Elastic moduli, yield stress, and ultimate stress of cancellous bone in the canine proximal femur

Elastic moduli, yield stress and ultimate compressive stress were determined for cancellous bone from the femoral head and neck regions of the canine femur. Unconfined compression tests were performed on 5 mm cubic samples which were cut from two femurs. Elastic moduli were measured in three orthogonal directions, and the yield stress and ultimate stress were measured along the proximal-distal axis. The results from this investigation support previous assumptions that the mechanical behavior of canine cancellous bone is qualitatively similar to human cancellous bone. The canine cancellous bone was observed to be anisotropic in elastic modulus. For two thirds of the cubic specimens tested, the elastic modulus was largest in the load-bearing, proximal-distal direction. A linear relationship between yield stress and elastic modulus was observed for canine bone, as is typical of human bone. A similar linear relationship between ultimate stress and elastic modulus was observed. Thus, for canine bone as well as for human bone, failure appears to be governed by a strain level which is position independent. The yield strain of 0.0259 and ultimate strain of 0.0288 for canine bone were both less than the yield strain of 0.0395 reported for human bone.     

Uniform stress state in bone structure with residual stress.

Residual stress and strain in living tissues have been investigated from the viewpoint of mechanical optimality maintained by adaptive remodeling. In this study, the residual stresses in the cortical-cancellous bone complex of bovine coccygeal vertebrae were examined. Biaxial strain gages were bonded onto the cortical surface, so that the gage axes were aligned in the cephalocaudal and circumferential directions. Strains induced by removal of the end-plate and the cancellous bone were recorded sequentially. The results revealed the existence of compressive residual stress in the cortical bone and tensile residual stress in the cancellous bone in both the cephalocaudal and the circumferential direction. The observed strains were examined on the basis of the uniform stress hypothesis using a three-bar model for the cephalocaudal direction and a three-layered cylinder model for the circumferential direction. In this model study, the magnitude of effective stresses, which is defined as the macroscopic stress divided by the area fraction of bone material, was found not to differ significantly between cephalocaudal and circumferential directions, although they were evaluated using independent models. These results demonstrate that the uniform stress state of the cortical-cancellous bone structure is consistent with results obtained in the cutting experiment when the existence of residual stress is taken into account.     

Finite element analysis of a hemi-pelvis: the effect of inclusion of cartilage layer on acetabular stresses and strain

An appropriate method of application of the hip-joint force and stress analysis of the pelvic bone, in particular the acetabulum, is necessary to investigate the changes in load transfer due to implantation and to calculate the reference stimulus for bone remodelling simulations. The purpose of the study is to develop a realistic 3D finite element (FE) model of the hemi-pelvis and to assess stress and strain distribution during a gait cycle. The FE modelling approach of the pelvic bone was based on CT scan data and image segmentation of cortical and cancellous bone boundaries. Application of hip-joint force through an anatomical femoral head having a cartilage layer was found to be more appropriate than a perfectly spherical head, thereby leading to more accurate stress–strain distribution in the acetabulum. Within the acetabulum, equivalent strains varied between 0.1% and 0.7% strain in the cancellous bone. High compressive (15–30 MPa) and low tensile (0–5 MPa) stresses were generated within the acetabulum. The hip-joint force is predominantly transferred from the acetabulum through the lateral cortex to the sacroiliac joint and the pubic symphysis. The study is useful to understand the load transfer within the acetabulum and for further investigations on acetabular prosthesis.     

Critical evaluation of known bone material properties to realize anisotropic FE-simulation of the proximal femur

PURPOSE: In a meta-analysis of the literature we evaluated the present knowledge of the material properties of cortical and cancellous bone to answer the question whether the available data are sufficient to realize anisotropic finite element (FE)-models of the proximal femur. MATERIAL AND METHOD: All studies that met the following criteria were analyzed: Young's modulus, tensile, compressive and torsional strengths, Poisson's ratio, the shear modulus and the viscoelastic properties had to be determined experimentally. The experiments had to be carried out in a moist environment and at room temperature with freshly removed and untreated human cadaverous femurs. All material properties had to be determined in defined load directions (axial, transverse) and should have been correlated to apparent density (g/cm(3)), reflecting the individually variable and age-dependent changes of bone material properties. RESULTS: Differences in Young's modulus of cortical [cancellous] bone at a rate of between 33% (58%) (at low apparent density) and 62% (80%) (at high apparent density), are higher in the axial than in the transverse load direction. Similar results have been seen for the compressive strength of femoral bone. For the tensile and torsional strengths, Poisson's ratio and the shear modulus, only ultimate values have been found without a correlation to apparent density. For the viscoelastic behaviour of bone only data of cortical bone and in axial load direction have been described up to now. CONCLUSIONS: Anisotropic FE-models of the femur could be realized for most part with the summarized material properties of bone if characterized by apparent density and load directions. Because several mechanical properties have not been correlated to these main criteria, further experimental investigations will be necessary in future.     

