Computational method for determination of bone and joint loads using bone density distributions

Personalized loading conditions for homogenized finite element analysis of the distal sections of the radius

The microstructure of trabecular bone is known to adapt its morphology in response to mechanical loads for achieving a biomechanical homeostasis. Based on this form-function relationship, previous investigators either simulated the remodeling of bone to predict the resulting density and architecture for a specific loading or retraced physiological loading conditions from local density and architecture. The latter inverse approach includes quantifying bone morphology using computed tomography and calculating the relative importance of selected load cases by minimizing the fluctuation of a tissue loading level metric. Along this concept, the present study aims at identifying an optimal, personalized, multiaxial load case at the distal section of the human radius using in vivo HR-pQCT-based isotropic, homogenized finite element (hFE) analysis. The dataset consisted of HR-pQCT reconstructions of the 20 mm most distal section of 21 human fresh-frozen radii. We simulated six different unit canonical load cases (FX palmar-dorsal force, FY ulnar-radial force, FZ distal-proximal force, MX moment about palmar-dorsal, MY moment about ulnar-radial, MZ moment about distal-proximal) using a simplified and efficient hFE method based on a single isotropic bone phase. Once we used a homogeneous mean density (shape model) and once the original heterogeneous density distribution (shape + density model). Using an analytical formulation, we minimized the deviation of the resulting strain tensors ε(x) to a hydrostatic compressive reference strain ε0, once for the 6 degrees of freedom (DOF) optimal (OPT) load case and for all individual 1 DOF load cases (FX, FY, FZ, MX, MY, MZ). All seven load cases were then extended in the nonlinear regime using the scaled displacements of the linear load cases as loading boundary conditions (MAX). We then compared the load cases and models for their objective function (OF) values, the stored energies and their ultimate strength using a specific torsor norm. Both shape and shape + density linear-optimized OPT models were dominated by a positive force in the z-direction (FZ). Transversal force DOFs were close to zero and mean moment DOFs were different depending on the model type. The inclusion of density distribution increased the influence and changed direction of MX and MY, while MZ was small in both models. The OPT load case had 12-15% lower objective function (OF) values than the FZ load case, depending on the model. Stored energies at the optimum were consistently 142-178% higher for the OPT load case than for the FZ load case. Differences in the nonlinear response maximum torsor norm ‖t‖ were heterogeneous, but consistently higher for OPT_MAX than FZ_MAX. We presented the proof of concept of an optimization procedure to estimate patient-specific loading conditions for hFE methods. In contrast to similar models, we included canonical load cases in all six DOFs and used a strain metric that favors hydrostatic compression. Based on a biomechanical analysis of the distal joint surfaces at the radius, the estimated load directions are plausible. For our dataset, the resulting OPT load case is close to the standard axial compression boundary conditions, usually used in HR-pQCT-based FE analysis today. But even using the present simplified hFE model, the optimized linear six DOF load case achieves a more homogeneous tissue loading and can absorb more than twice the energy than the standard uniaxial load case. The ultimate strength calculated with a torsor norm was consistently higher for the 6-DOF nonlinear model (OPT_MAX) than for the 1-DOF nonlinear uniaxial model (FZ_MAX). Defining patient-specific boundary conditions may decrease angulation errors during CT measurements and improve repeatability as well as reproducibility of bone stiffness and strength estimated by HR-pQCT-based hFE analysis. These results encourage the extension of the present method to anisotropic hFE models and their application to repeatability data sets to test the hypothesis of reduced angulation errors during measurement.

Quantitative CT of the knee in the IMI-APPROACH osteoarthritis cohort: Association of bone mineral density with radiographic disease severity, meniscal coverage and meniscal extrusion

OBJECTIVE

Osteoarthritis (OA) is a highly prevalent chronic condition. The subchondral bone plays an important role in onset and progression of OA making it a potential treatment target for disease-modifying therapeutic approaches. However, little is known about changes of periarticular bone mineral density (BMD) in OA and its relation to meniscal coverage and meniscal extrusion at the knee. Thus, the aim of this study was to describe periarticular BMD in the Applied Public-Private Research enabling OsteoArthritis Clinical Headway (APPROACH) cohort at the knee and to analyze the association with structural disease severity, meniscal coverage and meniscal extrusion.

