Determination of mechanical stiffness of bone by pQCT measurements: correlation with non-destructive mechanical four-point bending test data

Mechanical tests of bone provide valuable information about material and structural properties important for understanding bone pathology in both clinical and research settings, but no previous studies have produced applicable non-invasive, quantitative estimates of bending stiffness. The goal of this study was to evaluate the effectiveness of using peripheral quantitative computed tomography (pQCT) data to accurately compute the bending stiffness of bone. Normal rabbit humeri (N=8) were scanned at their mid-diaphyses using pQCT. The average bone mineral densities and the cross-sectional moments of inertia were computed from the pQCT cross-sections. Bending stiffness was determined as a function of the elastic modulus of compact bone (based on the local bone mineral density), cross-sectional moment of inertia, and simulated quasistatic strain rate. The actual bending stiffness of the bones was determined using four-point bending tests. Comparison of the bending stiffness estimated from the pQCT data and the mechanical bending stiffness revealed excellent correlation (R2=0.96). The bending stiffness from the pQCT data was on average 103% of that obtained from the four-point bending tests. The results indicate that pQCT data can be used to accurately determine the bending stiffness of normal bone. Possible applications include temporal quantification of fracture healing and risk management of osteoporosis or other bone pathologies.     

Analysis of creep strain during tensile fatigue of cortical bone

During fatigue tests of cortical bone specimens, at the unload portion of the cycle (zero stress) non-zero strains occur and progressively accumulate as the test progresses. This non-zero strain is hypothesised to be mostly, if not entirely, describable as creep. This work examines the rate of accumulation of this strain and quantifies its stress dependency. A published relationship determined from creep tests of cortical bone (Journal of Biomechanics 21 (1988) 623) is combined with knowledge of the stress history during fatigue testing to derive an expression for the amount of creep strain in fatigue tests. Fatigue tests on 31 bone samples from four individuals showed strong correlations between creep strain rate and both stress and "normalised stress" (sigma/E) during tensile fatigue testing (0-T). Combined results were good (r(2)=0.78) and differences between the various individuals, in particular, vanished when effects were examined against normalised stress values. Constants of the regression showed equivalence to constants derived in creep tests. The universality of the results, with respect to four different individuals of both sexes, shows great promise for use in computational models of fatigue in bone structures.     

The effect of sterilization on the mechanical properties of intact rabbit humeri in three-point bending, four-point bending and torsion

Load bearing bone allografts are used to replace the mechanical function of bone that has been removed or to augment bone that has been damaged in trauma. In order to minimize the risk of infection and immune response, the bone is delipidated and terminally sterilized prior to implantation. The optimal method for bone graft sterilization has been the topic of considerable research. Recently, supercritical carbon dioxide (SCCO₂) treatments have been shown to terminally sterilize bone against a range of bacteria and viruses. This study aimed to evaluate the effect of SCCO₂ treatment compared with two doses of gamma irradiation, on the mechanical properties of whole bone. Paired rabbit humeri were dissected and randomly assigned into either SCCO₂ control, SCCO₂ additive or gamma irradiation at 10 or 25 kGy treatment groups. The bones were mechanically tested in three-point and four-point bending and torsion, with the lefts acting as controls for the treated rights. Maximum load, energy to failure and stiffness were evaluated. This study found that SCCO₂ treatment with or without additive did not alter maximum load, energy to failure or stiffness significantly under any loading modality. Gamma irradiation had a deleterious dose dependant effect, with statistically significant decreases in all mechanical tests at 25 kGy; while at 10 kGy there were reductions in all loading profiles, though only reaching statistical significance in torsion. This study highlights the expediency of SCCO₂ treatment for bone allograft processing as terminal sterilization can be achieved while maintaining the intrinsic mechanical properties of the graft.     

Age-dependent fatigue behaviour of human cortical bone

Despite a general understanding that bone quality contributes to skeletal fragility, very little information exits on the age-dependent fatigue behavior of human bone. In this study four-point bending fatigue tests were conducted on aging bone in conjunction with the analysis of stiffness loss and preliminary investigation of nanoindentation based measurements of local tissue stiffness and histological evaluation of resultant tensile and compressive damage to identify the damage mechanism responsible for the increase in age-related bone fragility. The results obtained show that there is an exponential decrease in fatigue life with age, and old bone exhibits different modulus degradation profiles than young bone. In addition, this study provides preliminary evidence indicating that during fatigue loading, younger bone formed diffuse damage, lost local tissue stiffness on the tensile side. Older bone, in contrast, formed linear microcracks lost local tissue stiffness on the compressive side. Thus, the propensity of aging human bone to form more linear microcracks than diffuse damage may be a significant contributor to bone quality, and age related fragility in bone.     