The mechanical behaviour of intracondylar cancellous bone of the femur at different loading rates

The compressive properties of human cancellous bone of the distal intracondylar femur in its wet condition were determined. Specimens were obtained from six cadaveric femora and were tested at a strain rate of 0.002, 0.10 and 9.16 sec⁻¹. It was found that the compressive strength decreases with an increasing vertical distance from the joint. The highest compressive strength level was recorded in the posterior medial condyle. Correlations among the mechanical properties, the bulk specimen density and the bone mineral content yield (i) highly significant correlations between the compressive strength and the elastic modulus (ii) highly significant correlations between the compressive strength or the modulus of elasticity and the bulk specimen density (iii) a doubtful correlation between the compressive strength and the bone mineral content. All recorded graphs of the impact loaded specimens displayed several well defined stress peaks, unlike the graphs recorded at low loading rates. It can be concluded that upon impact loading the localized trabecular failure which is associated with each peak, does not affect the spongy bone's stress capacity in a detrimental way.     

Stress distribution and displacement analysis during an intermaxillary disjunction--a three-dimensional FEM study of a human skull

The goal of this study was to contribute to an understanding of how much expansion force is needed during a maxillary expansion (ME) and where bony reaction takes place. A finite element (FE) model of a dry human male skull was generated from CT scans. The FE model, which consists of cortical and cancellous bone and teeth, was loaded with the same force magnitudes, directions and working points as in rapid maxillary expansion (RME). A three-dimensional finite element stress analysis (FESA) of the forces and displacement was performed. The highest stress was observed in the maxilla in the region where the forces were applied, and spreads more or less throughout almost the whole frontal skull structures. The displacement distribution which causes stress in the skull is highly dependant on the thickness of the bone and its structure. All areas with high compressive and tensile stress are exactly the regions which determine the maximal amount of force to be used during the maxillary expansion and should be examined in case of any complication during a patient's treatment. Regions with significant compressive and tensile stress are the regions observed to have an increase in cellular activity. Further simulations with a given displacement (0.5mm) showed that displacement simulations need extra caution otherwise they will lead to very high forces which are not realistic in an orthodontic treatment.     

Yield behavior of bovine cancellous bone

The compressive yield strain was measured for 61 specimens of bovine cancellous bone from three distal femora. There was no significant relationship (p = 0.08, R2 = 0.051) between yield strain and the degree of trabecular orientation. There was a significant positive correlation (p less than 0.00001, R2 = 0.319) between yield strain and structural (apparent) density and significant negative correlation (p less than 0.0025, R2 = 0.145) between yield strain and bone density. Yield strain correlated best with bone solid volume fraction Vv (epsilon y = 0.592 +2- 1.446vv, R2 = 0.337). The quantity, yield strain, is highly dependent on specific definitions of the yield point and the point of zero strain. For this study the yield point was defined by a 0.0003 offset criterion, and the point of zero strain was defined as the point where the tangent at 15 percent of yield crosses zero. The results using these definitions were compared with results using yield strain values determined by other definitions of the yield point and zero strain. The correlations between yield strain and trabecular orientation, structural density and bone density changed very little for differing definitions of yield. The results suggest that yield strain in cancellous bone is isotropic or independent of textural anisotropy, so the yield behaviour may be characterized by a maximum strain yield criterion. The results also suggest that the primary mode of yield in cancellous bone is buckling of the trabeculae.     

