The open section effect in a long bone with a longitudinal defect - a theoretical modeling study

A longitudinal defect dramatically alters the stress distribution within a long bone. The altered stress distribution can influence the structural properties of the bone and the stimulus for repair and remodeling of the defect and the surrounding bone. For applied torsion, the defect interrupts the normal shear flow around the bone. Reversal of the shear flow along the inner cortex of the bone is the primary characteristic of the "open-section" effect. Stress concentration effects also produce large stresses at the defect corners. A finite element model of a femur mid-diaphysis with a rectangular defect in the posterior cortex was developed to quantify the femur stress distribution and torsional stiffness for defect widths ranging from one-tenth of the femur outer diameter (0.1 OD) to 0.3 OD, and defect lengths ranging from 0.5 to 5 OD. Defects with a length of 1 OD or shorter had little influence on the femur torsional stiffness or the femur shear-stress distribution. The torsional stiffness decreased most dramatically as the defect length increased from 2 to 3 OD, but began to approach an asymptote near 5 OD. Shear flow reversal peaked at the center of the defect for defects longer than 1 OD, and the magnitude of the reversal began to approach an asymptote near 5 OD. For each defect, the largest stresses within the bone, developed at the defect corners. The results indicate that the open-section effect decreases the torsional stiffness and stress concentration effects decrease the torsional strength of a long bone with a longitudinal defect.     

Incidence and mechanical significance of pneumatization in the long bones of birds

The incidence of pneumatization in avian long bones was studied, by direct observation, in a large sample of species. Only proximal bones (humerus and femur) presented pneumatization in the sample studied. The incidence obtained was related to the variation of the maximum cortical thickness and mechanical properties, such as bending strength and flexural Young's modulus. Cortical thickness, bending strength and flexural Young's modulus were significantly lower in pneumatized bones than in marrow-filled bones. Furthermore, some congruence was found between pneumatization and systematic groups when compared. In this sense, Charadriformes was the only order studied with total absence of long bone pneumatization. Results on cortical thickness appear to be in agreement with modelling predictions previously made and with results obtained on other groups of flying vertebrates. The possible selective advantage of reduction in cortical thickness in relation to flying is suggested.     

Influence of teeth, alveoli, and periodontal ligaments on torsional rigidity in human mandibles.

We investigated the influence of teeth, periodontal ligaments, and alveoli on the structural integrity of human mandibles loaded in torsion. Surface bone strain was recorded from the mandibular corpus below the first molar on each of four specimens. These specimens were loaded by an external force that caused primarily torsion about the long axis of the corpus, and bone strain was recorded under the following conditions: 1) all supporting structures intact, 2) all supporting structures intact and the M1 loaded by a simulated bite force, 3) M1 removed and 4) alveolar bone of the M1 removed. For comparative purposes, experiments were also designed to investigate the effects of intermittent holes on the torsional rigidity of a baboon femur. This permitted comparison of the mechanical behavior of the mandibles with that of a more homogeneous bony member. These experiments suggest that the presence of teeth within alveoli has a measurable role in the maintenance of torsional rigidity. The condition of the periodontal ligament also appears to influence these stress-bearing capabilities. Moreover, the alveolar bone supporting the teeth also provides structural support for countering torsional loads. For the specific case of corpus twisting, the mandible does not behave as a member with open or closed sections as predicted by theoretical models. The observed magnitudes of bone strain, however, conform more closely to the predictions generated by a closed-section model.     

Validating a voxel-based finite element model of a human mandible using digital speckle pattern interferometry

Finite element analysis is a powerful tool for predicting the mechanical behaviour of complex biological structures like bones, but to be confident in the results of an analysis, the model should be validated against experimental data. In such validation experiments, the strains in the loaded bones are usually measured with strain gauges glued to the bone surface, but the use of strain gauges on bone can be difficult and provides only very limited data regarding surface strain distributions. This study applies the full-field strain measurement technique of digital speckle pattern interferometry to measure strains in a loaded human mandible and compares the results with the predictions of voxel-based finite element models of the same specimen. It is found that this novel strain measurement technique yields consistent, reliable measurements. Further, strains predicted by the finite element analysis correspond well with the experimental data. These results not only confirm the usefulness of this technique for future validation studies in the field of bone mechanics, but also show that the modelling approach used in this study is able to predict the experimental results very accurately.     