DESIGN

Quantitative CT (QCT), MRI and radiographic examinations were acquired in 275 patients with knee osteoarthritis (OA). QCT was used to assess BMD at the femur and tibia, at the cortical bone plate (Cort) and at the epiphysis at three locations: subchondral (Sub), mid-epiphysis (Mid) and adjacent to the physis (Juxta). BMD was evaluated for the medial and lateral compartment separately and for subregions covered and not covered by the meniscus. Radiographs were used to determine the femorotibial angle and were evaluated according to the Kellgren and Lawrence (KL) system. Meniscal extrusion was assessed from 0 to 3.

RESULTS

Mean BMD differed significantly between each anatomic location at both the femur and tibia (p < 0.001) in patients with KL0. Tibial regions assumed to be covered with meniscus in patients with KL0 showed lower BMD at Sub (p < 0.001), equivalent BMD at Mid (p = 0.07) and higher BMD at Juxta (p < 0.001) subregions compared to regions not covered with meniscus. Knees with KL2-4 showed lower Sub (p = 0.03), Mid (p = 0.01) and Juxta (p < 0.05) BMD at the medial femur compared to KL0/1. Meniscal extrusion grade 2 and 3 was associated with greater BMD at the tibial Cort (p < 0.001, p = 0.007). Varus malalignment is associated with significant greater BMD at the medial femur and at the medial tibia at all anatomic locations.

CONCLUSION

BMD within the epiphyses of the tibia and femur decreases with increasing distance from the articular surface. Knees with structural OA (KL2-4) exhibit greater cortical BMD values at the tibia and lower BMD at the femur at the subchondral level and levels beneath compared to KL0/1. BMD at the tibial cortical bone plate is greater in patients with meniscal extrusion grade 2/3.

Quantitative Load Dependency Analysis of Local Trabecular Bone Microstructure to Understand the Spatial Characteristics in the Synthetic Proximal Femur

Analysis of the dependency of the trabecular structure on loading conditions is essential for understanding and predicting bone structure formation. Although previous studies have investigated the relationship between loads and structural adaptations, there is a need for an in-depth analysis of this relationship based on the bone region and load specifics. In this study, the load dependency of the trabecular bone microstructure for twelve regions of interest (ROIs) in the synthetic proximal femur was quantitatively analyzed to understand the spatial characteristics under seven different loading conditions. To investigate the load dependency, a quantitative measure, called the load dependency score (LDS), was established based on the statistics of the strain energy density (SED) distribution. The results showed that for the global model and epiphysis ROIs, bone microstructures relied on the multiple-loading condition, whereas the structures in the metaphysis depended on single or double loads. These results demonstrate that a given ROI is predominantly dependent on a particular loading condition. The results confirm that the dependency analysis of the load effects for ROIs should be performed both qualitatively and quantitatively.

A Density-Dependent Target Stimulus for Inverse Bone (Re)modeling with Homogenized Finite Element Models

Inverse bone (re)modeling (IBR) can infer physiological loading conditions from the bone microstructure. IBR scales unit loads, imposed on finite element (FE) models of a bone, such that the trabecular microstructure is homogeneously loaded and the difference to a target stimulus is minimized. Micro-FE (µFE) analyses are typically used to model the microstructure, but computationally more efficient, homogenized FE (hFE) models, where the microstructure is replaced by an equivalent continuum, could be used instead. However, also the target stimulus has to be translated from the tissue to the continuum level. In this study, a new continuum-level target stimulus relating relative bone density and strain energy density is proposed. It was applied using different types of hFE models to predict the physiological loading of 21 distal radii sections, which was subsequently compared to µFE-based IBR. The hFE models were able to correctly identify the dominant load direction and showed a high correlation of the predicted forces, but mean magnitude errors ranged from − 14.7 to 26.6% even for the best models. While µFE-based IBR can still be regarded as a gold standard, hFE-based IBR enables faster predictions, the usage of more sophisticated boundary conditions, and the usage of clinical images.

