Numerical estimation of bone density and elastic constants distribution in a human mandible

In this paper, we try to predict the distribution of bone density and elastic constants in a human mandible, based on the stress level produced by mastication loads using a mathematical model of bone remodelling. These magnitudes are needed to build finite element models for the simulation of the mandible mechanical behavior. Such a model is intended for use in future studies of the stability of implant-supported dental prostheses. Various models of internal bone remodelling, both phenomenological and more recently mechanobiological, have been developed to determine the relation between bone density and the stress level that bone supports. Among the phenomenological models, there are only a few that are also able to reproduce the level of anisotropy. These latter have been successfully applied to long bones, primarily the femur. One of these models is here applied to the human mandible, whose corpus behaves as a long bone. The results of bone density distribution and level of anisotropy in different parts of the mandible have been compared with various clinical studies, with a reasonable level of agreement.     

Finite element analysis of bone remodelling with piezoelectric effects using an open-source framework

Bone tissue exhibits piezoelectric properties and thus is capable of transforming mechanical stress into electrical potential. Piezoelectricity has been shown to play a vital role in bone adaptation and remodelling processes. Therefore, to better understand the interplay between mechanical and electrical stimulation during these processes, strain-adaptive bone remodelling models without and with considering the piezoelectric effect were simulated using the Python-based open-source software framework. To discretise numerical attributes, the finite element method (FEM) was used for the spatial variables and an explicit Euler scheme for the temporal derivatives. The predicted bone apparent density distributions were qualitatively and quantitatively evaluated against the radiographic scan of a human proximal femur and the bone apparent density calculated using a bone mineral density (BMD) calibration phantom, respectively. Additionally, the effect of the initial bone density on the resulting predicted density distribution was investigated globally and locally. The simulation results showed that the electrically stimulated bone surface enhanced bone deposition and these are in good agreement with previous findings from the literature. Moreover, mechanical stimuli due to daily physical activities could be supported by therapeutic electrical stimulation to reduce bone loss in case of physical impairment or osteoporosis. The bone remodelling algorithm implemented using an open-source software framework facilitates easy accessibility and reproducibility of finite element analysis made.

     

The many adaptations of bone

Studies concerned with the "adaptations" in bones usually deal with modelling taking place during the individual's lifetime. However, many adaptations are produced over evolutionary time. This survey samples some adaptations of bone that may occur over both length scales, and tries to show whether short- or long-term adaptation is important. (a) Woven and lamellar bone. Woven bone is less mechanically competent than lamellar bone but is frequently found in bones that grow quickly. (b) Stress concentrations in bone. Bone is full of cavities that potentially may act as stress concentrators. Usually these cavities are oriented to minimise their stress-concentrating effect. Furthermore, the "flow" of lamellae round the cavities will still further reduce their stress-concentrating effect, but the elastic anisotropy of bone will, contrarily, tend to enhance it in normal loading situations. (c) Stiffness versus toughness. The mineral content of bone is the main determinant of differences in mechanical properties. Different bones have different mineral contents that optimise the mix of stiffness and toughness needed. (d) Synergy of whole bone architecture and material properties. As bone material properties change during growth the architecture of the whole bone is modified concurrently, to produce an optimum mechanical behaviour of the whole bone. (e) Secondary remodelling. The formation of secondary osteones in general weakens bone. Various suggestions that have been put forward to account for secondary remodelling: enabling mineral homeostasis; removing dead bone; changing the grain of the bone; taking out microcracks. (f) The hollowness of bones. It is shown how the degree of hollowness is adapted to the life of the animal.     