Body size, body shape, and long bone strength in modern humans

To identify behaviorally significant differences in bone structure it is first necessary to control for the effects of body size and body shape. Here the scaling of cross-sectional geometric properties of long bone diaphyses with different "size" measures (bone length, body mass, and the product of bone length and body mass) are compared in two modern human populations with very different body proportions: Pecos Pueblo Amerindians and East Africans. All five major long bones (excluding the fibula) were examined. Mechanical predictions are that cortical area (axial strength) should scale with body mass, while section modulus (bending/torsional strength) should scale with the product of body mass and moment arm length. These predictions are borne out for section moduli, when moment arm length is taken to be proportional to bone length, except in the proximal femoral diaphysis, where moment arm length is proportional to mediolateral body breadth (as would be expected given the predominance of M-L bending loads in this region). Mechanical scaling of long bone bending/torsional strength is similar in the upper and lower limbs despite the fact that the upper limb is not weight-bearing. Results for cortical area are more variable, possibly due to a less direct dependence on mechanical factors. Use of unadjusted bone length alone as a "size" measure produces misleading results when body shape varies significantly, as is the case between many modern and fossil hominid samples. In such cases a correction factor for body shape should be incorporated into any "size" standardization.     

A torsional strength comparison of vascularized and nonvascularized bone grafts

Comparisons of torsional strength are made on the ulnae from the forelegs of short haired hounds where a nonvascularized graft was performed on one leg and a vascularized graft performed on the other. By using the forelegs of a dog as the experimental model and microsurgical techniques, a vascularized bone segment was used to graft a five centimeter nonunion in one leg and at the same time a conventional bone graft was performed on a similar nonunion in the other leg. Similar segments of rib bone were used for each graft. Torsional strength data are shown for nine experimental animals. A successful method for mounting the bones for testing of torsional strength in a torsion machine is given. In each case for which the bones healed properly, the vascularized bone graft proved to be significantly stronger in torsion.     

Long bone torsion: I. Effects of heterogeneity, anisotropy and geometric irregularity

The influences of heterogeneity, anisotropy and geometric irregularity on the unrestrained, linearly elastic torsional response of long bones are assessed. Longitudinal geometric variations contribute insignificantly to the torsional response for typical long bone geometries. Anisotropy, heterogeneity and transverse geometric irregularity significantly influence the torsional response. A procedure is discussed which uses an approximate means to characterize both heterogeneity and anisotropy in predicting the torsional response. The accuracy of circular and elliptical annulus models of the bone cross-sectional geometry are assessed by comparing the stress predictions of these simple models to those of finite element models of the bone geometry.     

Comparative analysis of bone remodelling models with respect to computerised tomography-based finite element models of bone

Subject-specific finite element models are an extensively used tool for the numerical analysis of the biomechanical behaviour of human bones. However, bone modelling is not an easy task due to the complex behaviour of bone tissue, involving non-homogeneous and anisotropic mechanical properties. Moreover, bone is a living tissue and therefore its microstructure and mechanical properties evolve with time in a known process called bone remodelling. This phenomenon has been widely studied, many being the numerical models that have been formulated to predict density distribution and its evolution in several bones. The aim of the present study is to assess the capability of a bone remodelling model to predict the bone density distribution of different types of human bone (femur, tibia and mandible) comparing the obtained results with the bone density estimated by means of computerised tomography. Good accuracy was observed for the bone remodelling predictions including the thickness of the cortical layer.     

Computed-tomography-based finite-element models of long bones can accurately capture strain response to bending and torsion

Finite element (FE) models of long bones constructed from computed-tomography (CT) data are emerging as an invaluable tool in the field of bone biomechanics. However, the performance of such FE models is highly dependent on the accurate capture of geometry and appropriate assignment of material properties. In this study, a combined numerical-experimental study is performed comparing FE-predicted surface strains with strain-gauge measurements. Thirty-six major, cadaveric, long bones (humerus, radius, femur and tibia), which cover a wide range of bone sizes, were tested under three-point bending and torsion. The FE models were constructed from trans-axial volumetric CT scans, and the segmented bone images were corrected for partial-volume effects. The material properties (Young's modulus for cortex, density-modulus relationship for trabecular bone and Poisson's ratio) were calibrated by minimizing the error between experiments and simulations among all bones. The R(2) values of the measured strains versus load under three-point bending and torsion were 0.96-0.99 and 0.61-0.99, respectively, for all bones in our dataset. The errors of the calculated FE strains in comparison to those measured using strain gauges in the mechanical tests ranged from -6% to 7% under bending and from -37% to 19% under torsion. The observation of comparatively low errors and high correlations between the FE-predicted strains and the experimental strains, across the various types of bones and loading conditions (bending and torsion), validates our approach to bone segmentation and our choice of material properties.     