Computational Analysis of Bone Remodeling in the Proximal Tibia Under Electrical Stimulation Considering the Piezoelectric Properties

The piezoelectricity of bone is known to play a crucial role in bone adaptation and remodeling. The application of an external stimulus such as mechanical strain or electric field has the potential to enhance bone formation and implant osseointegration. Therefore, in the present study, the objective is to investigate bone remodeling under electromechanical stimulation as a step towards establishing therapeutic strategies. For the first time, piezoelectric bone remodeling in the human proximal tibia under electro-mechanical loads was analyzed using the finite element method in an open-source framework. The predicted bone density distributions were qualitatively and quantitatively assessed by comparing with the computed tomography (CT) scan and the bone mineral density (BMD) calculated from the CT, respectively. The effect of model parameters such as uniform initial bone density and reference stimulus on the final density distribution was investigated. Results of the parametric study showed that for different values of initial bone density the model predicted similar but not identical final density distribution. It was also shown that higher reference stimulus value yielded lower average bone density at the final time. The present study demonstrates an increase in bone density as a result of electrical stimulation. Thus, to minimize bone loss, for example, due to physical impairment or osteoporosis, mechanical loads during daily physical activities could be partially replaced by therapeutic electrical stimulation.

Cancellous bone and theropod dinosaur locomotion. Part II-a new approach to inferring posture and locomotor biomechanics in extinct tetrapod vertebrates

This paper is the second of a three-part series that investigates the architecture of cancellous bone in the main hindlimb bones of theropod dinosaurs, and uses cancellous bone architectural patterns to infer locomotor biomechanics in extinct non-avian species. Cancellous bone is widely known to be highly sensitive to its mechanical environment, and therefore has the potential to provide insight into locomotor biomechanics in extinct tetrapod vertebrates such as dinosaurs. Here in Part II, a new biomechanical modelling approach is outlined, one which mechanistically links cancellous bone architectural patterns with three-dimensional musculoskeletal and finite element modelling of the hindlimb. In particular, the architecture of cancellous bone is used to derive a single 'characteristic posture' for a given species-one in which bone continuum-level principal stresses best align with cancellous bone fabric-and thereby clarify hindlimb locomotor biomechanics. The quasi-static approach was validated for an extant theropod, the chicken, and is shown to provide a good estimate of limb posture at around mid-stance. It also provides reasonable predictions of bone loading mechanics, especially for the proximal hindlimb, and also provides a broadly accurate assessment of muscle recruitment insofar as limb stabilization is concerned. In addition to being useful for better understanding locomotor biomechanics in extant species, the approach hence provides a new avenue by which to analyse, test and refine palaeobiomechanical hypotheses, not just for extinct theropods, but potentially many other extinct tetrapod groups as well.

Plausibility and parameter sensitivity of micro-finite element-based joint load prediction at the proximal femur

A micro-finite element-based method to estimate the bone loading history based on bone architecture was recently presented in the literature. However, a thorough investigation of the parameter sensitivity and plausibility of this method to predict joint loads is still missing. The goals of this study were (1) to analyse the parameter sensitivity of the joint load predictions at one proximal femur and (2) to assess the plausibility of the results by comparing load predictions of ten proximal femora to in vivo hip joint forces measured with instrumented prostheses (available from www.orthoload.com ). Joint loads were predicted by optimally scaling the magnitude of four unit loads (inclined [Formula: see text] to [Formula: see text] with respect to the vertical axis) applied to micro-finite element models created from high-resolution computed tomography scans ([Formula: see text]m voxel size). Parameter sensitivity analysis was performed by varying a total of nine parameters and showed that predictions of the peak load directions (range 10[Formula: see text]-[Formula: see text]) are more robust than the predicted peak load magnitudes (range 2344.8-4689.5 N). Comparing the results of all ten femora with the in vivo loading data of ten subjects showed that peak loads are plausible both in terms of the load direction (in vivo: [Formula: see text], predicted: [Formula: see text]) and magnitude (in vivo: [Formula: see text], predicted: [Formula: see text]). Overall, this study suggests that micro-finite element-based joint load predictions are both plausible and robust in terms of the predicted peak load direction, but predicted load magnitudes should be interpreted with caution.

