The elastic anisotropy of bone

In modeling the anisotropic properties of hydroxyapatite (HAp), Katz found that two kinds of phenomenological relationships held among the elastic stiffness coefficients. Firstly, there are three linear combinations--(c11 + c22 + c33), (c44 + c55 + c66), (c12 + c13 + c23)--which arise naturally when computing the isotropic averages of anisotropic crystal systems over all possible spatial orientations. Secondly, the degree of elastic anisotropy in such crystal systems is characterized by two specific factors: (a) the ratio of the linear compressibility along the unique axis to that perpendicular to it, (c11 + c12 - 2c23)(c33 - c13); and (b) the ratio of the two shear moduli, c44/c66. There have been a number of experiments in recent years which have used either mechanical methods or ultrasonic techniques to measure the anisotropic elastic properties of bovine and human cortical bone. Analyses of data from these experiments show that the above relationships also play a significant role in characterizing the elastic anisotropy in bone.     

Thermodynamic restrictions on the elastic constants of bone

The thermodynamic restrictions on the elastic coefficients of linear orthotropic elasticity and linear transversely isotropy elasticity are recorded and it is shown that previously reported data for the elastic orthotropic constants of bone satisfy these thermodynamic restrictions.     

Composite bounds on the elastic modulus of bone

Advances in diagnosis and treatment of some bone disorders can be made by understanding the linkage between mineral content and mechanical function. Bone is approximately half by volume a hydrated protein network, and the remainder is a biomineral analogue of hydroxyapatite. In the current work, paired measurements of mechanical properties, using nanoindentation, and of bone mineral volume fraction, computed from quantitative back-scattered electron imaging, were made on six different types of normal and outlier bone samples. Local elastic modulus was plotted against mineral fraction and compared with predictions of engineering bounds for a two-phase composite material. Experimental data spanning the composite bounds showed no one-to-one relationship between mechanical stiffness and bone composition, excluding the possibility of any single, simple composites model for bone at nanometer length-scales.     

The elastic and ultimate properties of compact bone tissue

The use of a tranversely isotropic model is tested for the elastic behavior of bovine and human bone and the five independent constants of this model are determined. The accuracy of the model is tested for eight cases by comparing the off-axis modulus predicted by a rotation of stiffness matrix with an experimentally determined off-axis modulus.

Ultimate properties are presented for bovine and human bone for tension, compression, and torsional loads. A Hankinson type failure criterion is proposed for off-axis ultimate stress and this predicted value compared with experimental values for nine cases.     

Predicting the structural integrity of bone defects repaired using bone graft materials

Bone defects create stress concentrations which can cause fracture under impact or cyclic loading. Defects are often repaired by filling them with a bone graft material; this will reduce the stress concentration, but not completely, because these materials have lower stiffness than bone. The fracture risk decreases over time as the graft material is replaced by living bone. Many new bone graft materials are being developed, using tissue engineering and other techniques, but currently there is no rational way to compare these materials and predict their effectiveness in repairing a given defect. This paper describes, for the first time, a theoretical model which can be used to predict failure by brittle fracture or fatigue, initiating at the defect. Preliminary results are presented, concentrating on the prediction of stress fracture during the crucial post-operative period. It is shown that the likelihood of fracture is strongly influenced by the shape of the defect as well as its size, and also by the level of post-operative exercise. The most important finding is that bone graft materials can be successful in preventing fracture even when their mechanical properties are greatly inferior to those of bone. Future uses of this technique include pre-clinical assessment of bone replacement materials and pre-operative planning in orthopaedic surgery.     

Predicting the structural integrity of bone defects repaired using bone graft materials

Bone defects create stress concentrations which can cause fracture under impact or cyclic loading. Defects are often repaired by filling them with a bone graft material; this will reduce the stress concentration, but not completely, because these materials have lower stiffness than bone. The fracture risk decreases over time as the graft material is replaced by living bone. Many new bone graft materials are being developed, using tissue engineering and other techniques, but currently there is no rational way to compare these materials and predict their effectiveness in repairing a given defect. This paper describes, for the first time, a theoretical model which can be used to predict failure by brittle fracture or fatigue, initiating at the defect. Preliminary results are presented, concentrating on the prediction of stress fracture during the crucial post-operative period. It is shown that the likelihood of fracture is strongly influenced by the shape of the defect as well as its size, and also by the level of post-operative exercise. The most important finding is that bone graft materials can be successful in preventing fracture even when their mechanical properties are greatly inferior to those of bone. Future uses of this technique include pre-clinical assessment of bone replacement materials and pre-operative planning in orthopaedic surgery.     

