A density distribution algorithm for bone incorporating local orthotropy,modal analysis and theories of cellular solids

An algorithm for bone remodeling is presented which allows for both a redistribution of density and a continuous change of principal material directions for the orthotropic material properties of bone. It employs a modal analysis to add density for growth and a local effective strain based analysis to redistribute density. General re-distribution functions are presented. The model utilizes theories of cellular solids to relate density and strength. The code predicts the same general density distributions and local orthotropy as observed in reality.     

Numerical modeling of bone tissue adaptation—A hierarchical approach for bone apparent density and trabecular structure

In this work, a three-dimensional model for bone remodeling is presented, taking into account the hierarchical structure of bone. The process of bone tissue adaptation is mathematically described with respect to functional demands, both mechanical and biological, to obtain the bone apparent density distribution (at the macroscale) and the trabecular structure (at the microscale). At global scale bone is assumed as a continuum material characterized by equivalent (homogenized) mechanical properties. At local scale a periodic cellular material model approaches bone trabecular anisotropy as well as bone surface area density. For each scale there is a material distribution problem governed by density-based design variables which at the global level can be identified with bone relative density. In order to show the potential of the model, a three-dimensional example of the proximal femur illustrates the distribution of bone apparent density as well as microstructural designs characterizing both anisotropy and bone surface area density. The bone apparent density numerical results show a good agreement with Dual-energy X-ray Absorptiometry (DXA) exams. The material symmetry distributions obtained are comparable to real bone microstructures depending on the local stress field. Furthermore, the compact bone porosity is modeled giving a transversal isotropic behavior close to the experimental data. Since, some computed microstructures have no permeability one concludes that bone tissue arrangement is not a simple stiffness maximization issue but biological factors also play an important role.     

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.     

Multiscale modeling of bone tissue with surface and permeability control

Natural biological materials usually present a hierarchical arrangement with various structural levels. The biomechanical behavior of the complex hierarchical structure of bone is investigated with models that address the various levels corresponding to different scales. Models that simulate the bone remodeling process concurrently at different scales are in development. We present a multiscale model for bone tissue adaptation that considers the two top levels, whole bone and trabecular architecture. The bone density distribution is calculated at the macroscale (whole bone) level, and the trabecular structure at the microscale level takes into account its mechanical properties as well as surface density and permeability. The bone remodeling process is thus formulated as a material distribution problem at both scales. At the local level, the biologically driven information of surface density and permeability characterizes the trabecular structure. The model is tested by a three-dimensional simulation of bone tissue adaptation for the human femur. The density distribution of the model shows good agreement with the actual bone density distribution. Permeability at the microstructural level assures interconnectivity of pores, which mimics the interconnectivity of trabecular bone essential for vascularization and transport of nutrients. The importance of this multiscale model relays on the flexibility to control the morphometric parameters that characterize the trabecular structure. Therefore, the presented model can be a valuable tool to define bone quality, to assist with diagnosis of osteoporosis, and to support the development of bone substitutes.     

Trabecular bone adaptation with an orthotropic material model.

Most bone adaptation algorithms, that attempt to explain the connection between bone morphology and loads, assume that bone is effectively isotropic. An isotropic material model can explain the bone density distribution, but not the structure and pattern of trabecular bone, which clearly has a mechanical significance. In this paper, an orthotropic material model is utilized to predict the proximal femur trabecular structure. Two hypotheses are combined to determine the local orientation and material properties of each element in the model. First, it is suggested that trabecular directions, which correspond to the orthotropic material axes, are determined locally by the maximal principal stress directions due to the multiple load cases (MLC) the femur is subject to. The second hypothesis is that material properties in each material direction can be determined using directional stimuli, thus extending existing adaptation algorithms to include directionality. An algorithm is utilized, where each iteration comprises of two stages. First, material axes are rotated to the direction of the largest principal stress that occurs from a multiple load scheme applied to the proximal femur. Next, material properties are modified in each material direction, according to a directional stimulus. Results show that local material directions correspond with known trabecular patterns, reproducing all main groups of trabeculae very well. The local directional stiffnesses, degree of anisotropy and density distribution are shown to conform to real femur morphology.     

