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PubMed1984

The relationship between elastic properties and microstructure of bovine cortical bone.

The relationship between the structure of bone on the microscopic level and its mechanical properties was investigated by comparing bovine cortical bone of two microstructural types. Measurements of elastic properties were made in three orthogonal directions on specimens of plexiform and Haversian bone, taken from the same level of the same bone, using extensional wave ultrasonic techniques. Differences in Young's modulus of elasticity measured in the three directions correspond to observed structural differences; plexiform bone is anisotropic, while Haversian bone appears to be transversely isotropic. The pattern of distribution of the two structural types within different aspects of the same bovine femur suggests that the level of osteonal remodeling is related to the pattern of mechanical stress, especially due to muscular activity, in the bone. This may occur because Haversian remodeling is stimulated by microcracks in regions of high stress and/or repeated stress cycles, thus replacing fatigue-damaged bone and preventing failure.

Animals↗
PubMed1985

The effect of Haversian remodeling on the tensile properties of human cortical bone.

The purpose of this study was to determine the effect of Haversian remodeling on the tensile properties of human cortical bone by testing specimens containing, as far a possible, a single type of bone tissue. Fifty-one specimens were prepared from sixteen fresh tibias, removed at autopsy. Age range was 19-35. Regions were selected so that the specimens would consist almost exclusively of either primary bone or Haversian bone. The ultimate tensile strength, ultimate strain and Young's modulus of elasticity were determined at a loading rate of 0.05 mm s-1. The primary bone specimens were found to have a significantly higher ultimate tensile strength and modulus of elasticity than those formed of Haversian bone.

Adult↗
PubMed1987

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.

Biomechanical Phenomena↗
PubMed1989

A discrete model for streaming potentials in a single osteon.

A mathematical model for streaming potentials in an osteon is proposed, taking into account the microstresses in the vicinity of the Haversian Canal. With the help of the finite element method, a boundary problem for the fluid pressure amplitude in the osteon is investigated when the bone sample is subjected to harmonic loading. A numerical analysis of the intra-osteonal potential is performed. It is found that there exists an azimuthal asymmetry which increases with the enlargement of the Haversian Canal. The results of the numerical modeling of the intra-osteonal potential are in accordance with the available experimental data.

Action Potentials↗
PubMed1989

Bone creep-fatigue damage accumulation.

Creep and fatigue tests were performed on human femoral cortical bone and the results were compared to a cumulative damage model for bone fracture. Fatigue tests in tension, compression, and reversed loading with a tensile mean stress were conducted at 2 Hz and 0.02 Hz. Load frequency had a strong influence on the number of cycles to failure but did not influence the total time to failure. Bone displayed poor creep-fracture properties in both tension and compression. The fracture surfaces of the tensile creep specimens are distinctly different than those of the compressive specimens. The results suggest that tensile cyclic loading creates primarily time-dependent damage and compressive cyclic loading creates primarily cycle-dependent damage. However, data for load histories involving both tensile and compressive loading indicate lower time to failure than predicted by a simple summation of time-dependent and cycle-dependent damage.

Adult↗
PubMed1992

Effect of groove on bone fracture toughness.

When testing for the effects of bone orientation on mode I fracture toughness, compact tension specimens are grooved with a V-notch to provide a crack guide. The effect of grooving on the expressions for the critical stress intensity factor (Kc) and the critical strain energy release rate (Gc) for mode I fracture toughness was investigated. Experiments were performed using grooved and ungrooved bovine compact tension specimens. The results indicate that the standard expression used to determine Kc for a compact tension specimen requires modification. The thickness (B) must be modified to account for the thickness between the grooves (Bn). The thickness used in the standard expression is replaced by an effective thickness written as (BBn)0.5. It was also found that the thickness between the grooves should be used in the standard formula for Gc.

Animals↗
PubMed1992

Collagen fiber orientation and geometry effects on the mechanical properties of secondary osteons.

The effects of collagen fiber orientation and osteon geometry on the mechanical properties of secondary osteons under axial compression/tension and combined loadings (compression, bending and torsion) were investigated using a composite-beam finite-element model. Three cross-sectional shapes of secondary osteons were studied to show the effect of geometry. The results of stiffness are presented using the tension and compression properties for each lamella. The model shows that the mechanical properties of osteons are enhanced in bending and torsion when collagen fibers are oriented within 30 degrees of the loading axis. Osteons with alternating lamellar orientation are not well adapted to resist torsional moments, but alternate collagen fiber orientation has virtually no effect on the bending stiffness of osteons. Fiber orientation affects the mechanical properties less significantly when osteons are non-circular. Collagen fiber orientation and osteon geometry interact to determine the mechanical behavior of the osteon, and may act in a compensatory manner in the adaptive process.