Finite element analysis of retroacetabular osteolytic defects following total hip replacement

Periprosthetic osteolysis in the retroacetabular region with cancellous bone loss is a recognized phenomenon in the long-term follow-up of total hip replacement. The effects on load transfer in the presence of defects are less well known. A finite element model incorporating a retroacetabular defect behind a cementless component was validated against a 4th generation sawbone pelvis. Computational predictions of surface strain and von Mises stresses were closely correlated to experimental findings. The presence of a cancellous defect increased von Mises stress in the cortical bone of the medial wall of the pelvis. At a load of 600 N this was under the predicted failure stress for cortical bone. Increases in the cup size relative to the acetabulum caused increased stress in the cortical bone of the lateral wall of the pelvis, adjacent to the acetabulum. We are confident that our modeling approach can be applied to patient specific defects to predict pelvis stress with large loads and a range of activities.     

In vitro sodium fluoride exposure decreases torsional and bending strength and increases ductility of mouse femora

Fluoride exposure in vivo can reduce the material strength of bone, an effect that has been attributed to a change in mineral structure. An in vitro model of fluoride exposure offers the potential to study directly the effects of fluoride on bone mineral. Previous investigators have reported that soaking bones in sodium fluoride in vitro reduces bone strength. However, long soaking times and the absence of physiological buffering ions from their treatment solutions may have caused mineral dissolution that contributed to the decrease in bone strength. Our objectives were to further characterize the effects of in vitro fluoride exposure on bone mechanical properties and to determine if the changes reported in previous studies of bovine cortical bone would be observed for whole rodent bones. We soaked 60 mouse femora in sodium fluoride solutions, with and without physiological buffering ions, and evaluated their torsional and bending properties. Fluoride soaked bones had a 30-fold increase in fluoride content and a 23% increase in water content compared to controls. These changes were associated with average reductions in ultimate load of 45%, reductions in rigidity of 70%, and increases in deformation to failure of 80%. The effect of fluoride was similar for bones treated in buffered and non-buffered solutions, and was observed in both torsion and bending. Our findings confirm those of previous studies and highlight the strong effect that in vitro fluoride exposure has on bone mechanical properties. The in vitro model of fluoride exposure offers a tool to further study the effects of ion substitution in bone.     

Cortical bone distribution in the femoral neck of strepsirhine primates

The thickness of the inferior and superior cortices of the femoral neck was measured on X-rays of 181 strepsirhine primate femora representing 24 species. Neck length, neck depth and neck-shaft angle were also measured. The strength of the femoral neck in frontal bending was estimated by modeling the neck as a hollow cylinder, with neck depth as the outer diameter and cortical thickness representing the superior and inferior shell dimensions. Results indicate that the inferior cortex is always thicker than the superior cortex. The ratio of superior to inferior cortical thickness is highly variable but distinguishes two of the three locomotor groups in the sample. Vertical clingers and leapers have higher ratios (i.e., a more even distribution of cortical bone) than quadrupeds. The slow climbers tend to have the lowest ratios, although they do not differ significantly from the leapers and quadrupeds. These results do not confirm prior theoretical expectations and reported data for anthropoid primates that link greater asymmetry of the cortical shell to more stereotypical hip excursions. The ratio of superior to inferior cortical thickness is unrelated to body mass, femoral neck length, and neck-shaft angle, calling into question whether the short neck of strepsirhine primates acts as a cantilever beam in bending. On the other hand, the estimated section moduli are highly correlated with body mass and neck length, a correlation that is driven primarily by body mass. In conclusion, we believe that an alternative interpretation to the cantilever beam model is needed to explain the asymmetry in bone distribution in the femoral neck, at least in strepsirhine primates (e.g., a thicker inferior cortex is required to reinforce the strongly curved inferior surface). As in prior studies of cross-sectional geometry of long bones, we found slightly positive allometry of cortical dimensions with body mass.     

