A morphological model of vertebral trabecular bone

In their micro-structures, typical natural cellular materials such as vertebral trabecular bone have a network of doubly tapered struts, thickening near the strut joints. However, past analytical models for vertebral trabecular bone do not take account of the effect of strut taper on the mechanical properties.This paper presents an analytical cell model comprised of doubly tapered struts to predict the global mechanical properties of vertebral trabecular bone. The predicted results for male, female, and both sexes fit the experimental data well. By considering several strut taper geometries, it is shown that the horizontal Young's modulus and the horizontal uniaxial collapse stress are, in some cases, approximately 1.8- and 2.2-fold higher, respectively, than those of the uniform strut model. This finding illustrates the importance of increased trabecular thickening near the strut joints (i) for improving the accuracy of calculating the mechanical properties and (ii) for the effective treatment of aged bone using drug therapy. It also highlights the need to combine trabecular architecture measurements with information about the morphology near the strut joints.     

The sensitivity of nonlinear computational models of trabecular bone to tissue level constitutive model

Microarchitectural finite element models have become a key tool in the analysis of trabecular bone. Robust, accurate, and validated constitutive models would enhance confidence in predictive applications of these models and in their usefulness as accurate assays of tissue properties. Human trabecular bone specimens from the femoral neck (n = 3), greater trochanter (n = 6), and lumbar vertebra (n = 1) of eight different donors were scanned by μ-CT and converted to voxel-based finite element models. Unconfined uniaxial compression and shear loading were simulated for each of three different constitutive models: a principal strain-based model, Drucker–Lode, and Drucker–Prager. The latter was applied with both infinitesimal and finite kinematics. Apparent yield strains exhibited minimal dependence on the constitutive model, differing by at most 16.1%, with the kinematic formulation being influential in compression loading. At the tissue level, the quantities and locations of yielded tissue were insensitive to the constitutive model, with the exception of the Drucker–Lode model, suggesting that correlation of microdamage with computational models does not improve the ability to discriminate between constitutive laws. Taken together, it is unlikely that a tissue constitutive model can be fully validated from apparent-level experiments alone, as the calculations are too insensitive to identify differences in the outcomes. Rather, any asymmetric criterion with a valid yield surface will likely be suitable for most trabecular bone models.     

A phenomenological model for predicting fatigue life in bovine trabecular bone

Cyclic loading of bone during daily activities can lead to fatigue degradation and increased risk of fracture in both the young and elderly population. Damage processes under cyclic loading in trabecular bone result in the reduction of the elastic modulus and accumulation of residual strain. These effects increase with increasing stress levels, leading to a progressive reduction in fatigue life. The present work analyzes the effect of stress and strain variation on the above damage processes in bovine trabecular bone, and develops a phenomenological model relating fatigue life to the imposed stress level. The elastic modulus reduction of the bone specimens was observed to depend on the maximum compressive strain, while the rate of residual strain accumulation was a function of the stress level. A model was developed for the upper and lower bounds of bone elastic modulus reduction with increasing number of cycles, at each stress range. The experimental observations were described well by the model. The model predicted the bounds of the fatigue life with change in fatigue stress. The decrease in the fatigue life with increasing stress was related to corresponding increases in the residual strain accumulation rates at the elevated stress levels. The model shows the validity of fatigue predictions from relatively few cyclic experiments, by combining trends observed in the monotonic and the cyclic tests. The model also presents a relatively simple procedure for predicting the endurance limit for bovine trabecular bone specimens.     

Microdamage accumulation in bovine trabecular bone in uniaxial compression

In this study we investigated how microdamage accumulated with increasing compressive strain in bovine trabecular bone. We found that little damage is created in the linear elastic region, up to -0.4 percent strain. At an average strain of -0.76 percent +/-0.25 percent, the stress-strain curve became nonlinear, and peaked at -1.91 percent +/-0.55 percent strain. Microdamage increases rapidly during the peak of the stress-strain curve, and a localized band of damage formed. At strains beyond the ultimate strain, the damaged band widened and the density of damage within the band increased. Microdamage occurred as groupings of cracks; the majority of damage occurred as regions of cross-hatching. All microdamage parameters increased with increasing maximum compressive strain. We also observed exponential relationships between crack numerical density and damage (1(o) - (o)Esec/E0) and between crack length density and damage.     