A meshless microscale bone tissue trabecular remodelling analysis considering a new anisotropic bone tissue material law

In this work, a novel anisotropic material law for the mechanical behaviour of the bone tissue is proposed. This new law, based on experimental data, permits to correlate the bone apparent density with the obtained level of stress. Combined with the proposed material law, a biomechanical model for predicting bone density distribution was developed, based on the assumption that the bone structure is a gradually self-optimising anisotropic biological material that maximises its own structural stiffness. The strain and the stress field required in the iterative remodelling process are obtained by means of an accurate meshless method, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). Comparing with other numerical approaches, the inclusion of the NNRPIM presents numerous advantages such as the high accuracy and the smoother stress and strain field distribution. The natural neighbour concept permits to impose organically the nodal connectivity and facilitates the analysis of convex boundaries and extremely irregular meshes. The viability and efficiency of the model were tested on several trabecular benchmark patch examples. The results show that the pattern of the local bone apparent density distribution and the anisotropic bone behaviour predicted by the model for the microscale analysis are in good agreement with the expected structural architecture and bone apparent density distribution.     

On the Use of Bone Remodelling Models to Estimate the Density Distribution of Bones. Uniqueness of the Solution

Bone remodelling models are widely used in a phenomenological manner to estimate numerically the distribution of apparent density in bones from the loads they are daily subjected to. These simulations start from an arbitrary initial distribution, usually homogeneous, and the density changes locally until a bone remodelling equilibrium is achieved. The bone response to mechanical stimulus is traditionally formulated with a mathematical relation that considers the existence of a range of stimulus, called dead or lazy zone, for which no net bone mass change occurs. Implementing a relation like that leads to different solutions depending on the starting density. The non-uniqueness of the solution has been shown in this paper using two different bone remodelling models: one isotropic and another anisotropic. It has also been shown that the problem of non-uniqueness is only mitigated by removing the dead zone, but it is not completely solved unless the bone formation and bone resorption rates are limited to certain maximum values.     

Comparison of HR-pQCT- and microCT-based finite element models for the estimation of the mechanical properties of the calcaneus trabecular bone

The calcaneus bone is formed of extensive trabecular bone and is therefore well suited to be used as an example of loaded bone to establish the ability of combining microfinite element (microFE) technique with high-resolution peripheral quantitative computed tomography (HR-pQCT) in determining its mechanical properties. HR-pQCT is increasingly used as a tool for in vivo bone clinical research, but its use has been limited to the distal radius and tibia. The goal of this study was to determine the applicability of HR-pQCT-derived microFE models of the calcaneus trabecular bone with 82 μm voxel size with reference to higher-resolution microCT-based models taken as gold standard. By comparing the outputs of microFE models generated from both HR-pQCT and microCT images of the trabecular bone of five calcaneus cadaveric specimens, it was found that the HR-pQCT-based models predicted mechanical properties for fracture load, total reaction force and von Mises stress are considerably different from microCT-based counterparts by 33, 64 and 70%, respectively. Also, the morphological analysis showed a comprehensive geometrical difference between HR-pQCT-based microFE models and their microCT-based equivalents. The results of the HR-pQCT-based models were found to have strong dependency on the threshold value chosen to binarise the images prior to finite element modelling. In addition, it was found that the voxel size has a strong impact on accuracy of imaged-based microFE models compared to other factors such as the presence of soft tissue and image scanning integration time. Therefore, although HR-pQCT has shown to be useful to predict overall structural and biomechanical changes, it is limited in providing local accurate biomechanical properties of trabecular bone and therefore should be used with caution when assessing bone remodelling through local changes of trabecular bone apposition and resorption in disease treatment monitoring.     

Compressive axial mechanical properties of rat bone as functions of bone volume fraction,apparent density and micro-ct based mineral density

Mechanical testing has been regarded as the gold standard to investigate the effects of pathologies on the structure–function properties of the skeleton. With recent advances in computing power of personal computers, virtual alternatives to mechanical testing are gaining acceptance and use. We have previously introduced such a technique called structural rigidity analysis to assess mechanical strength of skeletal tissue with defects. The application of this technique is predicated upon the use of relationships defining the strength of bone as a function of its density for a given loading mode. We are to apply this technique in rat models to assess their compressive skeletal response subjected to a host of biological and pharmaceutical stimulations. Therefore, the aim of this study is to derive a relationship expressing axial compressive mechanical properties of rat cortical and cancellous bone as a function of equivalent bone mineral density, bone volume fraction or apparent density over a range of normal and pathologic bones.We used bones from normal, ovariectomized and partially nephrectomized animals. All specimens underwent micro-computed tomographic imaging to assess bone morphometric and densitometric indices and uniaxial compression to failure.We obtained univariate relationships describing 71–78% of the mechanical properties of rat cortical and cancellous bone based on equivalent mineral density, bone volume fraction or apparent density over a wide range of density and common skeletal pathologies. The relationships reported in this study can be used in the structural rigidity analysis introduced by the authors to provide a non-invasive method to assess the compressive strength of bones affected by pathology and/or treatment options.     

