stress relaxation is a factor, porcine corneas cannot be used as an appropriate model for human corneas in mechanical property studies. It is very difficult to get enough specimens of human cornea, so we did the experiments for stress-strain relationship at a specific value of strain rate (corresponding to the velocity of loading 10mm/min), and for stress relaxation at a specific stretch ratio lambda=1.5.     

Load transfer along the bone–dental implant interface

In this paper the variation of normal and shear stresses along a path defined on the bone–dental implant interface is investigated. In particular, the effects of implant diameter, collar length and slope, body length, and the effects of four different types of external threads on the interfacial stress distribution are studied. The geometry of the bone is digitized from a CT scan of a mandibular incisor and the surrounding bone. The bone and the implant are assumed to be perfectly bonded. The finite element method with 2D plane strain assumption is used to compute interfacial stresses. Highest continuous interfacial stresses are encountered in the region where the implant collar engages the cortical region, and near the apex of the implant in the subcortical region. Stress concentrations in the interfacial stresses occur near the geometric discontinuities on the implant contour, and jumps in stress values occur where the elastic modulus of the bone transitions between the cortical and trabecular bone values. Among the six contour parameters, the slope and the length of the implant collar, and the implant diameter influence the interfacial stress levels the most, and the effects of changing these parameters are significantly noticed only in the cortical bone (alveolar ridge) area. External threads cause significant stress concentrations in interfacial stresses in otherwise smoothly varying regions. This work shows that the presence of external threads could cause significant variations in both normal and shear stresses along the bone–implant interface, but not reduction in shear stress as previously thought.     

Fracture analysis for biological materials with an expanded cohesive zone model

In this study, a theoretical framework for simulation of fracture of bone and bone-like materials is provided. An expanded cohesive zone model with thermodynamically consistent framework has been proposed and used to investigate the crack growth resistance of bone and bone-like materials. The reversible elastic deformation, irreversible plastic deformation caused by large deformation of soft protein matrix, and damage evidenced by the material separation and crack nucleation in the cohesive zone, were all taken into account in the model. Furthermore, the key mechanisms in deformation of biocomposites consisting of mineral platelets and protein interfacial layers were incorporated in the fracture process zone in this model, thereby overcoming the limitations of previous cohesive zone modeling of bone fracture. Finally, applications to fracture of cortical bone and human dentin were presented, which showed good agreement between numerical simulation and reported experiments and substantiated the effectiveness of the model in investigating the fracture behavior of bone-like materials.     

Shear strength and toughness of trabecular bone are more sensitive to density than damage

Microdamage occurs in trabecular bone under normal loading, which impairs the mechanical properties. Architectural degradation associated with osteoporosis increases damage susceptibility, resulting in a cumulative negative effect on the mechanical properties. Treatments for osteoporosis could be targeted toward increased bone mineral density, improved architecture, or repair and prevention of microdamage. Delineating the relative roles of damage and architectural degradation on trabecular bone strength will provide insight into the most beneficial targets. In this study, damage was induced in bovine trabecular bone samples by axial compression, and the effects on the mechanical properties in shear were assessed. The damaged shear modulus, shear yield stress, ultimate shear stress, and energy to failure all depended on induced damage and decreased as the architecture became more rod-like. The changes in ultimate shear strength and toughness were proportional to the decrease in shear modulus, consistent with an effective decrease in the cross-section of trabeculae based on cellular solid analysis. For typical ranges of bone volume fraction in human bone, the strength and toughness were much more sensitive to decreased volume fraction than to induced mechanical damage. While ultimately repairing or avoiding damage to the bone structure and increasing bone density both improve mechanical properties, increasing bone density is the more important contributor to bone strength.     

Mechanical behaviour of bio-inspired brittle-matrix nanocomposites under different strain rates using molecular dynamics

We study the generic mechanical behaviour of ceramic–ceramic nanocomposites inspired from biological materials. The nanocomposite models considered in our study are the regularly and stairwise staggered arrangements of stiff brittle platelets embedded in compliant brittle matrix. Molecular dynamics simulations are carried out to investigate the effect of strain rate on these nanocomposites. The variation in stress–strain behaviour and mechanical properties are analysed. The evolution of deformation processes is also investigated. Our results show the existence of different strain rate regimes separated by critical strain rate. Deformation mechanisms such as matrix cracking, crack bridging, interfacial debonding and hence platelet pullout are observed at lower strain rates. Amorphous deformation and direct debonding without matrix cracking are observed at higher strain rates.     