Effects of damage on the orthotropic material symmetry of bovine tibial trabecular bone

The macroscopic mechanical properties of trabecular bone can be predicted by its architecture using theoretical relationships between the elastic and architectural properties. Microdamage caused by overloading or fatigue decreases the apparent elastic moduli of trabecular bone requiring these relationships to be modified to predict the damaged elastic properties. In the case of isotropic damage, the apparent level elastic properties could be determined by multiplying all of the elastic constants by a single scalar factor. If the damage is anisotropic, the elastic constants may change by differing factors and the material coordinate system could become misaligned with the fabric coordinate system. High-resolution finite element models were used to simulate damage overloading on seven trabecular bone specimens subjected to pure shear strain in two planes. Comparison of the apparent elastic moduli of the specimens before and after damage showed that the reduction of the elastic moduli was anisotropic. This suggests that the microdamage within the specimens was inhomogeneous. However, after damage the specimens exhibited nearly orthotropic material symmetry as they did before damage. Changes in the orientation of the orthotropic material coordinate system were also small and occurred primarily in the transverse plane. Thus, while damage in trabecular bone is anisotropic, the material coordinate system remains aligned with the fabric tensor.     

Deficiencies in numerical models of anisotropic nonlinearly elastic materials

Incompressible nonlinearly hyperelastic materials are rarely simulated in finite element numerical experiments as being perfectly incompressible because of the numerical difficulties associated with globally satisfying this constraint. Most commercial finite element packages therefore assume that the material is slightly compressible. It is then further assumed that the corresponding strain-energy function can be decomposed additively into volumetric and deviatoric parts. We show that this decomposition is not physically realistic, especially for anisotropic materials, which are of particular interest for simulating the mechanical response of biological soft tissue. The most striking illustration of the shortcoming is that with this decomposition, an anisotropic cube under hydrostatic tension deforms into another cube instead of a hexahedron with non-parallel faces. Furthermore, commercial numerical codes require the specification of a ‘compressibility parameter’ (or ‘penalty factor’), which arises naturally from the flawed additive decomposition of the strain-energy function. This parameter is often linked to a ‘bulk modulus’, although this notion makes no sense for anisotropic solids; we show that it is essentially an arbitrary parameter and that infinitesimal changes to it result in significant changes in the predicted stress response. This is illustrated with numerical simulations for biaxial tension experiments of arteries, where the magnitude of the stress response is found to change by several orders of magnitude when infinitesimal changes in ‘Poisson’s ratio’ close to the perfect incompressibility limit of 1/2 are made.     

The effects of side-artifacts on the elastic modulus of trabecular bone

Determining accurate density-mechanical property relationships for trabecular bone is critical for correct characterization of this important structure-function relation. When testing any excised specimen of trabecular bone, an unavoidable experimental artifact originates from the sides of the specimen where peripheral trabeculae lose their vertical load-bearing capacity due to interruption of connectivity, a phenomenon denoted here as the 'side-artifact'. We sought in this study to quantify the magnitude of such side-artifact errors in modulus measurement and to do so as a function of the trabecular architecture and specimen size. Using parametric computational analysis of high-resolution micro-CT-based finite-element models of cores of elderly human vertebral trabecular bone, a specimen-specific correction factor for the side-artifact was quantified as the ratio of the side-artifact-free apparent modulus (Etrue) to the apparent modulus that would be measured in a typical experiment (Emeasured). We found that the width over which the peripheral trabeculae were mostly unloaded was between 0.19 and 0.58 mm. The side-artifact led to an underestimation error in Etrue of over 50% in some specimens, having a mean (+/-SD) of 27+/-11%. There was a trend for the correction factor to linearly increase as volume fraction decreased (p=0.001) and as mean trabecular separation increased (p<0.001). Further analysis indicated that the error increased substantially as specimen size decreased. Two methods used for correcting for the side-artifact were both successful in bringing Emeasured into statistical agreement with Etrue. These findings have important implications for the interpretation of almost all literature data on trabecular bone mechanical properties since they indicate that such properties need to be adjusted to eliminate the substantial effects of side-artifacts in order to provide more accurate estimates of in situ behavior.     

Errors induced by off-axis measurement of the elastic properties of bone

Misalignment between the axes of measurement and the material symmetry axes of bone causes error in anisotropic elastic property measurements. Measurements of Poisson's ratio were strongly affected by misalignment errors. The mean errors in the measured Young's moduli were 9.5 and 1.3 percent for cancellous and cortical bone, respectively, at a misalignment angle of 10 degrees. Mean errors of 1.1 and 5.0 percent in the measured shear moduli for cancellous and cortical bone, respectively, were found at a misalignment angle of 10 degrees. Although, cancellous bone tissue was assumed to have orthotropic elastic symmetry, the possibility of the greater symmetry of transverse isotropy was investigated. When the nine orthotropic elastic constants were forced to approximate the five transverse isotropic elastic constants, errors of over 60 percent were introduced. Therefore, it was concluded that cancellous bone is truly orthotropic and not transversely isotropic. A similar but less strong result for cortical bone tissue was obtained.     

The fabric dependence of the orthotropic elastic constants of cancellous bone

It has been proposed that the orthotropic elastic constants of cancellous bone depend upon a tensorial measure of anisotropy called fabric as well as the tissue's structural density. Cowin (1985, Mechanics Mater, 4, 137-147; 1986, J. biomech. Engng 108, 83-88) developed explicit relationships for the elastic constant, structural density and fabric relationship. In this study the orthotropic elastic moduli, structural density, and fabric components were measured for 11 cancellous bone specimens from five bovine femora and for 75 specimens from three human proximal tibiae and fitted to these relationships using a least squares analysis. The relationships explained between 72 and 94% of the variance in the elastic constants. The relationships between the elastic constants and squared or cubed power functions of structural density had better predictive value over the entire distribution of the data than did expressions with linear functions of structural density.     