Adult↗
PubMed1992

Acoustic microscope study of the elastic properties of fluorapatite and hydroxyapatite, tooth enamel and bone.

Measurements of Rayleigh velocity and attenuation were taken in single mineral crystals of hydroxyapatite and fluorapatite at angular intervals relative to their c axes, using an acoustic microscope. These results are compared with the values that were calculated using the elastic constants of apatite from Yoon and Newnham [(1969) Am. Miner. 54, 1193-1197.] and Katz and Ukraincik [(1971) J. Biomechanics 4, 221-227.]. The slowness curves of various wave modes are plotted and discussed in relation to cross-coupling effects that were found to cause instability in measurements of attenuation for the c axes direction. Velocity measurements were taken in specimens of tooth enamel and bone. Here comparisons are made with the values that were calculated by modelling the 'z' scan response of the microscope, using published data for the elastic and acoustic properties. Comparisons are also made with the measurements on single crystals, since apatite is a major component of enamel and bone.

Animals↗
PubMed1992

On the relationship between the microstructure of bone and its mechanical stiffness.

A recent study of bone structure shows that the plate-shaped carbonate apatite crystals in individual lamellae are arranged in layers across the lamellae, and that the orientation of these layers are different in alternate lamellae. Based on these findings, a new micromechanical model for the Young's modulus of bone is proposed, which accounts for the anisotropy and geometrical characteristics of the material. The model incorporates the platelet-like geometry of the basic reinforcing unit, the presence of alternating thin and thick lamellae, and the orientations of the crystal platelets in the lamellae. The thin and thick lamellae are modeled as orthotropic composite layers made up of thin rectangular apatite platelets within a collagen matrix, and classical orthotropic elasticity theory is used to calculate the Young's modulus of the lamellae. Bone is viewed as an assembly of such orthotropic lamellae bent into cylindrical structures, and having a constant, alternating angle between successive lamellae. The micromechanical model employs a modified rule-of-mixtures to account for the two types of lamellae. The model provides a curve similar to the published experimental data on the angular dependence of Young's modulus, including a local maximum at an angle between 0 and 90 degrees. A rigorous testing of the model awaits additional experimental data.

Animals↗
PubMedDate not supplied

Calculating the probability that microcracks initiate resorption spaces.

This paper explores the assumptions and limitations of the probability calculation that led to the conclusion by Burr et al. (1985) that microcracks initiate new remodeling events. It also corrects several minor errors in the calculation in the original manuscript. The results show that the probability that cracks and resorption spaces are associated depends heavily on a factor, F, that accounts for the possibility that some osteons that contain both a crack and a resorption space share a cement line with an adjacent osteon to which the crack more properly 'belongs.' F in turn depends on (1) the measurement criteria for cracks and resorption spaces, (2) the osteon population density in the bone, and (3) the mechanism by which cracks initiate remodeling. The theoretical maximum number of osteons that can contain both a crack and a resorption space (nmax) increases as the number of resorption spaces (r), the number of cracks (c), and F increase, but decreases as the osteon population density (d) increases. A larger nmax makes a direct association between cracks and resorption spaces more difficult to demonstrate experimentally.

Animals↗
PubMed1993

Compact bone: numerical simulation of mechanical characteristics.

One of the main difficulties encountered in the numerical simulation of the anisotropic elastic characteristics of compact bone is to account for the Haversian microstructure when determining the overall macroscopic behavior. Engineering analyses of such problems are usually based on 'homogenized approximations'. Compact bone is not exactly a composite material, but rather a heterogeneous medium which exhibits a multiscale composite structure. If the homogenized approximation is precise enough (and this is true for the mathematical theory of homogenization), it is then possible to simulate the macroscopic behavior from the microscopic mechanical characteristics. The present paper is devoted to such mathematical developments. Moreover, the 'inverse simulation' allows the computation of the microscopic stress fields in the haversian structure from the macroscopic stress fields, taking into account bone microstructure.

Biomechanical Phenomena↗
PubMed1993

Bone strain sensation via transmembrane potential changes in surface osteoblasts: loading rate and microstructural implications.

