A three-dimensional finite element analysis of the upper tibia

A three-dimensional finite element model of the proximal tibia has been developed to provide a base line for further modeling of prosthetic resurfaced tibiae. The geometry for the model was developed by digitizing coronal and transverse sections made with the milling machine, from one fresh tibia of average size. The load is equally distributed between the medial and lateral compartments over contact areas that were reported in the literature. An indentation test has been used to measure the stiffness and the ultimate strength of cancellous bone in four cadaver tibiae. These values provided the statistical basis for characterising the inhomogeneous distribution of the cancellous bone properties in the proximal tibia. All materials in the model were assumed to be linearly elastic and isotropic. Mechanical properties for the cortical bone and cartilage have been taken from the literature. Results have been compared with strain gage tests and with a two-dimensional axisymmetric finite element model both from the literature. Qualitative comparison between trabecular alignment, and the direction of the principal compressive stresses in the cancellous bone, showed a good relationship. Maximum stresses in the cancellous bone and cortical bone, under a load which occurs near stance phase during normal gait, show safety factors of approximately eight and twelve, respectively. The load sharing between the cancellous bone and the cortical bone has been plotted for the first 40 mm distally from the tibial eminence.     

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

Changes in the Fabric and Compliance Tensors of Cancellous Bone due to Trabecular Surface Remodeling,Predicted by a Digital Image-based Model

Fabric and compliance tensors of a cube of cancellous bone with a complicated three-dimensional trabecular structure were obtained for trabecular surface remodeling by using a digital image-based model combined with a large-scale finite element method. Using mean intercept length and a homogenization method, the fabric and compliance tensors were determined for the trabecular structure obtained in the computer remodeling simulation. The tensorial quantities obtained indicated that anisotropic structural changes occur in cancellous bone adapting to the compressive loading condition. There were good correlations between the fabric tensor, bone volume fraction, and compliance tensor in the remodeling process. The result demonstrates that changes in the structural and mechanical properties of cancellous bone are essentially anisotropic and should be expressed by tensorial quantities.     

Changes in the fabric and compliance tensors of cancellous bone due to trabecular surface remodeling, predicted by a digital image-based model

Fabric and compliance tensors of a cube of cancellous bone with a complicated three-dimensional trabecular structure were obtained for trabecular surface remodeling by using a digital image-based model combined with a large-scale finite element method. Using mean intercept length and a homogenization method, the fabric and compliance tensors were determined for the trabecular structure obtained in the computer remodeling simulation. The tensorial quantities obtained indicated that anisotropic structural changes occur in cancellous bone adapting to the compressive loading condition. There were good correlations between the fabric tensor, bone volume fraction, and compliance tensor in the remodeling process. The result demonstrates that changes in the structural and mechanical properties of cancellous bone are essentially anisotropic and should be expressed by tensorial quantities.     

Finite Element Analysis of Simulations of Cancellous Bone Resorption

A stochastic simulation of the resorption of cancellous bone has been developed and integrated with a finite element model to predict the resultant change in structural properties of bone as bone density decreases. The resorption represents the net imbalance of osteoclast and osteoblast activity that occurs in osteoporosis. A simple lattice structure of trabecular bone is considered, with an examination of the lattice geometry and discretization indicating that just five trabeculae need to be modelled. The results from the analysis show how the mechanical properties of the cancellous bone degrade with osteoporosis and demonstrate how the method can be used to predict the relationships between stiffness and density or porosity.     

Innervation of the human upper primary dentition: Implications for understanding tooth initiation and rethinking growth and eruption patterns

Recent comparisons of humans with apes and early fossil hominids have prompted renewed interest in the study of sequences of dental growth and development. Such comparisons, however, rely on certain assumptions about tooth development and dental homology and the biological reality of distinguishing “deciduous” from “permanent” teeth. In light of earlier suggestions by Schwartz that there might be a correlation between nerves and the stem progenitors of tooth classes, and thus between nerve branch number and number of tooth classes, we studied a large sample of ～ 3 month fetuses to elucidate the nature of nerve branching patterns and the development of the primary dentition (i.e., the “deciduous” incisors, canine, and molars, and the first “permanent” molar). Contrary to expectation, variation in nerve branch patterning was the rule. If nerve fibers do have a role in tooth development, it can only be at the time of initiation, with definitive innervation occurring late in tooth development. In taking into consideration the entire span of tooth development—from initiation to innervation to eruption—and the process by which successional teeth arise (each from the external dental epithelium of a predecessor tooth), we suggest that dividing tooth growth and eruption into patterns of the “deciduous” teeth vs. those of the “permanent” is artificial and that a more meaningful approach would be the study of the entire dentition.     