Computational simulation of internal bone remodelling around dental implants: a sensitivity analysis

This study aimed to predict the distribution of bone trabeculae, as a density change per unit time, around a dental implant based on applying a selected mathematical remodelling model. The apparent bone density change as a function of the mechanical stimulus was the base of the applied remodelling model that describes disuse and overload bone resorption. The simulation was tested in a finite element model of a screw-shaped dental implant in an idealised bone segment. The sensitivity of the simulation to different mechanical parameters was investigated; these included element edge length, boundary conditions, as well as direction and magnitude of the implant loads. The alteration in the mechanical parameters had a significant influence on density distribution and model stability, in particular at the cortical bone region. The remodelling model could succeed to achieve trabeculae-like structure around osseointegrated dental implants. The validation of this model to a real clinical case is required.     

Computational simulation of internal bone remodelling around dental implants: a sensitivity analysis

This study aimed to predict the distribution of bone trabeculae, as a density change per unit time, around a dental implant based on applying a selected mathematical remodelling model. The apparent bone density change as a function of the mechanical stimulus was the base of the applied remodelling model that describes disuse and overload bone resorption. The simulation was tested in a finite element model of a screw-shaped dental implant in an idealised bone segment. The sensitivity of the simulation to different mechanical parameters was investigated; these included element edge length, boundary conditions, as well as direction and magnitude of the implant loads. The alteration in the mechanical parameters had a significant influence on density distribution and model stability, in particular at the cortical bone region. The remodelling model could succeed to achieve trabeculae-like structure around osseointegrated dental implants. The validation of this model to a real clinical case is required.     

Subject-specific bone remodelling of the scapula

Finite element analyses, with increasing levels of detail and complexity, are becoming effective tools to evaluate the performance of joint replacement prostheses and to predict the behaviour of bone. As a first step towards the study of the complications of shoulder arthroplasty, the aim of this work was the development and validation of a 3D finite element model of an intact scapula for the prediction of the bone remodelling process based on a previously published model that attempts to follow Wolff's law. The boundary conditions applied include full muscle and joint loads taken from a multibody system of the upper limb based on the same subject whose scapula was here analysed. To validate the bone remodelling simulations, qualitative and quantitative comparisons between the predicted and the specimen's bone density distribution were performed. The results showed that the bone remodelling model was able to successfully reproduce the actual bone density distribution of the analysed scapula.     

A coupled mechano-biochemical model for bone adaptation

Bone remodelling is a fundamental biological process that controls bone microrepair, adaptation to environmental loads and calcium regulation among other important processes. It is not surprising that bone remodelling has been subject of intensive both experimental and theoretical research. In particular, many mathematical models have been developed in the last decades focusing in particular aspects of this complicated phenomenon where mechanics, biochemistry and cell processes strongly interact. In this paper, we present a new model that combines most of these essential aspects in bone remodelling with especial focus on the effect of the mechanical environment into the biochemical control of bone adaptation mainly associated to the well known RANKL-RANK-OPG pathway. The predicted results show a good correspondence with experimental and clinical findings. For example, our results indicate that trabecular bone is more severely affected both in disuse and disease than cortical bone what has been observed in osteoporotic bones. In future, the methodology proposed would help to new therapeutic strategies following the evolution of bone tissue distribution in osteoporotic patients.     