Material heterogeneity,microstructure, and microcracks demonstrate differential influence on crack initiation and propagation in cortical bone

The recent studies have shown that long-term bisphosphonate use may result in a number of mechanical alterations in the bone tissue including a reduction in compositional heterogeneity and an increase in microcrack density. There are limited number of experimental and computational studies in the literature that evaluated how these modifications affect crack initiation and propagation in cortical bone. Therefore, in this study, the entire crack growth process including initiation and propagation was simulated at the microscale by using the cohesive extended finite element method. Models with homogeneous and heterogeneous material properties (represented at the microscale capturing the variability in material property values and their distribution) as well as different microcrack density and microstructure were compared. The results showed that initiation fracture resistance was higher in models with homogeneous material properties compared to heterogeneous ones, whereas an opposite trend was observed in propagation fracture resistance. The increase in material heterogeneity level up to 10 different material property sets increased the propagation fracture resistance beyond which a decrease was observed while still remaining higher than the homogeneous material distribution. The simulation results also showed that the total osteonal area influenced crack propagation and the local osteonal area near the initial crack affected the crack initiation behavior. In addition, the initiation fracture resistance was higher in models representing bisphosphonate treated bone (low material heterogeneity, high microcrack density) compared to untreated bone models (high material heterogeneity, low microcrack density), whereas an opposite trend was observed at later stages of crack growth. In summary, the results demonstrated that tissue material heterogeneity, microstructure, and microcrack density influenced crack initiation and propagation differently. The findings also elucidate how possible modifications in material heterogeneity and microcrack density due to bisphosphonate treatment may influence the initiation and propagation fracture resistance of cortical bone.     

Prediction of bone density around orthopedic implants delivering bisphosphonate

The fixation of an orthopedic implant depends strongly upon its initial stability. Peri-implant bone may resorb shortly after the surgery. This resorption is directly followed by new bone formation and implants fixation strengthening, the so-called secondary fixation. If the initial stability is not reached, the resorption continues and the implant fixation weakens, which leads to implant loosening. Studies with rats and dogs have shown that a solution to prevent peri-implant resorption is to deliver bisphosphonate from the implant surface.The aims of the study were, first, to develop a model of bone remodeling around an implant delivering bisphosphonate, second, to predict the bisphosphonate dose that would induce the maximal peri-implant bone density, and third to verify in vivo that peri-implant bone density is maximal with the calculated dose.The model consists of a bone remodeling equation and a drug diffusion equation. The change in bone density is driven by a mechanical stimulus and a drug stimulus. The drug stimulus function and the other numerical parameters were identified from experimental data. The model predicted that a dose of 0.3 μg of zoledronate on the implant would induce a maximal bone density. Implants with 0.3 μg of zoledronate were then implanted in rat femurs for 3, 6 and 9 weeks. We measured that peri-implant bone density was 4% greater with the calculated dose compared to the dose empirically described as best.The approach presented in this paper could be used in the design and analysis processes of experiments in local delivery of drug such as bisphosphonate.     

Bone remodelling around a cementless glenoid component

Post-operative change in the mechanical loading of bone may trigger its (mechanically induced) adaptation and hamper the mechanical stability of prostheses. This is especially important in cementless components, where the final fixation is achieved by the bone itself. The aim of this study is, first, to gain insight into the bone remodelling process around a cementless glenoid component, and second, to compare the possible bone adaptation when the implant is assumed to be fully bonded (best case scenario) or completely loose (worst case scenario). 3D finite element models of a scapula with and without a cementless glenoid component were created. 3D geometry of the scapula, material properties, and several physiological loading conditions were acquired from or estimated for a specific cadaver. Update of the bone density after implantation was done according to a node-based bone remodelling scheme. Strain energy density for different loading conditions was evaluated, weighted according to their frequencies in activities of daily life and used as a mechanical stimulus for bone adaptation. The average bone density in the glenoid increased after implantation. However, local bone resorption was significant in some regions next to the bone-implant interface, regardless of the interface condition (bonded or loose). The amount of bone resorption was determined by the condition imposed to the interface, being slightly larger when the interface was loose. An ideal screw, e.g. in which material fatigue was not considered, was enough to keep the interface micromotions small and constant during the entire bone adaptation simulation.     

Computational modal analysis of a composite pelvic bone: convergence and validation studies

The purpose of the present study was to describe the structural density and geometry of the bone, as well as its sensitivity to the resolution of finite element discretisation. The study introduces a novel way to validate biomechanical model of the bone by experimental modal analysis. The structural density and geometry of the model was obtained from a composite bone. A detailed investigation of the weight dependence of the bone on the mesh resolution was performed to obtain the best match with the real weight of the tested bone. The computational model was compared with the experimental results obtained from the modal analysis. The overall changes of the modal properties and bone weight in the model caused by different mesh resolutions and order of approximation were below 10%, despite the bone was modelled with simple isotropic material properties. The experimental modal analysis shows a great potential to be a robust verification tool of computational biomechanical models because it provides boundary conditions–free results. The sensitivity analysis revealed that the linear approximation of the density field is not suitable for the modelling of the modal response of composite bone.     