A modelling approach demonstrating micromechanical changes in the tibial cemented interface due to in vivo service

Post-operative changes in trabecular bone morphology at the cement-bone interface can vary depending on time in service. This study aims to investigate how micromotion and bone strains change at the tibial bone-cement interface before and after cementation. This work discusses whether the morphology of the post-mortem interface can be explained by studying changes in these mechanical quantities. Three post-mortem cement-bone interface specimens showing varying levels of bone resorption (minimal, extensive and intermediate) were selected for this study Using image segmentation techniques, masks of the post-mortem bone were dilated to fill up the mould spaces in the cement to obtain the immediately post-operative situation. Finite element (FE) models of the post-mortem and post-operative situation were created from these segmentation masks. Subsequent removal of the cement layer resulted in the pre-operative situation. FE micromotion and bone strains were analyzed for the interdigitated trabecular bone. For all specimens micromotion increased from the post-operative to the post-mortem models (distally, in specimen 1: 0.1 to 0.5µm; specimen 2: 0.2 to 0.8µm; specimen 3: 0.27 to 1.62µm). Similarly bone strains were shown to increase from post-operative to post-mortem (distally, in specimen 1: -185 to -389µε; specimen 2: -170 to -824µε; specimen 3: -216 to -1024µε). Post-mortem interdigitated bone was found to be strain shielded in comparison with supporting bone indicating that failure of bone would occur distal to the interface. These results indicate that stress shielding of interdigitated trabeculae is a plausible explanation for resorption patterns observed in post-mortem specimens.

Subject-specific musculoskeletal loading of the tibia: Computational load estimation

The systematic development of subject-specific computer models for the analysis of personalized treatments is currently a reality. In fact, many advances have recently been developed for creating virtual finite element-based models. These models accurately recreate subject-specific geometries and material properties from recent techniques based on quantitative image analysis. However, to determine the subject-specific forces, we need a full gait analysis, typically in combination with an inverse dynamics simulation study. In this work, we aim to determine the subject-specific forces from the computer tomography images used to evaluate bone density. In fact, we propose a methodology that combines these images with bone remodelling simulations and artificial neural networks. To test the capability of this novel technique, we quantify the personalized forces for five subject-specific tibias using our technique and a gait analysis. We compare both results, finding that similar vertical loads are estimated by both methods and that the dominant part of the load can be reliably computed. Therefore, we can conclude that the numerical-based technique proposed in this work has great potential for estimating the main forces that define the mechanical behaviour of subject-specific bone.

Estimation of Local Bone Loads for the Volume of Interest

Computational bone remodeling simulations have recently received significant attention with the aid of state-of-the-art high-resolution imaging modalities. They have been performed using localized finite element (FE) models rather than full FE models due to the excessive computational costs of full FE models. However, these localized bone remodeling simulations remain to be investigated in more depth. In particular, applying simplified loading conditions (e.g., uniform and unidirectional loads) to localized FE models have a severe limitation in a reliable subject-specific assessment. In order to effectively determine the physiological local bone loads for the volume of interest (VOI), this paper proposes a novel method of estimating the local loads when the global musculoskeletal loads are given. The proposed method is verified for the three VOI in a proximal femur in terms of force equilibrium, displacement field, and strain energy density (SED) distribution. The effect of the global load deviation on the local load estimation is also investigated by perturbing a hip joint contact force (HCF) in the femoral head. Deviation in force magnitude exhibits the greatest absolute changes in a SED distribution due to its own greatest deviation, whereas angular deviation perpendicular to a HCF provides the greatest relative change. With further in vivo force measurements and high-resolution clinical imaging modalities, the proposed method will contribute to the development of reliable patient-specific localized FE models, which can provide enhanced computational efficiency for iterative computing processes such as bone remodeling simulations.