Dependence of yield strain of human trabecular bone on anatomic site

Understanding the dependence of human trabecular bone strength behavior on anatomic site provides insight into structure-function relationships and is essential to the increased success of site-specific finite element models of whole bones. To investigate the hypothesis that the yield strains of human trabecular bone depend on anatomic site, the uniaxial tensile and compressive yield properties were compared for cylindrical specimens from the vertebra (n=61), proximal tibia (n=31), femoral greater trochanter (n=23), and femoral neck (n=27) taken from 61 donors (67+/-15years). Test protocols were used that minimized end artifacts and loaded specimens along the main trabecular orientation. Yield strains by site (mean+/-S.D.) ranged from 0.70+/-0.05% for the trochanter to 0.85+/-0.10% for the femoral neck in compression, from 0.61+/-0.05% for the trochanter to 0.70+/-0.05% for the vertebra in tension, and were always higher in compression than tension (p<0.001). The compressive yield strain was higher for the femoral neck than for all other sites (p<0.001), as was the tensile yield strain for the vertebra (p<0.007). Analysis of covariance, with apparent density as the covariate, indicated that inter-site differences existed in yield stress even after adjusting statistically for density (p<0.035). Coefficients of variation in yield strain within each site ranged from only 5-12%, consistent with the strong linear correlations (r(2)=0.94-0.98) found between yield stress and modulus. These results establish that the yield strains of human trabecular bone can differ across sites, but that yield strain may be considered uniform within a given site despite substantial variation in elastic modulus and yield stress.     

Stress distribution and consolidation in cartilage constituents is influenced by cyclic loading and osteoarthritic degeneration

The understanding of load support mechanisms in cartilage has evolved with computational models that better mimic the tissue ultrastructure. Fibril-reinforced poroelastic models can reproduce cartilage behaviour in a variety of test conditions and can be used to model tissue anisotropy as well as assess stress and pressure partitioning to the tissue constituents. The goal of this study was to examine the stress distribution in the fibrillar and non-fibrillar solid phase and pressure in the fluid phase of cartilage in axisymmetric models of a healthy and osteoarthritic hip joint. Material properties, based on values from the literature, were assigned to the fibrillar and poroelastic components of cartilage and cancellous and subchondral compact bone regions. A cyclic load representing walking was applied for 25 cycles. Contact stresses in the fibrillar and non-fibrillar solid phase supported less than 1% of the contact force and increased only minimally with load cycles. Simulated proteoglycan depletion increased stresses in the radial and tangential collagen fibrils, whereas fibrillation of the tangential fibrils resulted in increased compressive stress in the non-fibrillar component and tensile stress in the radial fibrils. However neither had an effect on fluid pressure. Subchondral sclerosis was found to have the largest effect, resulting in increased fluid pressure, non-fibrillar compressive stress, tangential fibril stress and greater cartilage consolidation. Subchondral bone stiffening may play an important role in the degenerative cascade and may adversely affect tissue repair and regeneration treatments.     

Compressive and shear properties of commercially available polyurethane foams

BACKGROUND: The shear properties of rigid polyurethane (PU-R) foams, routinely used to simulate cancellous bone, are not well characterized. METHOD OF APPROACH: The present assessment of the shear and compressive properties of four grades of Sawbones "Rigid cellular" PU-R foam tested 20 mm gauge diameter dumb-bell specimens in torsion and under axial loading. RESULTS: Shear moduli ranged from 13.3 to 99.7 MPa, shear strengths from 0.7 MPa to 4.2 MPa. Compressive yield strains varied little with density while shear yield strains had peak values with "200 kgm-3" grade. CONCLUSIONS: PU-R foams may be used to simulate the elastic but not failure properties of cancellous bone.     

Mechanical testing of cancellous bone from the femoral head: experimental errors due to off-axis measurements

The aim of this study was to verify whether a misalignment between the testing direction and the trabecular main direction has a significant effect on the compressive behaviour of cancellous bone. Ten cylindrical specimens were extracted from femoral heads with a misalignment to the trabecular main direction of approximately 20 degrees. Each specimen was paired with a specimen extracted aligned with the main direction of the trabeculae on the basis of the closest bone volume fraction, obtaining two groups, one 'aligned' and one 'misaligned'. The average off-axis angle was 6.1 degrees and 21.6 degrees for the 'aligned' and 'misaligned' group, respectively. The specimens underwent micro-tomographic analysis, compressive testing, micro-indentation testing and ashing. No significant differences were found in histomorphometric parameters, hardness and ash density between the two groups, whereas significant differences were found in Young's modulus and ultimate stress: both parameters, measured for the 'misaligned' group, were about 40% lower than those measured for the 'aligned' group. These results demonstrate a great effect of the angle between the testing direction and the main direction of the trabecular structure on the compressive behaviour of cancellous bone. This angle should be reduced as much as possible (in the present work the average value was 6.6+/-3.3 degrees), in any case measured, and always reported together with the mechanical parameters of cancellous bone.     

