Concept and development of an orthotropic FE model of the proximal femur

PURPOSE: In contrast to many isotropic finite-element (FE) models of the femur in literature, it was the object of our study to develop an orthotropic FE "model femur" to realistically simulate three-dimensional bone remodelling. METHODS: The three-dimensional geometry of the proximal femur was reconstructed by CT scans of a pair of cadaveric femurs at equal distances of 2mm. These three-dimensional CT models were implemented into an FE simulation tool. Well-known "density-determined" bony material properties (Young's modulus; Poisson's ratio; ultimate strength in pressure, tension and torsion; shear modulus) were assigned to each FE of the same "CT-density-characterized" volumetric group.In order to fix the principal directions of stiffness in FE areas with the same "density characterization", the cadaveric femurs were cut in 2mm slices in frontal (left femur) and sagittal plane (right femur). Each femoral slice was scanned into a computer-based image processing system. On these images, the principal directions of stiffness of cancellous and cortical bone were determined manually using the orientation of the trabecular structures and the Haversian system. Finally, these geometric data were matched with the "CT-density characterized" three-dimensional femur model. In addition, the time and density-dependent adaptive behaviour of bone remodelling was taken into account by implementation of Carter's criterion. RESULTS: In the constructed "model femur", each FE is characterized by the principal directions of the stiffness and the "CT-density-determined" material properties of cortical and cancellous bone. Thus, on the basis of anatomic data a three-dimensional FE simulation reference model of the proximal femur was realized considering orthotropic conditions of bone behaviour. CONCLUSIONS: With the orthotropic "model femur", the fundamental basis has been formed to realize realistic simulations of the dynamical processes of bone remodelling under different loading conditions or operative procedures (osteotomies, total hip replacements, etc).     

Angular orientation of trabecular bone in the femoral head and its relationship to hip joint loads in leaping primates

The elastic properties and mechanical behavior of trabecular bone are largely determined by its three-dimensional (3D) fabric structure. Recent work demonstrating a correlation between the primary mechanical and material axes in trabecular bone specimens suggests that fabric orientation may be used to infer directional components of the material strength and, by extension, the hypothetical loading regime. Here we quantify the principal orientation of trabecular bone in the femoral head and relate these principal fabric directions to loading patterns during various locomotor behaviors. The proximal femora of a diverse sample of prosimians were scanned using a high-resolution X-ray computed tomography scanner with resolution of better than 50 mum. Spherical volumes of interest were defined within the femoral heads and the 3D fabric anisotropy was calculated using the mean intercept length and star volume distribution methods. In addition to differences in bone volume and anisotropy, significant differences were found in the spatial orientation of the principal trabecular axes depending on locomotor behavior. The principal orientations for leapers (Galago, Tarsius, Avahi) are relatively tightly clustered (alpha(95) confidence limit: 8.2; angular variance s: 18.2 degrees ) and oriented in a superoanterior direction, while those of nonleapers are more variable across a range of directions (alpha(95): 16.8; s: 42.0 degrees ). The mean principal directions are significantly different for leaping vs. nonleaping taxa. These results further suggest a relationship between bone microstructure in the hip joint and locomotor behavior and indicate a similarity of loading across leapers despite differences in kinematics and phylogeny.     

Relationships between loading history and femoral cancellous bone architecture

A theory relating bone maintenance to mechanical loading history has been applied to successfully predict the distribution of bone density and trabecular orientation in the adult proximal femur. The loading history was simulated by determining the stress fields in a two-dimensional finite element model exposed to various discrete loading cases and making assumptions about the relative number of loading cycles associated with each load case. The total stimulus to bone maintenance was then calculated by a linear superposition of the stimulus of each loading case. Based on the calculated total stimulus, the apparent density and material properties of each element were changed and the stress solutions were again determined. Using this iterative technique, the bone apparent density and orientation characteristics were predicted. The results indicate that the trabecular morphology of the femur can only be explained by considering the joint loadings from multiple directions. Contrary to the 'trajectorial theory' promoted by Wolff (The Law of Bone Remodelling, 1892), trabecular orientations predicted from our multiple-load analyses are not necessarily perpendicular and do not correspond to the principal stress directions of any one loading condition. Our predicted orientations correspond better to the drawing of bone trabecular morphology by von Meyer (Archs Anat. Physiol. wiss. Med. 34, 615-628, 1867) than to the classic drawing by Wolff and suggest that further study of the trajectorial theory is warranted.     