A structural model for the mechanical behavior of trabecular bone

A quantitative model is developed for trabecular bone by approximating the trabecular geometry with a hypothetical network of compact bone. For the region immediately beneath the articular cartilage in the distal end of the femur, finite element analyses were performed with a high speed computer, assuming a physiological static load. The results indicate that bending and buckling of trabeculae are considerable in any elastic deformation of the bone; that fatigue fracture in some fraction of suitably oriented trabeculae is inevitable in normal ambulation; and that the stiffness varies considerably with lateral position across the subchondral plate. The latter depends totally on trabecular arrangement and may play a role in joint function and degeneration. The adjustments necessary to bring the gross stiffness into agreement with experiment imply that the intertrabecular soft tissues are of no consequence to the mechanical properties and that the compact bone of which trabeculae are made is probably not as stiff as cortical bone.     

Simple constitutive model for a cortical bone

This short study presents a simple, one-dimensional constitutive model for the cortical bone with haversian structure. The model is developed within the general framework of the continuum damage theory. The kinetic equation is derived (rather than assumed a priori) through consideration of the irreversible changes of the mesostructure. As a consequence the analytical results closely approximate experimental measurements even though the theory does not introduce a single experimentally unidentifiable material parameter.     

Modeling modulus reduction in bovine trabecular bone damaged in compression.

Loading bone beyond its yield point creates microdamage, leading to reduction in stiffness. Previously, we related microdamage accumulation to changes in mechanical properties. Here, we develop a model that predicts stiffness loss based on the presence of microdamage. Modeling is done at three levels: (1) a single trabecula, (2) a cellular solid consisting of intact, damaged, and fractured trabeculae, and (3) a specimen with a localized damage band. Predictions of a reduced modulus agree well with experimental measured modulus reductions of post-yield compression of bovine trabecular bone. The predicted reduced modulus is relatively insensitive to changes in the input parameters.     

A 3D damage model for trabecular bone based on fabric tensors

Motivated by the role of damage in normal and pathological conditions of trabecular bone, a novel 3D constitutive law was developed that describes anisotropic elasticity and the rate-independent degradation in mechanical properties resulting from the growth of cracks or voids in the trabecular tissue. The theoretical model was formulated within the framework of continuum damage mechanics and based on two fabric tensors characterizing the local trabecular morphology. Experimental validation of the model was achieved by uniaxial and torsional testing of waisted bovine trabecular bone specimens. Strong correlations were found between cumulated permanent strain, reduction in elastic moduli and nonlinear postyield stress which support the hypothesis that these variables reflect the same underlying damage process.     

A novel mathematical model of bone remodelling cycles for trabecular bone at the cellular level

After an initial phase of growth and development, bone undergoes a continuous cycle of repair, renewal and optimisation by a process called remodelling. This paper describes a novel mathematical model of the trabecular bone remodelling cycle. It is essentially formulated to simulate a remodelling event at a fixed position in the bone, integrating bone removal by osteoclasts and formation by osteoblasts. The model is developed to construct the variation in bone thickness at a particular point during the remodelling event, derived from standard bone histomorphometric analyses. The novelties of the approach are the adoption of a predator-prey model to describe the dynamic interaction between osteoclasts and osteoblasts, using a genetic algorithm-based solution; quantitative reconstruction of the bone remodelling cycle; and the introduction of a feedback mechanism in the bone formation activity to co-regulate bone thickness. The application of the model is first demonstrated by using experimental data recorded for normal (healthy) bone remodelling to predict the temporal variation in the number of osteoblasts and osteoclasts. The simulated histomorphometric data and remodelling cycle characteristics compare well with the specified input data. Sensitivity studies then reveal how variations in the model's parameters affect its output; it is hoped that these parameters can be linked to specific biochemical factors in the future. Two sample pathological conditions, hypothyroidism and primary hyperparathyroidism, are examined to demonstrate how the model could be applied more broadly, and, for the first time, the osteoblast and osteoclast populations are predicted for these conditions. Further data are required to fully validate the model's predictive capacity, but this work shows it has potential, especially in the modelling of pathological conditions and the optimisation of the treatment of those conditions.     