On the influence of mechanical conditions in osteochondral defect healing

Despite the introduction of new surgical techniques, the treatment of cartilage defects remains challenging. Delay or complete failure of cartilage healing is associated with problems in biological regeneration. The influence of mechanical conditions on this process, however, remains unevaluated. Osteochondral defects were generated on the left femoral condyle in 18 Yucatan minipigs. After 4, 6 and 12 weeks the defect filling, trabecular orientation and bone density were compared to the intact contralateral side. The mechanical straining during this period was then analyzed using an adaptive finite element technique. Histologically, the osteochondral defects showed bone resorption at the base and bone formation from the circumference. At 12 weeks, the macroscopically healed specimens showed fibrous cartilage formation, a minimally organized trabecular structure and increased trabecular volume fraction compared to the controls (p < 0.002). The amount of cancellous, cartilagineous, and fibrous tissue and the defect size as measured in histomorphometric analysis for the three time points (4, 6 and 12 weeks) was comparable in magnitude to that predicted by finite element analysis. The simulated osteochondral healing process was not fully capable of re-establishing a hyaline-like cartilage layer. The correlation between simulation and histology allows identification of mechanical factors that appear to have a larger impact on the healing of osteochondral defects than previously considered.     

Finite element analysis of dental implant loading on atrophic and non-atrophic cancellous and cortical mandibular bone – a feasibility study

The first aim of this study was to assess displacements and micro-strain induced on different grades of atrophic cortical and trabecular mandibular bone by axially loaded dental implants using finite element analysis (FEA). The second aim was to assess the micro-strain induced by different implant geometries and the levels of bone-to-implant contact (BIC) on the surrounding bone. Six mandibular bone segments demonstrating different grades of mandibular bone atrophy and various bone volume fractions (from 0.149 to 0.471) were imaged using a micro-CT device. The acquired bone STL models and implant (Brånemark, Straumann, Ankylos) were merged into a three-dimensional finite elements structure. The mean displacement value for all implants was 3.1±1.2 µm. Displacements were lower in the group with a strong BIC. The results indicated that the maximum strain values of cortical and cancellous bone increased with lower bone density. Strain distribution is the first and foremost dependent on the shape of bone and architecture of cancellous bone. The geometry of the implant, thread patterns, grade of bone atrophy and BIC all affect the displacement and micro-strain on the mandible bone. Preoperative finite element analysis could offer improved predictability in the long-term outlook of dental implant restorations.     

Interstitial fluid flow in canaliculi as a mechanical stimulus for cancellous bone remodeling: in silico validation

Cancellous bone has a dynamic 3-dimensional architecture of trabeculae, the arrangement of which is continually reorganized via bone remodeling to adapt to the mechanical environment. Osteocytes are currently believed to be the major mechanosensory cells and to regulate osteoclastic bone resorption and osteoblastic bone formation in response to mechanical stimuli. We previously developed a mathematical model of trabecular bone remodeling incorporating the possible mechanisms of cellular mechanosensing and intercellular communication in which we assumed that interstitial fluid flow activates the osteocytes to regulate bone remodeling. While the proposed model has been validated by the simulation of remodeling of a single trabecula, it remains unclear whether it can successfully represent in silico the functional adaptation of cancellous bone with its multiple trabeculae. In the present study, we demonstrated the response of cancellous bone morphology to uniaxial or bending loads using a combination of our remodeling model with the voxel finite element method. In this simulation, cancellous bone with randomly arranged trabeculae remodeled to form a well-organized architecture oriented parallel to the direction of loading, in agreement with the previous simulation results and experimental findings. These results suggested that our mathematical model for trabecular bone remodeling enables us to predict the reorganization of cancellous bone architecture from cellular activities. Furthermore, our remodeling model can represent the phenomenological law of bone transformation toward a locally uniform state of stress or strain at the trabecular level.     