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.     

Consideration of multiple load cases is critical in modelling orthotropic bone adaptation in the femur

Functional adaptation of the femur has been investigated in several studies by embedding bone remodelling algorithms in finite element (FE) models, with simplifications often made to the representation of bone’s material symmetry and mechanical environment. An orthotropic strain-driven adaptation algorithm is proposed in order to predict the femur’s volumetric material property distribution and directionality of its internal structures within a continuum. The algorithm was applied to a FE model of the femur, with muscles, ligaments and joints included explicitly. Multiple load cases representing distinct frames of two activities of daily living (walking and stair climbing) were considered. It is hypothesised that low shear moduli occur in areas of bone that are simply loaded and high shear moduli in areas subjected to complex loading conditions. In addition, it is investigated whether material properties of different femoral regions are stimulated by different activities. The loading and boundary conditions were considered to provide a physiological mechanical environment. The resulting volumetric material property distribution and directionalities agreed with ex vivo imaging data for the whole femur. Regions where non-orthogonal trabecular crossing has been documented coincided with higher values of predicted shear moduli. The topological influence of the different activities modelled was analysed. The influence of stair climbing on the properties of the femoral neck region is highlighted. It is recommended that multiple load cases should be considered when modelling bone adaptation. The orthotropic model of the complete femur is released with this study.     

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.     

Variation of trabecular architecture in proximal femur of postmenopausal women

This investigation of microstructure in the human proximal femur probes the relationship between the parameters of the FRAX index of fracture risk and the parameters of bone microstructure. The specificity of fracture sites at the proximal femur raises the question of whether trabecular parameters are site-specific during post-menopause, before occurrence of fragility fracture. The donated proximal femurs of sixteen post-menopausal women in the sixth and seventh decades of life, free of metabolic pathologies and therapeutic interventions that could have altered the bone tissue, constituted the material of the study. We assessed bone mineral density of the proximal femurs by dual energy X-ray absorptiometry and then sectioned the femurs through the center of the femoral head and along the femoral neck axis. For each proximal femur, morphometry of trabeculae was conducted on the plane of the section divided into conventional regions and sub-regions consistent with the previously identified trabecular families that provide regions of relatively homogeneous microstructure. Mean trabecular width and percent bone area were calculated at such sites. Our findings indicate that each of mean trabecular width and percent bone area vary within each proximal femur independently from each other, with dependence on site. Both trabecular parameters show significant differences between pairs of sites. We speculate that a high FRAX index at the hip corresponds to a reduced percent bone area among sites that gives a more homogeneous and less site-specific quality to the proximal femur. This phenomenon may render the local tissue less able to carry out the expected mechanical function.     

Computational simulation of trabecular adaptation progress in human proximal femur during growth

There are a large number of clinical and experimental studies that analyzed trabecular architecture as a result of bone adaptation. However, only a limited amount of quantitative data is currently available on the progress of trabecular adaptation during growth. In this paper, we proposed a two-step numerical simulation method that predicts trabecular adaptation progress during growth using a recently developed topology optimization algorithm, design space optimization (DSO), under the hypothesis that the mechanisms of DSO are functionally equivalent to those of bone adaptation. We applied the proposed scheme to trabecular adaptation simulation in human proximal femur. For the simulation, the full trabecular architecture in human proximal femur was represented by a two-dimensional μFE model with 50 μm resolution. In Step 1, we determined a reference value that regulates trabecular adaptation in human proximal femur. In Step 2, we simulated trabecular adaptation in human proximal femur during growth with the reference value derived in Step 1. We analyzed the architectural and mechanical properties of trabecular patterns through iterations. From the comparison with experimental data in the literature, we showed that in the early growth stage trabecular adaptation was achieved mainly by increasing bone volume fraction (or trabecular thickness), while in the later stage of the development the trabecular architecture gained higher structural efficiency by increasing structural anisotropy with a relatively low level of bone volume fraction (or trabecular thickness). We demonstrated that the proposed numerical framework predicted the growing progress of trabecular bone that has a close correlation with experimental data.     