Development and implementation of a coupled computational muscle force optimization bone shape adaptation modeling method

Improved methods to analyze and compare the muscle-based influences that drive bone strength adaptation can aid in the understanding of the wide array of experimental observations about the effectiveness of various mechanical countermeasures to losses in bone strength that result from age, disuse, and reduced gravity environments. The coupling of gradient-based and gradientless numerical optimization routines with finite element methods in this work results in a modeling technique that determines the individual magnitudes of the muscle forces acting in a multisegment musculoskeletal system and predicts the improvement in the stress state uniformity and, therefore, strength, of a targeted bone through simulated local cortical material accretion and resorption. With a performance-based stopping criteria, no experimentally based or system-based parameters, and designed to include the direct and indirect effects of muscles attached to the targeted bone as well as to its neighbors, shape and strength alterations resulting from a wide range of boundary conditions can be consistently quantified. As demonstrated in a representative parametric study, the developed technique effectively provides a clearer foundation for the study of the relationships between muscle forces and the induced changes in bone strength. Its use can lead to the better control of such adaptive phenomena.

A mechanobiological model of epiphysis structures formation

Developing bone consists of epiphysis, metaphysis and diaphysis. The secondary ossification centre (SOC) appears and grows within the epiphysis, involving two histological stages. Firstly, cartilage canals appear; they carry hypertrophy factors towards the central area of the epiphysis. Canal growth and expansion is modulated by stress on the epiphysis. Secondly, the diffusion of hypertrophy factors causes SOC growth. Hypertrophy is regulated by biological and mechanical factors present within the epiphysis. The finite element method has been used for solving a coupled system of differential equations for modelling these histological stages of epiphyseal development. Cartilage canal spatial-temporal growth patterns were obtained as well as the SOC formation pattern. This model qualitatively agreed with experimental results reported by other authors.

Core decompression and osteonecrosis intervention rod in osteonecrosis of the femoral head: clinical outcome and finite element analysis

The osteonecrosis of the femoral head implies significant disability partly due to pain. After conventional core decompression using a 10-mm drill, patients normally are requested to be non-weight bearing for several weeks due to the risk of fracture. After core decompression using multiple small drillings, patients were allowed 50% weight bearing. The alternative of simultaneous implantation of a tantalum implant has the supposed advantage of unrestricted load bearing postoperatively. However, these recommendations are mainly based on clinical experience. The aim of this study was to perform a finite element analysis and confirm the results by clinical data after core decompression and after treatment using a tantalum implant. Postoperatively, the risk of fracture is lower after core decompression using multiple small drillings and after the implantation of a tantalum rod according to finite element analysis compared to core decompression of one 10-mm drill hole. According to the results of this study, a risk of fracture exists only during extreme loading. The long-term results reveal a superior performance for core decompression presumably due to the lack of complete bone ingrowth of the tantalum implant. In conclusion, core decompression using small drill holes seems to be superior compared to the tantalum implant and to conventional core decompression.