Tensile strength of bovine trabecular bone

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

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

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

Micromechanical modelling of cortical bone

Cortical bone is a heterogeneous material with a complex hierarchical microstructure. In this work, unit cell finite element models were developed to investigate the effect of microstructural morphology on the macroscopic properties of cortical bone. The effect of lacunar and vascular porosities, percentage of osteonal bone and orientation of the Haversian system on the macroscopic elastic moduli and Poisson's ratios was investigated. The results presented provide relationships for applying more locally accurate material properties to larger scale and whole bone models of varying porosity. Analysis of the effect of the orientation of the Haversian system showed that its effects should not be neglected in larger scale models. This study also provides insight into how microstructural features effect local distributions and cause a strain magnification effect. Limitations in applying the unit cell methodology approach to bone are also discussed.     

低强度脉冲超声波辐照与泡沫TiC/Ti对犬节段性骨缺修复的促进作用

目的探讨低强度脉冲超声波辐照对节段性骨缺损修复效果的影响。方法将直径12 mm长20mm泡沫TiC/Ti植入6只Beagle犬的左侧胫骨节段性骨缺损区。随机分为超声组和对照组,超声组采用低强度脉冲超声波辐照（频率1.5 MHz、强度30 mW/cm2、脉冲宽度200μs、脉冲周期1 kHz、20 min/次、1次/d）,对照组为不开功率源的假辐照,术后4、8周后分别行X线检查及骨密度测定,观察及分析材料周围骨愈合情况。结果 6只beagle犬均进入结果分析。术后4周超声组骨早期成熟度优于对照组,表现在材料周围骨痂影密度增高,骨痂影由两端向中央生长;对照组仅见骨痂区密度低,还可见部分骨痂缺如。术后8周超声组新生骨痂面积优于对照组,骨干结构相对稳定;对照组骨缺损区未闭合,在骨干两侧看到少量骨痂,愈合较差。骨密度测定结果显示,4周时超声组高于对照组,两组间存在统计学差异;8周时超声组略高于对照组,但两组间没有统计学差异。结论通过联合应用低强度脉冲超声波辐照与人工骨材料修复可提高新骨形成速度及骨组织密度,缩短节段性骨缺损的骨愈合时间。     

Study of skeleton gravitation physiology and problem of osteoporosis

Main osteoporosis definitions and some results of bone tissue research in Russian astronauts, patients, and healthy subjects, using modern osteodensitometry, are presented. Bone mineral density (BMD) was regularly decreased at lower segments of skeleton. In the skull bone and some other sites of upper part of skeleton, a tendency was revealed for an increase of the bone mineral content (BMC). The mean value of bone loss was within the normal range and not correlated with duration of space flight; it revealed a high individual variability and in some cases was clinically qualified as local osteopenia. On the ground of analysis of own results and animal and bone cultural experiments data in microgravity conditions, the described changes seem to be reflecting a deceleration of bone formation as an adaptive response of bone tissue to the mechanical unloading. The response is realized mainly on the tissue level. It does not exclude bone resorption activity as a result of changes in hierarchy of water and electrolytes metabolism as reflected by body fluid redistribution in cranial direction. The results obtained broaden our notions on pathogenesis of some types of osteoporosis in clinic.     

An improved method to assess torsional properties of rodent long bones

Torsion is an important testing modality commonly used to calculate structural properties of long bones. However, the effects of size and geometry must be excluded from the overall structural response in order to compare material properties of bones of different size, age and species. We have developed a new method to analyze torsional properties of bones using actual cross-sectional information and length-wise geometrical variations obtained by micro-computed topographic (μCT) imaging. The proposed method was first validated by manufacturing three rat femurs through rapid prototyping using a plastic with known material properties. The observed variations in calculated torsional shear modulus of the hollow elliptical model of mid-shaft cross-section (Ekeland et al.), multi-prismatic model of five true cross-sections (Levenston et al.) and multi-slice model presented in this study were 96%, ?7% and 6% from the actual properties of the plastic, respectively. Subsequently, we used this method to derive relationships expressing torsional properties of rat cortical bone as a function of μCT-based bone volume fraction or apparent density over a range of normal and pathologic bone densities. Results indicate that a regression model of shear modulus or shear strength and bone volume fraction or apparent density described at least 81% of the variation in torsional properties of normal and pathologic bones. Coupled with the structural rigidity analysis technique introduced by the authors, the relationships reported here can provide a non-invasive tool to assess fracture risk in bones affected by pathologies and/or treatment options.     

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.     