A Model of Bone Adaptation Using a Global Optimisation Criterion Based on the Trajectorial Theory of Wolff

Julius Wolff originally proposed that trabecular bone was influenced by mechanical stresses during the formative processes of growth and repair such that trabeculae were required to intersect at right angles. In this work, we have developed an analytical parametric microstructural model, which captures this restriction. Using homogenisation theory, a global material model was obtained. An optimal structure constructed of the homogenised material could then be found by optimising a cost function accounting for both the structural stiffness and the biological cost associated with metabolic maintenance of the bone tissue. The formulation was applied to an example problem of the proximal femur. Optimal densities and orientations were obtained for single load cases. The situation of multiple loads was also considered. In this case, we observe that the alignment of principal strains with the material orthotropy direction is, in general, not possible for all load cases. Thus less restrictive microstructures (nonorthotropic) will yield higher structural stiffnesses than strictly orthotropic microstructures.     

Three-dimensional micro-level computational study of Wolff's law via trabecular bone remodeling in the human proximal femur using design space topology optimization

The law of bone remodeling, commonly referred to as Wolff's Law, asserts that the internal trabecular bone adapts to external loadings, reorienting with the principal stress trajectories to maximize mechanical efficiency creating a naturally optimum structure. The goal of the current study was to utilize an advanced structural optimization algorithm, called design space optimization (DSO), to perform a micro-level three-dimensional finite element bone remodeling simulation on the human proximal femur and analyse the results to determine the validity of Wolff's hypothesis. DSO optimizes the layout of material by iteratively distributing it into the areas of highest loading, while simultaneously changing the design domain to increase computational efficiency. The result is a "fully stressed" structure with minimized compliance and increased stiffness. The large-scale computational simulation utilized a 175 μm mesh resolution and the routine daily loading activities of walking and stair climbing. The resulting anisotropic trabecular architecture was compared to both Wolff's trajectory hypothesis and natural femur samples from literature using a variety of visualization techniques, including radiography and computed tomography (CT). The results qualitatively revealed several anisotropic trabecular regions, that were comparable to the natural human femurs. Quantitatively, the various regional bone volume fractions from the computational results were consistent with quantitative CT analyses. The global strain energy proceeded to become more uniform during optimization; implying increased mechanical efficiency was achieved. The realistic simulated trabecular geometry suggests that the DSO method can accurately predict bone adaptation due to mechanical loading and that the proximal femur is an optimum structure as the Wolff hypothesized.     

Computational simulation of simultaneous cortical and trabecular bone change in human proximal femur during bone remodeling

In this study, we developed a numerical framework that computationally determines simultaneous and interactive structural changes of cortical and trabecular bone types during bone remodeling, and we investigated the structural correlation between the two bone types in human proximal femur. We implemented a surface remodeling technique that performs bone remodeling in the exterior layer of the cortical bone while keeping its interior area unchanged. A micro-finite element (μFE) model was constructed that represents the entire cortical bone and full trabecular architecture in human proximal femur. This study simulated and compared the bone adaptation processes of two different structures: (1) femoral bone that has normal cortical bone shape and (2) perturbed femoral bone that has an artificial bone lump in the inferomedial cortex. Using the proposed numerical method in conjunction with design space optimization, we successfully obtained numerical results that resemble actual human proximal femur. The results revealed that actual cortical bone, as well as the trabecular bone, in human proximal femur has structurally optimal shapes, and it was also shown that a bone abnormality that has little contribution to bone structural integrity tends to disappear. This study also quantitatively determined the structural contribution of each bone: when the trabecular adaptation was complete, the trabecular bone supported 54% of the total load in the human proximal femur while the cortical bone carried 46%.     

Mapping anisotropy of the proximal femur for enhanced image based finite element analysis

Finite element (FE) models of bone derived from quantitative computed tomography (QCT) rely on realistic material properties to accurately predict bone strength. QCT cannot resolve bone microarchitecture, therefore QCT-based FE models lack the anisotropy apparent within the underlying bone tissue. This study proposes a method for mapping femoral anisotropy using high-resolution peripheral quantitative computed tomography (HR-pQCT) scans of human cadaver specimens. Femur HR-pQCT images were sub-divided into numerous overlapping cubic sub-volumes and the local anisotropy was quantified using a ‘direct-mechanics’ method. The resulting directionality reflected all the major stress lines visible within the trabecular lattice, and provided a realistic estimate of the alignment of Harvesian systems within the cortical compartment. QCT-based FE models of the proximal femur were constructed with isotropic and anisotropic material properties, with directionality interpolated from the map of anisotropy. Models were loaded in a sideways fall configuration and the resulting whole bone stiffness was compared to experimental stiffness and ultimate strength. Anisotropic models were consistently less stiff, but no statistically significant differences in correlation were observed between material models against experimental data. The mean difference in whole bone stiffness between model types was approximately 26%, suggesting that anisotropy can still effect considerable change in the mechanics of proximal femur models. The under prediction of whole bone stiffness in anisotropic models suggests that the orthotropic elastic constants require further investigation. The ability to map mechanical anisotropy from high-resolution images and interpolate information into clinical-resolution models will allow testing of new anisotropic material mapping strategies.     