Novel materials for bone and cartilage regeneration

Materials that enhance bone and cartilage regeneration promise to be valuable in both research and clinical applications. Both natural and synthetic polymers can be used to create scaffolds that support cells and incorporate cues which guide tissue repair. Recently, electrospinning, peptide self-assembly and biomineralisation have been employed to fabricate nanostructured scaffolds that better mimic the complex extracellular environment found within tissues, in vivo. The incorporation of peptide motifs recognised by cell receptors and the use of recombinant DNA technology have enabled the creation of scaffolds with new levels of biofunctionality. Advances in materials design will enhance our ability to create highly tailored cellular environments for bone and cartilage regeneration.     

A two-parameter model of the effective elastic tensor for cortical bone

Multiscale models of cortical bone elasticity require a large number of parameters to describe the organization and composition of the tissue. We hypothesize that the macro-scale anisotropic elastic properties of different bones can be modeled retaining only two variable parameters, and setting the others to universal values identical for all bones. Cortical bone is regarded as a two-phase composite material: a dense mineralized matrix (ultrastructure) and a soft phase (pores). The ultrastructure is assumed to be a homogeneous and transversely isotropic tissue whose elastic properties in different directions are mutually dependent and can be scaled with a single parameter driving the overall rigidity. This parameter is taken to be the volume fraction of mineral f(ha). The pore network is modeled as an ensemble of water-filled cylinders and described only by the porosity p. The effective macroscopic elasticity tensor C(ij)(f(ha),p) is calculated with a multiscale micromechanics approach starting from existing models. The modeled stiffness coefficients compare favorably to four literature datasets which were chosen because they provide the full stiffness tensors of groups of human samples. Since the physical counterparts of f(ha) and p were unknown for the datasets, their values which allow the best fit of experimental tensors by the modeled ones were determined by optimization. Optimum values of f(ha) and p are found to be unique and realistic. These results suggest that a two-parameter model may be sufficient to model the elasticity of different samples of human femora and tibiae. Such a model would in particular be useful in large-scale parametric studies of bone mechanical response.     

Physical characteristics of bone composite materials

To study the effects of varying mineral content and various trace elements in bone composities on its electrical behavior and possible use in design of transducers, various physical, dielectric, piezoelectric, and electromechanical parameters have been measured. For electrical characterization of various such composites in the high-frequency region (1–108 MHz), variation of impedance (Z), phase angle (tan ), and relative output voltage with frequency has been examined. Furtherfore, the Curie temperature has been determined and the temperature variation of capacitance and loss factor (tan ) studied (24–225°C). Two types of bone composites were prepared and studied. First, powdered collagen and apatite obtained from full bone were mixed intimately in various proportions by weight to prepare eleven bone compositions. Second, such bone materials were made to contain 5–10% various doping foreign additives (A1Br₃, Na₂CO₃, SrCO₃, LiCO₃, Sb₂O₃, ZnO, Nb₂O₅, piezoelectric ceramic (PZT), and Pb(NO₃)₂. It has been observed that a bone composition of 50% collagen + 50% apatite has possible piezoelectric application and other compositions [85% collagen + 15% apatite, 90% collagen + 10% ZnO, and 90% bone + 10% Ba(OH)₂] have a sharp rise in capacitance near the Curie temperature. The Curie temperature is generally shifted towards higher values by additives. It is expected that such results will be relevant in characterizing bone behavior.     

Identification of passive elastic joint moments in the lower extremities.

Musculotendon actuators produce active and passive moments at the joints they span. Due to the existence of bi-articular muscles, the passive elastic joint moments are influenced by the angular positions of adjacent joints. To obtain quantitative information about this passive elastic coupling between lower limb joints, we examined the passive elastic joint properties of the hip, knee, and ankle joint of ten healthy subjects. Passive elastic joint moments were found to considerably depend on the adjacent joint angles. We present a simple mathematical model that describes these properties on the basis of a double-exponential expression. The model can be implemented in biomechanical models of the lower extremities, which are generally used for the simulation of multi-joint movements such as standing-up, walking, running, or jumping.     

Instrumental neutron activation analysis of rib bone samples and of bone reference materials

The instrumental neutron activation analysis method was used for the determination of trace elements in rib bone samples taken from autopsies of accident victims. The elements Br, Ca, Cl, Cr, Fe, Mg, Mn, Na, P, Sr, Rb, and Zn were determined in cortical tissues by using short and long irradiations with thermal neutron flux of the IEA-R1m nuclear reactor. The reference materials NIST SRM 1400 Bone Ash and NIST SRM 1486 Bone Meal were also analyzed in order to evaluate the precision and the accuracy of the results. It was verified that lyophilization is the most convenient process for drying bone samples because it does not cause any element losses. Comparisons were made between the results obtained for rib samples and the literature values as well as between the results obtained for different ribs from a single individual and for bones from different individuals.