Material changes in osteoporotic human cancellous bone following infiltration with acrylic bone cement for a vertebral cement augmentation

Bone cement infiltration can be effective at mechanically augmenting osteoporotic vertebrae. While most published literature describes the gain in mechanical strength of augmented vertebrae, we report the first measurements of viscoelastic material changes of cancellous bone due to cement infiltration. We infiltrated cancellous core specimen harvested from osteoporotic cadaveric spines with acrylic bone cement. Bone specimen before and after cement infiltration were subjected to identical quasi-static and relaxation loading in confined and free compression. Testing data were fitted to a linear viscoelastic model of compressible material and the model parameters for cement, native cancellous bone, and cancellous bone infiltrated (composite) with cement were identified. The fitting demonstrated that the linear viscoelastic model presented in this paper accurately describes the mechanical behaviour of cement and bone, before and after infiltration. Although the composite specimen did not completely adopt the properties of bulk bone cement, the stiffening of cancellous bone due to cement infiltration is considerable. The composite was, for example, 8.5 times stiffer than native bone. The local stiffening of cancellous bone in patients may alter the load transfer of the augmented motion segment and may be the cause of subsequent fractures in the vertebrae adjacent to the ones infiltrated with cement. The material model and parameters in this paper, together with an adequate finite-element model, can be helpful to investigate the load shift, the mechanism for subsequent fractures, and filling patterns for ideal cement infiltration.     

Mandibular bone remodeling induced by dental implant

The ability to assess the effects of an implant on bone remodeling is of particular importance to prosthesis placement planning and associated treatment assurance. Prediction of on-going bone responses will enable us to improve the performance of a restoration. Although the bone remodeling for long bones had been extensively studied, there have been relatively few reports for dental scenarios despite its increasing significance with more and more dental implant placements. This paper aimed to develop a systematic protocol to assess mandibular bone remodeling induced by dental implantation, which extends the remodeling algorithms established for the long bones into dental settings. In this study, a 3D model for a segment of a human mandible was generated from in vivo CT scan images, together with a titanium implant embedded to the mandible. The results examined the changes in bone density and stiffness as a result of bone remodeling over a period of 48 months. Resonance frequency analysis was also performed to relate natural frequencies to bone remodeling. The density contours are qualitatively compared with clinical follow-up X-ray images, thereby providing validity for the bone remodeling algorithm presented in dental bone analysis.     

Monochromatic synchrotron radiation muCT reveals disuse-mediated canal network rarefaction in cortical bone of growing rat tibiae.

The purpose of this study was to demonstrate the ability of computed microtomography based on monochromatic synchrotron radiation (SRmuCT) in microstructural analysis of cortical bone. Tibial diaphyses of growing rats (14 wk, n = 8) undergoing unilateral sciatic neurectomy 8 wk before study were imaged with spatial volume resolution of 5.83 x 5.83 x 5.83 microm3 by SRmuCT (20 keV) at the synchrotron radiation facility (SPring-8). Reconstructed image data were translated into local mineral densities by using a calibrated linear relationship between linear absorption coefficients and concentrations of homogeneous K2HPO4 solution. Pure bone three-dimensional images, produced by simple thresholding at a bone mineral density of 0.82 g/cm3, were analyzed for macro- and microscopic structural properties. In neurectomized hindlimbs, cortical canal network rarefaction as well as bone atrophy were found. The former was characterized by 30% smaller porosity, 11% smaller canal density in transverse section, and 38% smaller canal connectivity density than those in contralateral bone. On the other hand, no difference was found in bone mineral density between neurectomized and intact hindlimbs (1.37 vs. 1.36 g/cm3). In conclusion, SRmuCT is a promising method for the three-dimensional analysis of cortical microstructure and the degree of mineralization in small animals.     

Bone remodelling around the tibia due to total ankle replacement: effects of implant material and implant–bone interfacial conditions

Abstract

One of the major causes of implant loosening is due to excessive bone resorption surrounding the implant due to bone remodelling. The objective of the study is to investigate the effects of implant material and implant–bone interface conditions on bone remodelling around tibia bone due to total ankle replacement. Finite element models of intact and implanted ankles were developed using CT scan data sets. Bone remodelling algorithm was used in combination with FE analysis to predict the bone density changes around the ankle joint. Dorsiflexion, neutral, and plantar flexion positions were considered, along with muscle force and ligaments. Implant–bone interfacial conditions were assumed as debonded and bonded to represent non-osseointegration and fully osseointegration at the porous coated surface of the implant. To investigate the effect of implant material, three finite element models having different material combinations of the implant were developed. For model 1, tibial and talar components were made of Co–Cr–Mo, and meniscal bearing was made of UHMWPE. For model 2, tibial and talar components were made of ceramic and meniscal bearing was made of UHMWPE. For model 3, tibial and talar components were made of ceramic and meniscal bearing was made of CFR-PEEK. Changes in implant material showed no significant changes in bone density due to bone remodelling. Therefore, ceramic appears to be a viable alternative to metal and CFR-PEEK can be used in place of UHMWPE. This study also indicates that proper bonding between implant and bone is essential for long-term survival of the prosthetic components.