Mechanical significance of femoral head trabecular bone structure in Loris and Galago evaluated using micromechanical finite element models

Work on the interspecific and intraspecific variation of trabecular bone in the proximal femur of primates demonstrates important architectural variation between animals with different locomotor behaviors. This variation is thought to be related to the processes of bone adaptation whereby bone structure is optimized to the mechanical environment. Micromechanical finite element models were created for the proximal femur of the leaping Galago senegalensis and the climbing and quadrupedal Loris tardigradus by converting bone voxels from high-resolution X-ray computed tomography scans of the femoral head to eight-noded brick elements. The resulting models had approximately 1.8 million elements each. Loading conditions representing takeoff phase of a leap and more generalized load orientations were applied to the models, and the models were solved using the iterative "row-by-row" matrix-vector multiplication algorithm. The principal strain and Von Mises stress results for the leaping model were similar for both species at each load orientation. Similar hip joint reaction forces in the range of 4.9 x to 12 x body weight were calculated for both species under each loading condition, but the hip reaction values estimated for Loris were higher than predicted based on locomotor behavior. These results suggest that functional adaptation to hip joint loading may not fully explain the differences in femoral head trabecular bone structure in Galago and Loris. The finite element method represents a unique and useful tool for analyzing the functional adaptation of trabecular bone in a diversity of animals and for reconstructing locomotor behavior in extinct taxa.     

Computational study of Wolff's law with trabecular architecture in the human proximal femur using topology optimization

In the field of bone adaptation, it is believed that the morphology of bone is affected by its mechanical loads, and bone has self-optimizing capability; this phenomenon is well known as Wolff's law of the transformation of bone. In this paper, we simulated trabecular bone adaptation in the human proximal femur using topology optimization and quantitatively investigated the validity of Wolff's law. Topology optimization iteratively distributes material in a design domain producing optimal layout or configuration, and it has been widely and successfully used in many engineering fields. We used a two-dimensional micro-FE model with 50 microm pixel resolution to represent the full trabecular architecture in the proximal femur, and performed topology optimization to study the trabecular morphological changes under three loading cases in daily activities. The simulation results were compared to the actual trabecular architecture in previous experimental studies. We discovered that there are strong similarities in trabecular patterns between the computational results and observed data in the literature. The results showed that the strain energy distribution of the trabecular architecture became more uniform during the optimization; from the viewpoint of structural topology optimization, this bone morphology may be considered as an optimal structure. We also showed that the non-orthogonal intersections were constructed to support daily activity loadings in the sense of optimization, as opposed to Wolff's drawing.     

Computer simulation of trabecular remodeling in human proximal femur using large-scale voxel FE models: Approach to understanding Wolff's law

Ever since Julius Wolff proposed the law of bone transformation in the 19th century, it has been widely known that the trabecular structure of cancellous bone adapts functionally to the loading environment. To understand the mechanism of Wolff's law, a three-dimensional (3D) computer simulation of trabecular structural changes due to surface remodeling was performed for a human proximal femur. A large-scale voxel finite element model was constructed to simulate the structural changes of individual trabeculae over the entire cancellous region. As a simple remodeling model that considers bone cellular activities regulated by the local mechanical environment, nonuniformity of local stress was assumed to drive the trabecular surface remodeling to seek a uniform stress state. Simulation results demonstrated that cell-scale (～10 μm) remodeling in response to mechanical stimulation created complex 3D trabecular structures of the entire bone-scale (～10 cm), as illustrated in the reference of Wolff. The bone remodeling reproduced the characteristic anisotropic structure in the coronal cross section and the isotropic structures in other cross sections. The principal values and axes of a structure characterized by fabric ellipsoids corresponded to those of the apparent stress of the structure. The proposed large-scale computer simulation indicates that in a complex mechanical environment of a hierarchical bone structure of over 10⁴ length scale (from ～10 μm to ～10 cm), a simple remodeling at the cellular/trabecular levels creates a highly complex and functional trabecular structure, as characterized by bone density and orientation.     