Dependence of mechanical compressive strength on local variations in microarchitecture in cancellous bone of proximal human femur

Human cancellous bone is a heterogeneous material. Despite this, most of the published studies report correlations between mechanical properties and morphometric parameters averaged on the whole specimen. This work investigated whether local variations in morphometric parameters were linked to the localized failure regions of cancellous bone. Additionally, it was examined whether local values of morphometric parameters can predict the ultimate stress better than the average bone volume fraction (BV/TV). Cylindrical cancellous bone specimens extracted along the primary compressive group of human femoral heads were studied. These were microCT-imaged to assess the morphometric parameters, compressed to determine the ultimate stress, and rescanned by microCT to visualize the failure region. Failure involved slightly less than half of the free height of the specimens. Significant differences were found in the morphometric parameters calculated in the failure and in the non-failure regions. The cross-sections containing minimum BV/TV values were those most often located inside the failure region (83%, p<0.001). Regression analysis confirmed that variations in BV/TV best describe variations in ultimate stress (R2=0.84) out of the averaged morphometric parameters. The prediction of ultimate stress increased when minimum or maximum values of the morphometric parameters were taken, with the highest prediction found by considering the minimum BV/TV (R2=0.95). In conclusion, due to the heterogeneity of cancellous bone, there may exist regions characterized by a different microarchitecture, where the bone is weaker and consequently is more likely to fail. These regions mostly contain minimum values in BV/TV, which were found to predict ultimate stress better than average BV/TV.     

A theoretical model to predict distribution of the fabric tensor and apparent density in cancellous bone

 The adaptation of cancellous bone to mechanical forces is well recognized. Theoretical models for predicting cancellous bone architecture have been developed and have mainly focused on the distribution of trabecular mass or the apparent density. The purpose of this study was to develop a theoretical model which can simultaneously predict the distribution of trabecular orthotropy/orientation, as represented by the fabric tensor, along with apparent density. Two sets of equations were derived under the assumption that cancellous bone is a biological self-optimizing material which tends to minimize strain energy. The first set of equations provide the relationship between the fabric tensor and stress tensor, and have been verified to be consistent with Wolff’s law of trabecular architecture, that is, the principal directions of the fabric tensor coincide with the principal stress trajectories. The second set of equations yield the apparent density from the stress tensor, which was shown to be identical to those obtained based on local optimization with strain energy density of true bone tissue as the objective function. These two sets of equations, together with elasticity field equations, provide a complete mathematical formulation for the adaptation of cancellous bone. Received: 25 February 1997/Revised version: 23 September 1997     

A three-dimensional finite element stress analysis for tunnel placement and buttons in anterior cruciate ligament reconstructions

This communication reports the results of a three-dimensional finite element (FE) model of stresses in a surgically altered femur and tibia. The model incorporated a novel approach in implementing orthotropic and inhomogeneous bone properties and non-uniform distributed loading. Cortical, cancellous, and subchondral bone of the femur and the tibia were modeled. Mechanical properties for the cortical and cancellous bone were mapped from published data characterizing the anisotropy and inhomogeneity of the bone properties. Mesh adequacy was determined using stress convergence and strain energy error convergence. Qualitatively, the results of the study compare well with experimental principal compressive strains from the literature. With respect to tunnel placement in anterior cruciate ligament reconstruction, the model predicted stress-shielding at the postero-lateral region of the tunnel wall, and increased stress at the postero-medial region of the tunnel wall. The stresses in the cancellous bone beneath the tunnel were, in general, lower than those above the tunnel. Prolonged stress shielding leads to bone resorption of the posterior tunnel wall leading to tunnel enlargement, and possible compromise of the ACL reconstruction. The stresses on the femoral cortex produced from a button-type fixation were noticeable for low levels of loading; the stress levels were very similar in models incorporating bone properties of patients aged 45 and 65. Repeated compression of the femoral cortex at these stress levels may cause microdamage to the cortex eventually resulting in fatigue failure.     

A musculoskeletal model of the equine forelimb for determining surface stresses and strains in the humerus--part I. Mathematical modeling