Body size and joint posture in primates

Body mass has been shown in experimental and comparative morphological studies to have a significant effect on joint posture in major limb joints. The generalizability of experimental studies is limited by their use of small sample sizes and limited size ranges. In contrast, while comparative morphological studies often have increased sample sizes, the connection between joint posture and morphological variables is often indirect. The current study infers joint postures for a large sample of primates using an experimentally validated method, and tests whether larger primates use more extended joint postures than smaller species. Postures are inferred through the analysis of patterns of subchondral bone apparent density on the medial femoral condyle. Femora from 94 adult wild-shot individuals of 28 species were included. Apparent density measurements were obtained from CT scans using AMIRA software, and the angular position of the anterior-most extent of the region of maximum apparent density on the medial femoral condyle was recorded. In general, the hypothesis that larger-bodied primates use more extended knee posture was supported, but it should be noted that considerable variation exists, particularly at small body sizes. This indicates that smaller species are less constrained by their body size, and their patterns of apparent density are consistent with a wide range of knee postures. The size-related increase in inferred joint posture was observed in most major groups of primates, and this observation attests to the generalizability of Biewener's model that relates body size and joint posture.

Modeling of Dynamic Fracture and Damage in Two-Dimensional Trabecular Bone Microstructures Using the Cohesive Finite Element Method

Trabecular bone fracture is closely related to the trabecular architecture, microdamage accumulation, and bone tissue properties. Micro-finite-element models have been used to investigate the elastic and yield properties of trabecular bone but have only seen limited application in modeling the microstructure dependent fracture of trabecular bone. In this research, dynamic fracture in two-dimensional (2D) micrographs of ovine (sheep) trabecular bone is modeled using the cohesive finite element method. For this purpose, the bone tissue is modeled as an orthotropic material with the cohesive parameters calculated from the experimental fracture properties of the human cortical bone. Crack propagation analyses are carried out in two different 2D orthogonal sections cut from a three-dimensional 8mm diameter cylindrical trabecular bone sample. The two sections differ in microstructural features such as area fraction (ratio of the 2D space occupied by bone tissue to the total 2D space), mean trabecula thickness, and connectivity. Analyses focus on understanding the effect of the rate of loading as well as on how the rate variation interacts with the microstructural features to cause anisotropy in microdamage accumulation and in the fracture resistance. Results are analyzed in terms of the dependence of fracture energy dissipation on the microstructural features as well as in terms of the changes in damage and stresses associated with the bone architecture variation. Besides the obvious dependence of the fracture behavior on the rate of loading, it is found that the microstructure strongly influences the fracture properties. The orthogonal section with lesser area fraction, low connectivity, and higher mean trabecula thickness is more resistant to fracture than the section with high area fraction, high connectivity, and lower mean trabecula thickness. In addition, it is found that the trabecular architecture leads to inhomogeneous distribution of damage, irrespective of the symmetry in the applied loading with the fracture of the entire bone section rapidly progressing to bone fragmentation once the accumulated damage in any trabeculae reaches a critical limit.

Bone density spatial patterns in the distal radius reflect habitual hand postures adopted by quadrupedal primates

Primates adopt diverse hand postures during terrestrial and above-branch quadrupedal locomotion--knuckle-walking, digitigrady, and palmigrady--that incorporate varying degrees of wrist dorsiflexion (i.e., extension). Although relationships between hand postures, wrist joint range of motion, and the external properties of wrist bones (e.g., surface morphology) have been examined, the relationship between hand postures and the internal properties of wrist bones (e.g., bone density) remains largely unexplored. Because articular joint surfaces transmit mechanical loads between conjoining limb bones, measures of density (e.g., magnitudes and patterns) in the subchondral cortical plate of bone of the distal radius can be used to evaluate load regimes experienced by the wrist joint in different hand postures. We assessed apparent (i.e. optical) density patterns in several extant catarrhine primate taxa partitioned into different hand posture groups: knuckle-walking apes, digitigrade monkeys, and palmigrade monkeys. Computed tomography osteoabsorptiometry (CT-OAM) was used to construct maximum intensity projection (MIP) maps of apparent densities. High apparent density areas were characterized relative to a dorsal-volar reference plane and compared across hand posture groups. All groups had large percentage areas of high apparent density in the dorsal region of the distal radial articular surface. Only knuckle-walking apes, however, had a large percentage area of high apparent density in the volar region of the distal radial articular surface. These patterns are consistent with radiocarpal articulations in specific hand postures as evidenced by available radiographic data and suggest that the different habitual hand postures adopted by monkeys and African apes during quadrupedal locomotion have different stereotypic loading patterns. This has implications for understanding the functional morphology and evolution of knuckle-walking and digitigrade hand postures in primates.