Effects of thresholding techniques on microCT-based finite element models of trabecular bone

Microimaging based finite element analysis is widely used to predict the mechanical properties of trabecular bone. The choice of thresholding technique, a necessary step in converting grayscale images to finite element models, can significantly influence the predicted bone volume fraction and mechanical properties. Therefore, we investigated the effects of thresholding techniques on microcomputed tomography (micro-CT) based finite element models of trabecular bone. Three types of thresholding techniques were applied to 16-bit micro-CT images of trabecular bone to create three different models per specimen. Bone volume fractions and apparent moduli were predicted and compared to experimental results. In addition, trabecular tissue mechanical parameters and morphological parameters were compared among different models. Our findings suggest that predictions of apparent mechanical properties and structural properties agree well with experimental measurements regardless of the choice of thresholding methods or the format of micro-CT images.     

Incorporating tissue anisotropy and heterogeneity in finite element models of trabecular bone altered predicted local stress distributions

Trabecular bone is composed of organized mineralized collagen fibrils, which results in heterogeneous and anisotropic mechanical properties at the tissue level. Recently, biomechanical models computing stresses and strains in trabecular bone have indicated a significant effect of tissue heterogeneity on predicted stresses and strains. However, the effect of the tissue-level mechanical anisotropy on the trabecular bone biomechanical response is unknown. Here, a computational method was established to automatically impose physiologically relevant orientation inherent in trabecular bone tissue on a trabecular bone microscale finite element model. Spatially varying tissue-level anisotropic elastic properties were then applied according to the bone mineral density and the local tissue orientation. The model was used to test the hypothesis that anisotropy in both homogeneous and heterogeneous models alters the predicted distribution of stress invariants. Linear elastic finite element computations were performed on a 3 mm cube model isolated from a microcomputed tomography scan of human trabecular bone from the distal femur. Hydrostatic stress and von Mises equivalent stress were recorded at every element, and the distributions of these values were analyzed. Anisotropy reduced the range of hydrostatic stress in both tension and compression more strongly than the associated increase in von Mises equivalent stress. The effect of anisotropy was independent of the spatial redistribution high compressive stresses due to tissue elastic heterogeneity. Tissue anisotropy and heterogeneity are likely important mechanisms to protect bone from failure and should be included for stress analyses in trabecular bone.     

A contact model with ingrowth control for bone remodelling around cementless stems.

This work presents a computational model for bone remodelling around cementless stems. The problem is formulated as a material optimisation problem considering the bone and stem surfaces to be in contact. To emphasise the behaviour of the bone/stem interface, the computer model detects the existence of bone ingrowth during the remodelling; consequently, the contact conditions are changed for a better interface simulation. The trabecular bone is modelled as a strictly orthotropic material with equivalent properties computed by homogenisation. The distribution of bone relative density is obtained by the minimisation of a function that considers both the bone structural stiffness and the biological cost associated with metabolic maintenance of bone tissue. The situation of multiple load conditions is considered. The remodelling law, obtained from the necessary conditions for an optimum, is derived analytically from the optimisation problem and solved numerically using a suitable finite element mesh. The formulation is applied to an implanted femur. Results of bone density and ingrowth distribution are obtained for different coating conditions. Bone ingrowth does not occur over the entire coated surfaces. Indeed, we observed regions where separation or high relative displacement occurs that preclude bone ingrowth attachment. This prediction of the model is consistent with clinical observations of bone ingrowth. Thus, this model, which detect bone ingrowth and allow modification of the interface conditions, are useful for analysis of existing stems as well as design optimisation of coating extent and location on such stems.     

Biomimesis and biomorphic transformations: new concepts applied to bone regeneration

In the last decades the activity of material scientists was more and more directed to the development of biomimetic scaffolds, able to drive and address cell activity towards proper differentiation and the repair of diseased human tissues. In case of bone, this requires the synthesis of three-dimensional constructs able to exchange chemical signals promoting osteogenesis and to progressively be resorbed during the formation and remodelling of new bone. Besides, particularly for the regeneration of extensive portions of bone, a morphological and mechanical biomimesis is also required, to allow cell colonization and formation of a proper vascularization tree. The healing of load-bearing bones also requires scaffolds with a hierarchically organized morphology, to provide improved biomechanical behaviour and allow a proper mechano-transduction of the mechanical stimuli down to the cell level. The present paper is an overview of the current technologies and devices developed in the last decade for the regeneration of bone tissue. In particular, novel biomimetic and biomorphic scaffolds, obtained by the controlled transformation of native ligneous structures, promise to adequately face the problem of obtaining complex hierarchical structures, not achievable otherwise by any currently existing manufacturing techniques.     