An adaptation model for trabecular bone at different mechanical levels

Background  

Bone has the ability to adapt to mechanical usage or other biophysical stimuli in terms of its mass and architecture, indicating that a certain mechanism exists for monitoring mechanical usage and controlling the bone's adaptation behaviors. There are four zones describing different bone adaptation behaviors: the disuse, adaptation, overload, and pathologic overload zones. In different zones, the changes of bone mass, as calculated by the difference between the amount of bone formed and what is resorbed, should be different.     

Finite element analysis of a three-dimensional open-celled model for trabecular bone

Based on a regular array of cubic unit cells, each containing a body-centered spherical void, we created an idealized three-dimensional model for both subchondral trabecular bone and a class of porous foams. By considering only face-to-face stacking of unit cells, the inherent symmetry was such that, except at the surface, the displacements and stresses within any one unit cell were representative of the entire porous structure. Using prescribed displacements the model was loaded in both uniaxial compressive strain and uniaxial shear strain. Based on the response to these loads, we found the tensor of elastic constants for an equivalent homogeneous elastic solid with cubic symmetry. We then compared the predicted modulus with our experimental values for bovine trabecular bone and literature values for an open-celled latex rubber foam.     

Quantification of a rat tail vertebra model for trabecular bone adaptation studies

A feedback controlled loading apparatus for the rat tail vertebra was developed to deliver precise mechanical loads to the eighth caudal vertebra (C8) via pins inserted into adjacent vertebrae. Cortical bone strains were recorded using strain gages while subjecting the C8 in four cadaveric rats to mechanical loads ranging from 25 to 100 N at 1 Hz with a sinusoidal waveform. Finite element (FE) models, based on micro computed tomography, were constructed for all four C8 for calculations of cortical and trabecular bone tissue strains. The cortical bone strains predicted by FE models agreed with strain gage measurements, thus validating the FE models. The average measured cortical bone strain during 25-100 N loading was between 298 +/- 105 and 1210 +/- 297 microstrain (muepsilon). The models predicted average trabecular bone tissue strains ranging between 135 +/- 35 and 538 +/- 138 mu epsilon in the proximal region, 77 +/- 23-307 +/- 91 muepsilon in the central region, and 155 +/- 36-621 +/- 143 muepsilon in the distal region for 25-100 N loading range. Although these average strains were compressive, it is also interesting that the trabecular bone tissue strain can range from compressive to tensile strains (-1994 to 380 mu epsilon for a 100 N load). With this novel approach that combines an animal model with computational techniques, it could be possible to establish a quantitative relationship between the microscopic stress/strain environment in trabecular bone tissue, and the biosynthetic response and gene expression of bone cells, thereby study bone adaptation.     

A constitutive formulation of vascular tissue mechanics including viscoelasticity and softening behaviour

Nearly all soft tissues, among which the vascular tissue is included, present a certain degree of viscoelastic response. This behaviour may be attributed in part to fluid transport within the solid matrix, and to the friction between its fluid and solid constituents. After being preconditioned, the tissue displays highly repetitive behaviour, so that it can be considered pseudo-elastic, that is, elastic but behaving differently in loading and unloading. Because of this reason, very few constitutive laws accounting for the viscoelastic behaviour of the tissue have been developed. Nevertheless, the consideration of this inelastic effect is of crucial importance in surgeries—like vascular angioplasty—where the mentioned preconditioning cannot be considered since non-physiological deformation is applied on the vessel which, in addition, can cause damage to the tissue. A new constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these two inelastic phenomena is presented here and used to fit experimental stress–stretch curves from simple tension loading–unloading tests and relaxation test on porcine and ovine vascular samples.     