The effect of recovery time and test conditions on viscoelastic measures of tensile damage in cortical bone

Stiffness degradation and strength degradation are often measured to monitor and characterize the effects of damage accumulation in bone. Based on evidence that these properties could be affected by not only damage magnitude but also test conditions, the present study investigated the effect of hold condition and recovery time on measures of tensile damage. Machined human femoral cortical bone specimens were subjected to tensile tests consisting of a pre-damage diagnostic loading cycle, a damage loading cycle and post-damage cycle. Controlled variables were recovery time (1, 10, and 100 min) and hold condition (zero load or zero strain) after the damage cycle. Damage measures were calculated as the ratio of each post-damage cycle to the pre-damage value for loading modulus, secant modulus, unloading modulus, stress relaxation and strain (stress) recovery at 1 min post-diagnostic time. The damage cycle caused reductions in all measures, and some measures varied with recovery time and hold condition. Apparent modulus degradation for both hold conditions decreased with recovery time. Stress relaxation was unaffected by recovery time for both hold conditions. Zero-strain hold conditions resulted in lower values for degradation of modulus and change of relaxation. Stress or strain recovery after the damage cycle was evident through 100 min, but 90% of the recovery occurred within 10 min. The results demonstrate that choice of test conditions can influence the apparent magnitude of damage effects. They also indicate that 10 min recovery time was sufficient to stabilize most measures of the damage state.     

Volume effects on fatigue life of equine cortical bone

Materials, including bone, often fail due to loading in the presence of critical flaws. The relative amount, location, and interaction of these flaws within a stressed volume of material play a role in determining the failure properties of the structure. As materials are generally imperfect, larger volumes of material have higher probabilities of containing a flaw of critical size than do smaller volumes. Thus, larger volumes tend to fail at fewer cycles compared with smaller volumes when fatigue loaded to similar stress levels. A material is said to exhibit a volume effect if its failure properties are dependent on the specimen volume. Volume effects are well documented in brittle ceramics and composites and have been proposed for bone. We hypothesized that (1) smaller volumes of cortical bone have longer fatigue lives than similarly loaded larger volumes and (2) that compared with microstructural features, specimen volume was able to explain comparable amounts of variability in fatigue life. In this investigation, waisted rectangular specimens (n=18) with nominal cross-sections of 3×4 mm and gage lengths of 10.5, 21, or 42 mm, were isolated from the mid-diaphysis of the dorsal region of equine third metacarpal bones. These specimens were subjected to uniaxial load controlled fatigue tests, with an initial strain range of 4000 microstrain. The group having the smallest volume exhibited a trend of greater log fatigue life than the larger volume groups. Each volume group exhibited a significant positive correlation between the logarithm of fatigue life and the cumulative failure probability, indicating that the data follow the two-parameter Weibull distribution. Additionally, log fatigue life was negatively correlated with log volume, supporting the hypothesis that smaller stressed volumes of cortical bone possess longer fatigue lives than similarly tested larger stressed volumes.     

Some mechanical properties of goose femoral cortical bone

The ultimate compressive strength and modulus of elasticity of femoral cortical bone from adult geese (Anser anser), were determined by sex and by quadrant by compressing small right circular cylinders which were 2.4 mm in height and 0.8 mm in diameter. The average ultimate compressive strength was 183 +/- 29 MPa. The average modulus of elasticity was 13.2 +/- 3.4 GPa. The bending strength and bending modulus of elasticity were determined by a three point bend test on rectangular prisms which had the approximate dimensions 0.75 mm X 0.75 mm X 25 mm. The average bending strength was 263 +/- 44 MPa while the average bending modulus was 19.6 +/- 3.1 GPa. The calcium content was determined by atomic absorption spectrophotometry and no correlation was found with the mechanical properties. The histology of the cortical bone was examined both quantitatively and qualitatively. A unique type of Haversian bone is described. Goose bone was found to be morphologically similar to adolescent human bone and to have mechanical properties similar to those of adult human bone.     

Finite element prediction of fatigue damage growth in cancellous bone

Cyclic stresses applied to bones generate fatigue damage that affects the bone stiffness and its elastic modulus. This paper proposes a finite element model for the prediction of fatigue damage accumulation and failure in cancellous bone at continuum scale. The model is based on continuum damage mechanics and incorporates crack closure effects in compression. The propagation of the cracks is completely simulated throughout the damaged area. In this case, the stiffness of the broken element is reduced by 98% to ensure no stress-carrying capacities of completely damaged elements. Once a crack is initiated, the propagation direction is simulated by the propagation of the broken elements of the mesh. The proposed model suggests that damage evolves over a real physical time variable (cycles). In order to reduce the computation time, the integration of the damage growth rate is based on the cycle blocks approach. In this approach, the real number of cycles is reduced (divided) into equivalent blocks of cycles. Damage accumulation is computed over the cycle blocks and then extrapolated over the corresponding real cycles. The results show a clear difference between local tensile and compressive stresses on damage accumulation. Incorporating stiffness reduction also produces a redistribution of the peak stresses in the damaged region, which results in a delay in damage fracture.     