Biomimesis and biomorphic transformations: New concepts applied to bone regeneration

In the last decades the activity of material scientists was more and more directed to the development of biomimetic scaffolds, able to drive and address cell activity towards proper differentiation and the repair of diseased human tissues. In case of bone, this requires the synthesis of three-dimensional constructs able to exchange chemical signals promoting osteogenesis and to progressively be resorbed during the formation and remodelling of new bone. Besides, particularly for the regeneration of extensive portions of bone, a morphological and mechanical biomimesis is also required, to allow cell colonization and formation of a proper vascularization tree. The healing of load-bearing bones also requires scaffolds with a hierarchically organized morphology, to provide improved biomechanical behaviour and allow a proper mechano-transduction of the mechanical stimuli down to the cell level. The present paper is an overview of the current technologies and devices developed in the last decade for the regeneration of bone tissue. In particular, novel biomimetic and biomorphic scaffolds, obtained by the controlled transformation of native ligneous structures, promise to adequately face the problem of obtaining complex hierarchical structures, not achievable otherwise by any currently existing manufacturing techniques.     

Fundamental subchondral bone changes in spontaneous knee osteoarthritis

Osteoarthritis has an unknown aetiology, and tissue samples from early stage human osteoarthritis tissue cannot be reliably obtained. Therefore understanding the development of OA relies on using animal models: such as the spontaneous changes seen in the Dunkin-Hartley guinea pig strain, which are biochemically, histologically and radiologically similar to human OA. We investigated the role of bone change in early OA development using the non-OA developing Bristol strain-2 as control from 3 to 36 weeks by standard microfocal X-ray imaging and histological techniques. The patella, tibia and femur epiphyseal region and immediate subchondral area were analysed for bone density at all ages. We found that both radiological and histological osteoarthritis scores increased progressively for the Dunkin-Hartley, but not for the BS2 demonstrating its value as a control. The Dunkin-Hartley had a higher bone density and greater subchondral bone thickness from 24 weeks of age. We conclude that prior to any gross osteoarthritis pathology the Dunkin-Hartley are undergoing subchondral bone remodelling, thus demonstrating the fundamental role of early bone remodelling in the development of osteoarthritis.     

Effect of including damage at the tissue level in the nonlinear homogenisation of trabecular bone

Being able to predict bone fracture or implant stability needs a proper constitutive model of trabecular bone at the macroscale in multiaxial, non-monotonic loading modes. Its macroscopic damage behaviour has been investigated experimentally in the past, mostly with the restriction of uniaxial cyclic loading experiments for different samples, which does not allow for the investigation of several load cases in the same sample as damage in one direction may affect the behaviour in other directions. Homogenised finite element models of whole bones have the potential to assess complicated scenarios and thus improve clinical predictions. The aim of this study is to use a homogenisation-based multiscale procedure to upscale the damage behaviour of bone from an assumed solid phase constitutive law and investigate its multiaxial behaviour for the first time. Twelve cubic specimens were each submitted to nine proportional strain histories by using a parallel code developed in-house. Evolution of post-elastic properties for trabecular bone was assessed for a small range of macroscopic plastic strains in these nine load cases. Damage evolution was found to be non-isotropic, and both damage and hardening were found to depend on the loading mode (tensile, compression or shear); both were characterised by linear laws with relatively high coefficients of determination. It is expected that the knowledge of the macroscopic behaviour of trabecular bone gained in this study will help in creating more precise continuum FE models of whole bones that improve clinical predictions.     

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.