A model is developed in which osteoblasts can sense the strains applied to a small region of bone through electrical coupling between adjacent cells. The stress-generated potentials within bone are assumed to occur through streaming potentials, and the coupled network of osteocytes is assumed to act in a manner similar to the classical cable model for nerve cells. In a one-dimensional model, the linear poroelastic equations for motion of the fluid are solved analytically for sinusoidally varying imposed strains, and the streaming potentials are predicted from the fluid flow. The changes in the osteocyte and osteoblast transmembrane potential are given by an analytical solution to the governing equations, and the dependence of the transmembrane potential changes (TPC) on position, loading rate, manner of loading (compression versus bending), and on the degree of cellular coupling is discussed. The model correctly predicts the rate dependence of remodelling established by other investigators. The influence of the electrical parameters within the model indicate that further study of the cellular coupling in bone can yield important new information on bone remodelling.

Animals↗
PubMed1993

The effect of temperature, stress and microstructure on the creep of compact bovine bone.

Creep tests of 117 compact bovine bone specimens were conducted at three temperatures (25, 37, and 43 degrees C), with applied stresses between 71 and 115 MPa. Following testing, the amount of secondary haversian bone in the gage region of the specimens was estimated. The resulting steady-state creep rates (epsilon) were fit to an Arrhenius (e-Qc/RT) model (where Qc is the activation energy for the mechanism(s) controlling creep, R is the gas constant, and T is the absolute temperature) of the type used to describe the classic steady-state creep behavior of metals, ceramics, and metamorphic rocks. The empirical model developed was epsilon = 5.6 x 10(-9) e4.6F sigma 5.2 e-5330/T, where epsilon is the estimated mean steady-state creep rate, F is the volume fraction of secondary haversian bone, sigma is the applied stress, and T is the absolute temperature. There was a positive, significant association between the estimated mean steady-state creep rate and F, sigma, and T. Qc was determined to be 44.3 kJ mol-1, a reasonable value when compared to activation energies for creep in ceramics. It is hypothesized that permanent deformation during creep of compact bovine bone is primarily due to damage mechanisms associated with dislocations in the hydroxyapatite mineral lattice structure.

Animals↗
PubMed1995

Fracture toughness of human bone under tension.

The longitudinal fracture toughnesses of human cortical bone were compared to those of bovine cortical bone to test the hypothesis that although human osteonal bone is significantly weaker and more compliant than primary (plexiform) bone, it is not less tough than primary bone. The fracture toughness indices, critical strain energy release rate (Gc) and critical stress intensity factor (Kc), were determined for human Haversian bone and bovine bone under tension (Mode I) loading using the compact tension method. The effects of thickness, crack growth range and anisotropy on fracture indices for slow stable crack growth in cortical bone were determined. Plane strain assumptions required for application of linear elastic fracture mechanics (LEFM) to bone were investigated. Longitudinal oriented fracture toughness tests were used to assess the crack inhibiting effect of human bone microstructure on fracture resistance. Human bone Kc calculated from the stress concentration formula for 2 and 3 mm thick specimens equaled 4.32 and 4.05 MN m-3/2, respectively. Human bone Gc calculated from the compliance method equaled 827 N m-1 for 2 mm thick specimens and 595 N m-1 for 3 mm thick specimens. It was found that crack growth range, thickness and material assumptions affect fracture toughness. Kc calculated from Gc using an anisotropic relation provided the lowest estimate of Kc and equaled 3.31 MN m-3/2 for 2 mm thick specimens and 2.81 MN m-3/2 for 3 mm thick specimens. Both Kc and Gc were significantly reduced after being adjusted to ASTM standard thickness using ratios determined from bovine bone. The fracture toughness of bovine bone relative to human bone ranged from 1.08 to 1.66. This was compared to the longitudinal strength of bovine bone relative to the longitudinal strength of human bone which is approximately equal to 1.5. We found that even though human bone is significantly weaker than bovine bone, relative to its strength, the toughness of human and bovine bone are roughly similar, but the data were not sufficiently definitive to answer the question of which is tougher.

Aged↗
PubMed1995

Viscoelastic properties of bone as a function of water content.

Stress relaxation of bovine femur was investigated as a function of water content, phi. As found for bone and bone collagen [Sasaki et al. (1993) J. Biomech. 26, 1369-1376], all the relaxation curves measured were described by a linear combination of a Kohlrausch-Williams-Watts (KWW) function and a simple exponential decay (Debye) function: G(t)/Gi = A1 exp[- (t/tau 1)beta] + A2 exp(= t/tau 2), A1 + A2 = 1, 0 < or = beta < or = 1, where Gi is an initial value of the relaxation shear modulus G(t), A1 and A2 are portions of KWW and Debye relaxations, respectively, and tau 1 and tau 2 are relaxation times of respective relaxations. Shear modulus values in the relaxation described by the KWW function (KWW relaxation) depend remarkably on phi while those in Debye relaxation are almost constant for increasing phi. phi dependencies of A1, tau 1 and beta are explained by assuming that the elementary process for the KWW relaxation would be a rearranging process of local disorders in the collagen molecular array. The relaxation rate for the Debye relaxation (= 1/tau 2) decreases linearly with phi. This linear relation between tau 2-1 and phi was well described on the basis of the concept of non-elasticity of a solid by the nuclearion of microcracks at the area of stress concentration.