Characterization of in vivo strain in the rat tibia during external application of a four-point bending load.

This paper describes a technique for characterizing strains and stresses induced in vivo in the rat tibia during application of an external four-point bending load. An external load was applied through the muscle and soft tissue with a four-point bending device, to induce strain in a 11 mm section of the right tibiae of ten adult female Sprague-Dawley rats. Induced strains were measured in vivo on the lateral surface of the tibia. Inter-rat difference, leg positioning and strain gage placement were evaluated as sources of variability of applied strains. Beam bending theory was used to predict externally induced in vivo strains. Finite element analysis was used to quantify the magnitude of shear stresses induced by this type of loading. There was a linear relationship between applied load and induced in vivo strains. In vivo strains induced by external loading were linearly correlated (R2 = 0.87) with the strains calculated using beam bending theory. The finite element analysis predicted shear stresses at less than 10% of the longitudinal stresses resulting from four-point bending. Strains predicted along the tibia by finite element analysis and beam bending theory were well-correlated. Inter-rat variability due to tibia size and shape difference was the most important source of variation in induced strain (CV = 21.6%). Leg positioning was less important (CV = 9.5%).     

Sensitivity of multiple damage parameters to compressive overload in cortical bone

Damage accumulation plays a key role in weakening bones prior to complete fracture and in stimulating bone remodeling. The goal of this study was to characterize the degradation in the mechanical properties of cortical bone following a compressive overload. Longitudinally oriented, low-aspect ratio specimens (n=24) of bovine cortical bone were mechanically tested using an overload-hold-reload protocol. No modulus reductions greater than 5% were observed following overload magnitudes less than 0.73% strain. For each specimen, changes in strength and Poisson's ratio were greater (p=0.02) than that in modulus by 10.8- and 26.6-fold, respectively, indicating that, for the specimen configuration used in this study, longitudinal elastic modulus is one of the least sensitive properties to a compressive overload. Residual strains were also proportionately greater by 6.4-fold (p=0.01) in the transverse than axial direction. These results suggest that efforts to relate microcrack density and morphology to changes in compressive mechanical properties of cortical bone may benefit from considering alternative parameters to modulus reductions.     

Micro-biomechanics vs macro-biomechanics in cortical bone. A micromechanical investigation of femurs deformed by bending

Previous studies have shown that longitudinal and transverse lamellae in compact bone signal the presence of tensile and compressive forces, respectively. On this basis an investigation has been carried out to ascertain the distribution of lamellae in femoral shafts that have been deformed by bending. To do this, a series of undecalcified plane-parallel sections 100 microns thick were prepared, and the proportion of transversely oriented collagen and crystallites in the lamellae was measured, using circularly polarized light as an illuminating source and a Quantimet 720 image-analyzing computer. It has been concluded that the distribution of the two types of lamellae is in accordance with the need to compensate for the deformity produced by the bending of the bones, and that an obvious relationship exists between the macroscopic features of the femurs and their modified microscopic structures.     

A damage model for nonlinear tensile behavior of cortical bone.

To describe the time-dependent nonlinear tensile behavior observed in experimental studies of cortical bone, a damage model was developed using two internal state variables (ISV's). One ISV is a damage parameter that represents the loss of stiffness. A rule for the evolution of this ISV was defined based on previously observed creep behavior. The second ISV represents the inelastic strain due to viscosity and internal friction. The model was tested by simulating experiments in tensile and bending loading. Using average values from previous creep studies for parameters in the damage evolution rule, the model tended to underestimate the maximum nonlinear strains and to overestimate the nonlinear strain accumulated after load reversal in the tensile test simulations. Varying the parameters for the individual tests produced excellent fits to the experimental data. Similarly, the model simulations of the bending tests could produce excellent fits to the experimental data. The results demonstrate that the 2-ISV model combining damage (stiffness loss) with slip and viscous behavior could capture the nonlinear tensile behavior of cortical bone in axial and bending loading.     

A comparison of the fatigue behavior of human trabecular and cortical bone tissue

The fatigue properties of trabecular bone tissue (single trabeculae) and similarly sized cortical bone specimens from human tibia were experimentally determined on a microstructural level using four-point bending cyclic tests, and they were compared based on modulus, mineral density, and microstructural characteristics. The results showed that trabecular specimens had significantly lower moduli and lower fatigue strength than cortical specimens, despite their higher mineral density values. Fracture surface and microdamage analyses illustrated different fracture and damage patterns between trabecular and cortical bone tissue, depending upon their microstructural characteristics. Based on the results from mechanical tests and qualitative observations, a possible mechanical role of the cement lines in trabecular tissue microfracture was suggested.