Stress relaxation behaviour of trabecular bone specimens

The present study defines several conditions under which stress relaxation tests can be performed and investigates the viscoelastic behaviour of trabecular bone in compression through a series of stress relaxation tests at three strain levels and in three loading directions of each cubic specimen. A visoelastic model is proposed to characterize the behaviour of trabecular bone and a spectrum of relaxation times is determined. Trabecular bone from the femoral head is non-linearly viscoelastic and displays anisotropic behaviour, which cannot be more symmetric elastically than orthotropic.     

Errors in the orientation of the principal stress axes if bone tissue is modeled as isotropic

The error in the prediction of the orientation of the principal axes of stress in bone tissue is determined in the case when the tissue is modeled as elastically isotropic rather than as orthotropic, the probable symmetry of bone tissue. Results are two-dimensional and assume the same underlying strain state for both the orthotropic and isotropic cases. The maximum error is 45 degrees, and the typical error is generally significant.     

Optimal Mass Distribution Prediction for Human Proximal Femur with Bi-modulus Property

Simulation of the mass distribution in a human proximal femur is important to provide a reasonable therapy scheme for a patient with osteoporosis. An algorithm is developed for prediction of optimal mass distribution in a human proximal femur under a given loading environment. In this algorithm, the bone material is assumed to be bi-modulus, i.e., the tension modulus is not identical to the compression modulus in the same direction. With this bi-modulus bone material, a topology optimization method, i.e., modified SIMP approach, is employed to determine the optimal mass distribution in a proximal femur. The effects of the difference between two moduli on the final material distribution are numerically investigated. Numerical results obtained show that the mass distribution in bi-modular bone materials is different from that in traditional isotropic material. As the tension modulus is less than the compression modulus for bone tissues, the amount of mass required to support tension loads is greater than that required by isotropic material for the same daily activities including one-leg stance, abduction and adduction.     

Application of an anisotropic bone-remodelling model based on a damage-repair theory to the analysis of the proximal femur before and after total hip replacement

In this work, a new model for internal anisotropic bone remodelling is applied to the study of the remodelling behaviour of the proximal femur before and after total hip replacement (THR). This model considers bone remodelling under the scope of a general damage-repair theory following the principles of continuum damage mechanics. A "damage-repair" tensor is defined in terms of the apparent density and Cowin's "fabric tensor", respectively, associated with porosity and directionality of the trabeculae. The different elements of a thermodynamically consistent damage theory are established, including resorption and apposition criteria, evolution law and rate of remodelling. All of these elements were introduced and discussed in detail in a previous paper (García, J. M., Martinez, M. A., Doblaré, M., 2001. An anisotrophic internal-external bone adaptation model based on a combination of CAO and continuum damage mechanics technologies. Computer Methods in Biomechanics and Biomedical Engineering 4(4), 355-378.), including the definition of the proposed mechanical stimulus and the qualitative properties of the model. In this paper, the fundamentals of the proposed model are briefly reviewed and the computational aspects of its implementation are discussed. This model is then applied to the analysis of the remodelling behaviour of the intact femur obtaining densities and mass principal values and directions very close to the experimental data. The second application involved the proximal femoral extremity after THR and the inclusion of an Exeter prosthesis. As a result of the simulation process, some well-known features previously detected in medical clinics were recovered, such as the stress yielding effect in the proximal part of the implant or the enlargement of the cortical layer at the distal part of the implant. With respect to the anisotropic properties, bone microstructure and local stiffness are known to tend to align with the stress principal directions. This experimental fact is mathematically proved in the framework of this remodelling model and clearly shown in the results corresponding to the intact femur. After THR the degree of anisotropy decreases tending, specifically in the proximal femur, to a more isotropic behaviour.     

A novel approach to estimate trabecular bone anisotropy using a database approach

Continuum finite element (FE) models of bones have become a standard pre-clinical tool to estimate bone strength. These models are usually based on clinical CT scans and material properties assigned are chosen as isotropic based only on the density distribution. It has been shown, however, that trabecular bone elastic behavior is best described as orthotropic. Unfortunately, the use of orthotropic models in FE analysis derived from CT scans is hampered by the fact that the measurement of a trabecular orientation (fabric) is not possible from clinical CT images due to the low resolution of such images. In this study, we explore the concept of using a database (DB) of high-resolution bone models to derive the fabric information that is missing in clinical images. The goal of this study was to investigate if models with fabric derived from a relatively small database can already produce more accurate results than isotropic models.