The role of fiber-matrix interactions in a nonlinear fiber-reinforced strain energy model of tendon

The objective of this study was to develop a nonlinear and anisotropic three-dimensional mathematical model of tendon behavior in which the structural components of fibers, matrix, and fiber-matrix interactions are explicitly incorporated and to use this model to infer the contributions of these structures to tendon mechanical behavior. We hypothesized that this model would show that: (i) tendon mechanical behavior is not solely governed by the isotropic matrix and fiber stretch, but is also influenced by fiber-matrix interactions; and (ii) shear fiber-matrix interaction terms will better describe tendon mechanical behavior than bulk fiber-matrix interaction terms. Model versions that did and did not include fiber-matrix interaction terms were applied to experimental tendon stress-strain data in longitudinal and transverse orientations, and the R2 goodness-of-fit was evaluated. This study showed that models that included fiber-matrix interaction terms improved the fit to longitudinal data (R2(toe) = 0.88, R2(Lin) = 0.94) over models that only included isotropic matrix and fiber stretch terms (R2(Toe) = 0.36, R2(Lin) = 0.84). Shear fiber-matrix interaction terms proved to be responsible for the best fit to data and to contribute to stress-strain nonlinearity. The mathematical model of tendon behavior developed in this study showed that fiber-matrix interactions are an important contributor to tendon behavior The more complete characterization of mechanical behavior afforded by this mathematical model can lead to an improved understanding of structure-function relationships in soft tissues and, ultimately, to the development of tissue-engineered therapies for injury or degeneration.     

Tendon exhibits complex poroelastic behavior at the nanoscale as revealed by high-frequency AFM-based rheology

Tendons transmit load from muscle to bone by utilizing their unique static and viscoelastic tensile properties. These properties are highly dependent on the composition and structure of the tissue matrix, including the collagen I hierarchy, proteoglycans, and water. While the role of matrix constituents in the tensile response has been studied, their role in compression, particularly in matrix pressurization via regulation of fluid flow, is not well understood. Injured or diseased tendons and tendon regions that naturally experience compression are known to have alterations in glycosaminoglycan content, which could modulate fluid flow and ultimately mechanical function. While recent theoretical studies have predicted tendon mechanics using poroelastic theory, no experimental data have directly demonstrated such behavior. In this study, we use high-bandwidth AFM-based rheology to determine the dynamic response of tendons to compressive loading at the nanoscale and to determine the presence of poroelastic behavior. Tendons are found to have significant characteristic dynamic relaxation behavior occurring at both low and high frequencies. Classic poroelastic behavior is observed, although we hypothesize that the full dynamic response is caused by a combination of flow-dependent poroelasticity as well as flow-independent viscoelasticity. Tendons also demonstrate regional dependence in their dynamic response, particularly near the junction of tendon and bone, suggesting that the structural and compositional heterogeneity in tendon may be responsible for regional poroelastic behavior. Overall, these experiments provide the foundation for understanding fluid-flow-dependent poroelastic mechanics of tendon, and the methodology is valuable for assessing changes in tendon matrix compressive behavior at the nanoscale.     

Tensile properties of the mandibular condylar cartilage

Mandibular condylar cartilage plays a crucial role in temporomandibular joint (TMJ) function, which includes facilitating articulation with the temporomandibular joint disc and reducing loads on the underlying bone. The cartilage experiences considerable tensile forces due to direct compression and shear. However, only scarce information is available about its tensile properties. The present study aims to quantify the biomechanical characteristics of the mandibular condylar cartilage to aid future three-dimensional finite element modeling and tissue engineering studies. Porcine condylar cartilage was tested under uniaxial tension in two directions, anteroposterior and mediolateral, with three regions per direction. Stress relaxation behavior was modeled using the Kelvin model and a second-order generalized Kelvin model, and collagen fiber orientation was determined by polarized light microscopy. The stress relaxation behavior of the tissue was biexponential in nature. The tissue exhibited greater stiffness in the anteroposterior direction than in the mediolateral direction as reflected by higher Young's (2.4 times), instantaneous (1.9 times), and relaxed (1.9 times) moduli. No significant differences were observed among the regional properties in either direction. The predominantly anteroposterior macroscopic fiber orientation in the fibrous zone of condylar cartilage correlated well with the biomechanical findings. The condylar cartilage appears to be less stiff and less anisotropic under tension than the anatomically and functionally related TMJ disc. The anisotropy of the condylar cartilage, as evidenced by tensile behavior and collagen fiber orientation, suggests that the shear environment of the TMJ exposes the condylar cartilage to predominantly but not exclusively anteroposterior loading.     

The relation between surface roughness and interfacial shear strength for bone-anchored implants. A mathematical model.

A rough bone implant surface was conceptualized as being built up of pits of different sizes and of different shapes. Hypotheses were formulated regarding the mechanical strength of the interfacial bone based upon the present knowledge of the character of the tissues adjacent to endosseous implants and the mechanical characteristics of different bone constituents. A surface roughness parameter was derived, the pit effectivity factor (fpe), which describes how effective the individual pits of the rough surface are as retention elements with regard to shear. Another surface roughness parameter was defined, the pit density factor (fpd), the value of which depends upon how densely packed the pits are. The interfacial shear strength of a rough implant surface with known microgeometry can be estimated by means of these two surface roughness parameters. The effectiveness of pits of different sizes and of different shapes was investigated using this model.