A theoretical framework for strain-related trabecular bone maintenance and adaptation

It is assumed that density and morphology of trabecular bone is partially controlled by mechanical forces. How these effects are expressed in the local metabolic functions of osteoclast resorption and osteoblast formation is not known. In order to investigate possible mechano-biological pathways for these mechanisms we have proposed a mathematical theory (Nature 405 (2000) 704). This theory is based on hypothetical osteocyte stimulation of osteoblast bone formation, as an effect of elevated strain in the bone matrix, and a role for microcracks and disuse in promoting osteoclast resorption. Applied in a 2-D Finite Element Analysis model, the theory explained the formation of trabecular patterns. In this article we present a 3-D FEA model based on the same theory and investigated its potential morphological predictability of metabolic reactions to mechanical loads. The computations simulated the development of trabecular morphological details during growth, relative to measurements in growing pigs, reasonably realistic. They confirmed that the proposed mechanisms also inherently lead to optimal stress transfer. Alternative loading directions produced new trabecular orientations. Reduction of load reduced trabecular thickness, connectivity and mass in the simulation, as is seen in disuse osteoporosis. Simulating the effects of estrogen deficiency through increased osteoclast resorption frequencies produced osteoporotic morphologies as well, as seen in post-menopausal osteoporosis. We conclude that the theory provides a suitable computational framework to investigate hypothetical relationships between bone loading and metabolic expressions.     

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.     

A linearized and incompressible constitutive model for arteries

In many biomechanical studies, blood vessels can be modeled as pseudoelastic orthotropic materials that are incompressible (volume-preserving) under physiological loading. To use a minimum number of elastic constants to describe the constitutive behavior of arteries, we adopt a generalized Hooke's law for the co-rotational Cauchy stress and a recently proposed logarithmic-exponential strain. This strain tensor absorbs the material nonlinearity and its trace is zero for volume-preserving deformations. Thus, the relationships between model parameters due to the incompressibility constraint are easy to analyze and interpret. In particular, the number of independent elastic constants reduces from ten to seven in the orthotropic model. As an illustratory study, we fit this model to measured data of porcine coronary arteries in inflation-stretch tests. Four parameters, n (material nonlinearity), Young's moduli E? (circumferential), E? (axial), and E? (radial) are necessary to fit the data. The advantages and limitations of this model are discussed.     

A three-dimensional elastic plastic damage constitutive law for bone tissue

Motivated by mechanical analysis of bones and bone-implant systems, a 3D constitutive law describing the macroscopic mechanical behaviour of both cortical and trabecular bone in cyclic (not fatigue) overloads is developed. The proposed model which mathematical formulation is established within the framework of generalized standard materials accounts for three distinct material evolution modes where elastic, plastic and damage aspects are closely related. The anisotropic elasticity of bone is described by a morphology-based model and distinct damage behaviour in tension and compression by a halfspacewise generalized Hill criterion. The plastic criterion is based on the intact elastic compliance tensor. The algorithm applies three distinct projections based on the relationship between the internal variables and criteria. Their respective consistent tangent operators are presented. Numerical resolutions of several boundary value problems and a biomechanical application are presented to illustrate the potential of the constitutive model and demonstrate the expected quadratic convergence of the algorithm.     

Artificial composite bone as a model of human trabecular bone: the implant-bone interface

The use of artificial bones in implant testing has become popular due to their low variability and ready availability. However, friction coefficients, which are critical to load transfer in uncemented implants, have rarely been compared between human and artificial bone, particularly for wet and dry conditions. In this study, the static and dynamic friction coefficients for four commercially used titanium surfaces (polished, Al(2)O(3) blasted, plasma sprayed, beaded) acting on the trabecular component of artificial bones (Sawbones) were compared to those for human trabecular bone. Artificial bones were tested in dry and wet conditions and normal interface stress was varied (0.25, 0.5, 1.0MPa). Friction coefficients were mostly lower for artificial bones than real bone. In particular, static friction coefficients for the dry polished surface were 20% of those for real bone and 42-61% for the dry beaded surface, with statistical significance (alpha<0.05). Less marked differences were observed for dynamic friction coefficients. Significant but non-systematic effects of normal stress or wet/dry condition on friction coefficients were observed within each surface type. These results indicate that the use of artificial bone models for pre-clinical implant testing that rely on interface load transfer with trabecular bone for mechanical integrity can be particularly sensitive to surface finish and lubrication conditions.