The discrete nature of trabecular bone microarchitecture affects implant stability

Small endosseous implants, such as screws, are important components of modern orthopedics and dentistry. Hence they have to reliably fulfill a variety of requirements, which makes the development of such implants challenging. Finite element analysis is a widely used computational tool used to analyze and optimize implant stability in bone. For these purposes, bone is generally modeled as a continuum material. However, bone failure and bone adaptation processes are occurring at the discrete level of individual trabeculae; hence the assessment of stresses and strains at this level is relevant. Therefore, the aim of the present study was to investigate how peri-implant strain distribution and load transfer between implant and bone are affected by the continuum assumption. We performed a computational study in which cancellous screws were inserted in continuum and discrete models of trabecular bone; axial loading was simulated. We found strong differences in bone-implant stiffness between the discrete and continuum bone model. They depended on bone density and applied boundary conditions. Furthermore, load transfer from the screw to the surrounding bone differed strongly between the continuum and discrete models, especially for low-density bone. Based on our findings we conclude that continuum bone models are of limited use for finite element analysis of peri-implant mechanical loading in trabecular bone when a precise quantification of peri-implant stresses and strains is required. Therefore, for the assessment and improvement of trabecular bone implants, finite element models which accurately represent trabecular microarchitecture should be used.     

Trabecular shear stress in human vertebral cancellous bone: intra- and inter-individual variations

Correlation of the mean and standard deviation of trabecular stresses has been proposed as a mechanism by which a strong relationship between the apparent strength and stiffness of cancellous bone can be achieved. The current study examined whether the relationship between the mean and standard deviation of trabecular von Mises stresses can be generalized for any group of cancellous bone. Cylindrical human vertebral cancellous bone specimens were cut in the infero-superior direction from T12 of 23 individuals (inter-individual group). Thirty nine additional specimens were prepared similarly from the T4-T12 and L2-L5 vertebrae of a 63 year old male (intra-individual group). The specimens were scanned by micro-computed tomography (microCT) and trabecular von Mises stresses were calculated using finite element modeling. The expected value, standard deviation and coefficient of variation of the von Mises stress were calculated form a three-parameter Weibull function fitted to von Mises stress data from each specimen. It was found that the average and standard deviation of trabecular von Mises shear stress were: (i) correlated with each other, supporting the idea that high correlation between the apparent strength and stiffness of cancellous bone can be achieved through controlling the trabecular level shear stress variations, (ii) dependent on anatomical site and sample group, suggesting that the variation of stresses are correlated to the mean stress to different degrees between vertebrae and individuals, and (iii) dependent on bone volume fraction, consistent with the idea that shear stress is less well controlled in bones with low BV/TV. The conversion of infero-superior loading into trabecular von Mises stresses was maximum for the tissue at the junction of the thoracic and lumbar spine (T12-L1) consistent with this junction being a common site of vertebral fracture.     

Ontogeny and homology of the dentition in dasyurid marsupials: Development inSminthopsis virginiae

Histological analysis of an ontogenetic series of the dasyurid marsupial,Sminthopsis virginiae, from birt to 60 days old, was undertaken to assess the developmental homologies of the deciduous and successional teeth. This period covers the time from the initiation of all teeth as epithelial buds up until the time of early eruption of some teeth. In addition, two older specimens, aged 81 and 97 days, were examined to provide additional information on the state of differentiation of the unerupted third premolar. In the postcanine dentition, only a single tooth position, dP3, was characterized by the later development of a replacing successional tooth (P3), following developmental pathways identical to those in eutherian mammals. In contrast, the anterior dentition is characterized by the formation of rudimentary, nonerupting deciduous incisors and canines, and by the accelerated development of normal, erupting successional incisors and canines in both jaws. Comparison of relative developmental stages for each tooth position throughout its preeruptive ontogeny suggests thatheterochrony (both developmental acceleration and retardation) has played an important role in the evolutionary history of the dasyurid dentition. Differing aspects of this phenomenon are identified and discussed for the anterior dentition, the anterior two premolars, P3, and the lower molars. Further evidence is presented to corroborate the identification of the anterior two premolars in the adult as dP1 and dP2, based on the relative retardation of their initiation and their lack of successor tooth germs. This developmental heterochrony has probably occurred in all three-premolared marsupials.     