Accuracy of specimen-specific nonlinear finite element analysis for evaluation of radial diaphysis strength in cadaver material

The feasibility of a user-specific finite element model for predicting the in situ strength of the radius after implantation of bone plates for open fracture reduction was established. The effect of metal artifact in CT imaging was characterized. The results were verified against biomechanical test data. Fourteen cadaveric radii were divided into two groups: (1) intact radii for evaluating the accuracy of radial diaphysis strength predictions with finite element analysis and (2) radii with a locking plate affixed for evaluating metal artifact. All bones were imaged with CT. In the plated group, radii were first imaged with the plates affixed (for simulating digital plate removal). They were then subsequently imaged with the locking plates and screws removed (actual plate removal). Fracture strength of the radius diaphysis under axial compression was predicted with a three-dimensional, specimen-specific, nonlinear finite element analysis for both the intact and plated bones (bones with and without the plate captured in the scan). Specimens were then loaded to failure using a universal testing machine to verify the actual fracture load. In the intact group, the physical and predicted fracture loads were strongly correlated. For radii with plates affixed, the physical and predicted (simulated plate removal and actual plate removal) fracture loads were strongly correlated. This study demonstrates that our specimen-specific finite element analysis can accurately predict the strength of the radial diaphysis. The metal artifact from CT imaging was shown to produce an overestimate of strength.     

A new cortical thickness mapping method with application to an in vivo finite element model

Finite element modelling of musculoskeletal systems, with geometrical structures constructed from computed tomography (CT) scans, is a useful and powerful tool for biomechanical studies. The use of CT scans from living human subjects, however, is still limited. Accurate reconstruction of thin cortical bone structures from CT scans of living human subjects is especially problematic, due to low CT resolution that results from mandatory low radiation doses and/or involuntary movements of the subject. In this study, a new method for mapping cortical thickness is described. Using the method, cortical thickness measurements of a coxal (pelvis) bone obtained from CT scans of a cadaver were mapped to the coxal geometry as obtained through CT scans of a live human subject, resulting in accurate cortical thickness while maintaining geometric fidelity of the live subject. The mapping procedure includes shape-preserving parameterisation, mesh movement and interpolation of thickness using a search algorithm. The methodology is applicable to modelling of other bones where accurate cortical thickness is needed and for which such data exist.     

Long bone torsion: II. A combined experimental and computational method for determining an effective shear modulus

A technique is established which allows an effective torsional shear modulus to be determined for long bones, while remaining nondestructive to whole bone specimens. Strain gages are bonded to the diaphysis of the bone. Strains are then recorded under pure torsional loads. Theoretical stress predictions are combined with experimental strain recordings to arrive at a modulus value. Shear modulus calculations for four canine radii are reported using theoretical stress predictions from circular, elliptical and finite element models of the transverse bone geometry. The effective shear modulus, obtained from an average of the shear moduli determined at strain gage locations, serves to average the heterogeneous shear modulus distribution over the cross section. The shear modulus obtained is that associated with the "circumferential" direction in transverse planes.     

A comparison of stress distributions for different surgical procedures, screw dimensions and orientations for a Temporomandibular joint implant

Finite element analysis is a useful analytical tool for the design of biomedical implants. The aim of this study was to investigate the behavior of temporomandibular joint implants with multiple design variables of the screws used for fixation of the implant. A commercially available implant with full mandible was analyzed using a finite element software package. The effects of different design variables such as orientation, diameter and stem length of the screws on the stress distribution in bone for two different surgical procedures were investigated. Considering the microstrain in bone as a principal factor, the acceptable ranges for screw diameter and length were determined. Parallel orientation of the screws performed better from a stress point of view when compared to the zig-zag orientation. Sufficient contact between the implant collar and mandibular condyle was shown to reduce the peak stresses which may lead to long term success. The distance between screw holes in the parallel orientation was much closer when compared to the zig-zag orientation. However, the stresses in bone near the screw hole area for the parallel orientation were within acceptable limits.     