Algorithms↗
PubMed1995

Material and compositional properties of selectively demineralized cortical bone.

Timed immersion in buffered ethylenediamine-tetraacetic acid (EDTA) was used to selectively alter the mineral content at each level in the cortical bone structural hierarchy. The effects on the mechanical behavior were investigated using a combination of experimental techniques which provide collectively a wide range of resolution (5 microns to 3 mm). Optical microscopy and histological analysis demonstrated a heterogeneous structure consisting of a mineralized tissue core surrounded by a layer of demineralized tissue (collagen) whose thickness varied depending on the immersion time. The mechanical behaviors of treated samples with (intact) and without (core) the surrounding demineralized layer were evaluated using three-point flexure. Overall, the intact specimens became significantly less brittle with increased immersion time in buffered-EDTA. For the core specimens, there was a systematic decrease in the elastic flexural properties (E, sigma e, epsilon e). The site-specific properties of the specimens were determined using microhardness testing, scanning acoustic microscopy, and wavelength dispersive analysis. The mineralization and site-specific properties of the mineralized cores were not significantly affected by buffered-EDTA immersion; however, histomorphometric analysis showed a decrease in the mineralized volume fraction via widening of the pre-existing vascular channels. The experimental hierarchy was effective in discerning site-specific property changes and the localized heterogeneities resulting from the buffered-EDTA treatment. Based on the results of this study, buffered-EDTA treatment can be used to facilitate the determination of material and physical properties of intact and demineralized tissues within a single cortical bone sample.

Animals↗
PubMed1994

Prediction of bone adaptation using damage accumulation.

The adaptation of bones to a change in function has been recognised for many centuries, but only recently have mathematical laws been proposed to describe it. One proposed mathematical law is based on the hypothesis that, after a change in load, the strain in the bone microstructure is regulated to a homeostatic equilibrium. Strain-adaptive remodelling has been used successfully to simulate bone adaptation around orthopaedic implants but the predictive capabilities are constrained because many empirical constants are required in the remodelling law (a reference stimulus, a zone of equilibrium stresses or 'lazy zone' and a parameter transducing macroscopic stresses to a tissue level stimulus). An alternative approach has been proposed. It is that bone adapts to attain an optimal strength by regulating the damage generated in its microstructural elements. The question is raised whether or not a mathematical law to predict the time course of bone adaptation can be derived for damage-adaptive remodelling in a similar way to the mathematical laws based on a strain stimulus. In the present study, the hypotheses required to develop damage-adaptive remodelling laws are proposed and a remodelling law to predict the time course of bone adaptation is derived. It is shown that this is an integral remodelling law which accounts naturally for the stress history to which the tissue has been exposed since formation. A simulation of the adaptive response of a bone diaphysis under a change in torsional load shows that the law gives physically reasonable predictions. The initial remodelling prediction is similar to strain-adaptive remodelling. However, in the later stages of remodelling, the predictions differ from strain-adaptive remodelling in that direct convergence to a homeostatic strain is not predicted. Instead, undershoot (in the case of a reduction in load) and overshoot (in the case of an increase in load) are predicted.

Adaptation, Physiological↗
PubMed1994

The torsional properties of single selected osteons.

The aim of the present investigation is to determine the mechanical behavior of single selected osteons loaded by torsion along their axis. Two osteon types were chosen: (a) fully calcified 'longitudinal' osteons, whose fibers (and, consequently, crystallites) have a marked longitudinal spiral course in successive lamellae, (b) fully calcified 'alternate' osteons, whose fibers (and, consequently, crystallites) have a marked longitudinal spiral course in one lamella, and an apparently transverse or circular course in the next. The osteon samples, which were cylindrically shaped, and whose extremities fitted into two rectangular lugs, were prepared on a microturning lathe. The principle of the specially manufactured torsional device was that of a pendulum set oscillating around a very sharp wedge so as to reduce friction to negligible levels. The results indicate that longitudinal osteons are those most resistant to torsional loading. The low resistance to torsional loading of the lamellae whose fibers apparently have an almost transversal course is discussed with reference to recent views on the microstructure of this type of lamella.

Adult↗