The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone.

Conceptually, the elastic characteristics of cancellous bone could be predicted directly from the trabecular morphology--or architecture--and by the elastic properties of the tissue itself. Although hardly any experimental evidence exists, it is often implicitly assumed that tissue anisotropy has a negligible effect on the apparent elastic properties of cancellous bone. The question addressed in this paper is whether this is actually true. If it is, then micromechanical finite element analysis (micro-FEA) models, representing trabecular architecture, using an 'effective isotropic tissue modulus' should be able to predict apparent elastic properties of cancellous bone. To test this, accurate multi-axial compressive mechanical tests of 29 whale bone specimens were simulated with specimen-specific micro-FEA computer models built from true three-dimensional reconstructions. By scaling the micro-FEA predictions by a constant tissue modulus, 92% of the variation of Young's moduli determined experimentally could be explained. The correlation even increased to 95% when the micro-FEA moduli were scaled to the isotropic tissue moduli of individual specimens. Excellent agreement was also found in the elastic symmetry axes and anisotropy ratios. The prediction of Poisson's ratios was somewhat less precise at 85% correlation. The results support the hypothesis; for practical purposes, the concept of an 'effective isotropic tissue modulus' concept is a viable one. They also suggest that the value of such a modulus for individual cases might be inferred from the average tissue density, hence the degree of mineralization. Future studies must clarify how specific the tissue modulus should be for different types of bone if adequate predictions of elastic behavior are to be made in this way.     

Trabecular surface remodeling simulation for cancellous bone using microstructural voxel finite element models

A computational simulation method for three-dimensional trabecular surface remodeling was proposed, using voxel finite element models of cancellous bone, and was applied to the experimental data. In the simulation, the trabecular microstructure was modeled based on digital images, and its morphological changes due to surface movement at the trabecular level were directly expressed by removing/adding the voxel elements from/to the trabecular surface. A remodeling simulation at the single trabecular level under uniaxial compressive loading demonstrated smooth morphological changes even though the trabeculae were modeled with discrete voxel elements. Moreover, the trabecular axis rotated toward the loading direction with increasing stiffness, simulating functional adaptation to the applied load. In the remodeling simulation at the trabecular structural level, a cancellous bone cube was modeled using a digital image obtained by microcomputed tomography (microCT), and was uniaxially compressed. As a result, the apparent stiffness against the applied load increased by remodeling, in which the trabeculae reoriented to the loading direction. In addition, changes in the structural indices of the trabecular architecture coincided qualitatively with previously published experimental observations. Through these studies, it was demonstrated that the newly proposed voxel simulation technique enables us to simulate the trabecular surface remodeling and to compare the results obtained using this technique with the in vivo experimental data in the investigation of the adaptive bone remodeling phenomenon.     

Orthotropic properties of cancellous bone modelled as parameterized cellular material

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of “equivalent” trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.     

Orthotropic properties of cancellous bone modelled as parameterized cellular material

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of "equivalent" trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.     

A structural model for the mechanical behavior of trabecular bone

A quantitative model is developed for trabecular bone by approximating the trabecular geometry with a hypothetical network of compact bone. For the region immediately beneath the articular cartilage in the distal end of the femur, finite element analyses were performed with a high speed computer, assuming a physiological static load. The results indicate that bending and buckling of trabeculae are considerable in any elastic deformation of the bone; that fatigue fracture in some fraction of suitably oriented trabeculae is inevitable in normal ambulation; and that the stiffness varies considerably with lateral position across the subchondral plate. The latter depends totally on trabecular arrangement and may play a role in joint function and degeneration. The adjustments necessary to bring the gross stiffness into agreement with experiment imply that the intertrabecular soft tissues are of no consequence to the mechanical properties and that the compact bone of which trabeculae are made is probably not as stiff as cortical bone.     