Knowledge of the forces that act upon the equine humerus while the horse is standing and the resulting strains experienced by the bone is useful for the prevention and treatment of fractures and for assessing the proximolateral aspect of the bone as a site for obtaining autogenous bone graft material. The first objective was to develop a mathematical model to predict the loads on the proximal half of the humerus created by the surrounding musculature and ground reaction forces while the horse is standing. The second objective was to calculate surface bone stresses and strains at three cross sections on the humerus corresponding to the donor site for bone grafts, a site predisposed to stress fracture, and the middle of the diaphysis. A three-dimensional mathematical model employing optimization techniques and asymmetrical beam analysis was used to calculate shoulder muscle forces and surface strains on the proximal and mid-diaphyseal aspects of the humerus. The active shoulder muscles, which included the supraspinatus, infraspinatus, subscapularis, and short head of the deltoid, produced small forces while the horse is standing; all of which were limited to 4.3% of their corresponding maximum voluntary contraction. As a result, the strains calculated at the proximal cross sections of the humerus were small, with maximum compressive strains of -104microepsilon at the cranial aspect of the bone graft donor cross section. The middle of the diaphysis experienced larger strain magnitudes with compressive strains at the lateral and the caudal aspects and tensile strains at the medial and cranial aspects (-377microepsilon and 258microepsilon maximum values, respectively) while the horse is standing. Small strains at the donor bone graft site do not rule out using this location to harvest bone graft tissue, although strains while rising to a standing position during recovery from anesthesia are unknown. At the site common to stress fractures, small strains imply that the stresses seen by this region while the horse is standing, although applied for long periods of time, are not a cause of fracture in this location. Knowing the specific regions of the middle of the diaphysis of the humerus that experience tensile and compressive strains is valuable in determining optimum placement of internal fixation devices for the treatment of complete fractures.     

A physical model for simulating fusion of impaction-grafted morselized cancellous bone

This article describes a method to simulate fusion of morselized cancellous bone. The morselized cancellous bone is mixed with an epoxy adhesive, in empirically determined proportions. The mixture is then impacted into a construct. When the epoxy cures, the morselized material fuses into a cohesive, contiguous structure with a compressive modulus equivalent to that of intact cancellous bone. This model can be used to study biomechanical aspects of fused impaction grafts.     

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.     

The effect of stress concentration on bone remodeling: theoretical predictions

Theoretical predictions of internal bone remodeling around an elliptical hole are studied. The internal remodeling theory due to Cowin and Hegedus is employed. The bone is modeled as an initially homogeneous adaptive elastic plate with an elliptical hole under a superposed steady compressive load. It is shown that there will exist a final inhomogeneous remodeling distribution around the hole that will disappear away from the hole. The remodeling is such that the compressive stress concentration causes the bone structure to evolve to one of greater density and stiffer elastic coefficients. The speed of remodeling around the hole and its variation with respect to distance is investigated and discussed. It is shown that the rate of bone reinforcement in the area of compressive stress concentration is much higher than the rate of bone resorption in the area of existing tensile stress. Special cases of a circular hole and vertical and horizontal slots are studied and discussed.     

On Wolff's law of trabecular architecture.

Several studies suggest that the yield strain in cancellous bone may be uniformly distributed and isotropic. Yield strain was reported to be independent of textural anisotropy in bovine cancellous bone [Turner, J. biomech. Engng 111, 1-5 (1989)] and it is plausible that yield strain is isotropic in human cancellous bone as well. In this paper, it is hypothesized that uniform, isotropic strain represents a goal of cancellous bone adaptation, i.e. cancellous bone alters its structure to maintain uniform, isotropic peak strains. Therefore, textural anisotropy must exactly cancel the anisotropy of the peak principal stresses imposed upon cancellous bone. When evaluating the relationships between mechanical properties of cancellous bone and trabecular architecture, it was found that over 90% of the variance of yield strength can be explained by one term--rho 2H3 (where rho is apparent density and H is the normalized anisotropy (fabric) constant). Furthermore, this single term explains 70-78% of the variance in Young's modulus of cancellous bone. Based upon these findings, it was postulated that fabric adaptation goes as Hi/Hj = [ sigma i/sigma j[, where Hi and Hj are fabric eigenvalues in the i- and the j-direction and sigma i and sigma j are peak principal stresses.     

Tensile strength of bovine trabecular bone

Data on the tensile and compressive properties of trabecular bone are needed to define input parameters and failure criteria for modeling total joint replacements. To help resolve differences in reports comparing tensile and compressive properties of trabecular bone, we have developed new methods, based on porous foam technology, for tensile testing of fresh/frozen trabecular bone specimens. Using bovine trabecular bone from an isotropic region from the proximal humerus as a model material, we measured ultimate strengths in tension and compression for two groups of 24 specimens each. The average ultimate strength in tension was 7.6 +/- 2.2 (95% C.I.) MPa and in compression was 12.4 +/- 3.2 MPa. This difference was statistically significant (p = 0.013) and was not related to density differences between the test groups (p = 0.28). Strength was related by a power-law function of the local apparent density, but, even accounting for density influences, isotropic bovine trabecular bone exhibits significantly lower strengths in tension than in compression.