Loading Mode Interactions in Simulations of Long Bone Cross-Sectional Adaptation

Although many bone adaptation theories have been formulated to address both trabecular and cortical adaptation, most applications have focused on trabecular adaptation. Thus far, no thorough investigations of the influence of different types of loading on predicted patterns of long bone cross-sectional adaptation have been reported. In the current study, we present a new model for long bone cross-sectional adaptation that incorporates axial, bending and torsional loading components. We found that bending moments have a strong potential to modulate cross-sectional geometry, but can produce unforseen (and unrealistic) geometric instabilities. Torsional moments have the ability to suppress these instabilities, suggesting that torsion may play a more significant role in guiding long bone development than previously recognized. Our results also call into question the concept of strict "remodeling equilibrium," suggesting that long bones do not necessarily approach a state of uniform mechanical stimulation. This modeling approach provides an additional perspective on experimental studies, and may lead to a greater understanding of the interaction between mechanics and biology in long bone adaptation.     

Loading Mode Interactions in Simulations of Long Bone Cross-Sectional Adaptation

Although many bone adaptation theories have been formulated to address both trabecular and cortical adaptation, most applications have focused on trabecular adaptation. Thus far, no thorough investigations of the influence of different types of loading on predicted patterns of long bone cross-sectional adaptation have been reported. In the current study, we present a new model for long bone cross-sectional adaptation that incorporates axial, bending and torsional loading components. We found that bending moments have a strong potential to modulate cross-sectional geometry, but can produce unforseen (and unrealistic) geometric instabilities. Torsional moments have the ability to suppress these instabilities, suggesting that torsion may play a more significant role in guiding long bone development than previously recognized. Our results also call into question the concept of strict “remodeling equilibrium,” suggesting that long bones do not necessarily approach a state of uniform mechanical stimulation. This modeling approach provides an additional perspective on experimental studies, and may lead to a greater understanding of the interaction between mechanics and biology in long bone adaptation.     

Virtual structural analysis of tibial fracture healing from low-dose clinical CT scans

Quantitative assessment of bone fracture healing remains a significant challenge in orthopaedic trauma research. Accordingly, we developed a new technique for assessing bone healing using virtual mechano-structural analysis of computed tomography (CT) scans. CT scans from 19 fractured human tibiae at 12 weeks after surgery were segmented and prepared for finite element analysis (FEA). Boundary conditions were applied to the models to simulate a torsion test that is commonly used to access the structural integrity of long bones in animal models of fracture healing. The output of each model was the virtual torsional rigidity (VTR) of the healing zone, normalized to the torsional rigidity of each patient’s virtually reconstructed tibia. This provided a structural measure to track the percentage of healing each patient had undergone. Callus morphometric measurements were also collected from the CT scans. Results showed that at 12 weeks post-op, more than 75% of patients achieved a normalized VTR (torsional rigidity relative to uninjured bone) of 85% or above. The predicted intact torsional rigidities compared well with published cadaveric data. Across all patients, callus volume and density were weakly and non-significantly correlated with normalized VTR and time to clinical union. Conversely, normalized VTR was significantly correlated with time to union (R² = 0.383, p = 0.005). This suggests that fracture scoring methods based on the visual appearance of callus may not accurately predict mechanical integrity. The image-based structural analysis presented here may be a useful technique for assessment of bone healing in orthopaedic trauma research.     

The effect of defect size on the stress concentration and fracture characteristics for a tubular torsional model with a transverse hole

The maximum stress location and crack resistance of a tubular torsional model with varying transverse circular defects were determined by the use of experimental and global-local finite element modeling techniques. The experimental results showed that the reduction in torsional strength was inversely proportional to defect size. In addition, the maximum stress location around the defect was closely related to the normalized defect diameter. By measuring the shifted angle associated with each defect ratio, a linear relationship, delta theta = -6.28 + 0.55*(d/D), was determined. Finite element results indicated that the stress concentration factor, Kg, for a single-cortex defect is similar to that of a double-cortex defect of identical dimension. Application of the strain energy density (SED) theory proposed by Sih and Oliveira Faria (Fracture Mechanics Methodology, Martinus Nijhoff, The Hague, 1984), indicated that the fracture toughness, KIC, for large defects was greater than that for small defects. This implies that tubular structures with large defects have a greater resistance to crack initiation and growth.