Functional adaptation of cancellous bone in human proximal femur predicted by trabecular surface remodeling simulation toward uniform stress state

Two-dimensional simulation of trabecular surface remodeling was conducted for a human proximal femur to investigate the structural change of cancellous bone toward a uniform stress state. Considering that a local mechanical stimulus plays an important role in cellular activities in bone remodeling, local stress nonuniformity was assumed to drive trabecular structural change to seek a uniform stress state. A large-scale pixel-based finite element model was used to simulate structural changes of individual trabeculae over the entire bone. As a result, the initial structure of trabeculae changed from isotropic to anisotropic due to trabecular microstructural changes caused by surface remodeling according to the mechanical environment in the proximal femur. Under a single-loading condition, it was shown that the apparent structural property evaluated by fabric ellipses corresponded to the apparent stress state in cancellous bone. As is observed in the actual bone, a distributed trabecular structure was obtained under a multiple-loading condition. Through these studies, it was concluded that trabecular surface remodeling toward a local uniform stress state at the trabecular level could naturally bring about functional adaptation phenomenon at the apparent tissue level. The proposed simulation model would be capable of providing insight into the hierarchical mechanism of trabecular surface remodeling at the microstructural level up to the apparent tissue level.     

NACOB presentation to ASB Young Scientist Award: Postdoctoral. The impact of boundary conditions and mesh size on the accuracy of cancellous bone tissue modulus determination using large-scale finite-element modeling. North American Congress on Biomechanics.

The apparent properties of cancellous bone are determined by a combination of both hard tissue properties and microstructural organization. A method is desired to extract the underlying hard tissue properties from simple mechanical tests, free from the complications of microstructure. It has been suggested that microCT voxel-based large-scale finite element models could be employed to accomplish this goal (van Rietbergen et al., 1995, Journal of Biomechanics, 28, 69-81). This approach has recently been implemented and it is becoming increasingly popular as finite element models increase in size and sophistication (Fyhrie et al., 1997, Proceedings of the 43rd Annual Meeting of the Orthopaedic Research Society, San Francisco, CA, p. 815; van Rietbergen et al., 1997, Proceedings of the 43rd Annual Meeting of the Orthopaedic Research Society, San Francisco, CA, p. 62). However, no direct quantitative measurements of the accuracy of this method applied to porous structures such as cancellous bone have been made. This project demonstrates the feasibility of this approach by quantifying its best-case accuracy in determining the trabecular hard tissue modulus of analogues fabricated of a material with known material properties determined independently by direct testing. In addition we were able to assess the impact of mesh size and boundary conditions on accuracy. We found that the assumption of a frictionless boundary condition in the parallel plate compression loading configuration was a significant source of error that could be overcome with the use of rigid end-caps similar to those used by Keaveny et al. (1997 Journal of Orthopaedic Research, 15(1), 101-110). In conclusion, we found that this approach is an effective method for determining the average trabecular hard tissue properties of human cancellous bone with an expected practical accuracy level better than 5%.     

Elastic properties of cancellous bone derived from finite element models of parameterized microstructure cells

Evaluation of constitutive properties of cancellous bone and their relationships to microstructural parameters is a crucial issue in analysis of stresses and strains in bone tissues and simulation of their remodelling. Known limitations of experimental methods as well as of the micro-FE techniques make the analysis and homogenization of 'equivalent' trabecular microstructures an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. Two cell types are analysed: cube- and prism-based. The models are fully three-dimensional, have realistic curvilinear shapes and are parameterized with three shape parameters. Variation of the parameters allows to imitate most of the typical microstructure patterns observed in real bones, along with variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Six static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Multi-point boundary conditions imposed on the models ensure mutual fitting of deformed neighbouring cells. Values of computed stresses allow to determine all coefficients of elastic orthotropic stiffness matrix. Results have a form of tabularized functions of elastic constants versus the shape parameters. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement, though evidently better in the case of the prism-based cell model.