Density-based load estimation using two-dimensional finite element models: a parametric study

A parametric investigation was conducted to determine the effects on the load estimation method of varying: (1) the thickness of back-plates used in the two-dimensional finite element models of long bones, (2) the number of columns of nodes in the outer medial and lateral sections of the diaphysis to which the back-plate multipoint constraints are applied and (3) the region of bone used in the optimization procedure of the density-based load estimation technique. The study is performed using two-dimensional finite element models of the proximal femora of a chimpanzee, gorilla, lion and grizzly bear. It is shown that the density-based load estimation can be made more efficient and accurate by restricting the stimulus optimization region to the metaphysis/epiphysis. In addition, a simple method, based on the variation of diaphyseal cortical thickness, is developed for assigning the thickness to the back-plate. It is also shown that the number of columns of nodes used as multipoint constraints does not have a significant effect on the method.

Sensitivity of Vertebral Compressive Strength to Endplate Loading Distribution

The sensitivity of vertebral body strength to the distribution of axial forces along the endplate has not been comprehensively evaluated. Using quantitative computed tomography-based finite element models of 13 vertebral bodies, an optimization analysis was performed to determine the endplate force distributions that minimized (lower bound) and maximized (upper bound) vertebral strength for a given set of externally applied axial compressive loads. Vertebral strength was also evaluated for three generic boundary conditions: uniform displacement, uniform force, and a nonuniform force distribution in which the interior of the endplate was loaded with a force that was 1.5 times greater than the periphery. Our results showed that the relative difference between the upper and lower bounds on vertebral strength was 14.2±7.0%(mean±SD). While there was a weak trend for the magnitude of the strength bounds to be inversely proportional to bone mineral density (R2=0.32, p=0.02), both upper and lower bound vertebral strength measures were well predicted by the strength response under uniform displacement loading conditions (R2=0.91 and R2=0.99, respectively). All three generic boundary conditions resulted in vertebral strength values that were statistically indistinguishable from the loading condition that resulted in an upper bound on strength. The results of this study indicate that the uncertainty in strength arising from the unknown condition of the disc is dependent on the condition of the bone (whether it is osteoporotic or normal). Although bone mineral density is not a good predictor of strength sensitivity, vertebral strength under generic boundary conditions, i.e., uniform displacement or force, was strongly correlated with the relative magnitude of the strength bounds. Thus, explicit disc modeling may not be necessary.

A contact algorithm for density-based load estimation

An algorithm, which includes contact interactions within a joint, has been developed to estimate the dominant loading patterns in joints based on the density distribution of bone. The algorithm is applied to the proximal femur of a chimpanzee, gorilla and grizzly bear and is compared to the results obtained in a companion paper that uses a non-contact (linear) version of the density-based load estimation method. Results from the contact algorithm are consistent with those from the linear method. While the contact algorithm is substantially more complex than the linear method, it has some added benefits. First, since contact between the two interacting surfaces is incorporated into the load estimation method, the pressure distributions selected by the method are more likely indicative of those found in vivo. Thus, the pressure distributions predicted by the algorithm are more consistent with the in vivo loads that were responsible for producing the given distribution of bone density. Additionally, the relative positions of the interacting bones are known for each pressure distribution selected by the algorithm. This should allow the pressure distributions to be related to specific types of activities. The ultimate goal is to develop a technique that can predict dominant joint loading patterns and relate these loading patterns to specific types of locomotion and/or activities.

A comparison of the femoral head and neck trabecular architecture of Galago and Perodicticus using micro-computed tomography (microCT)

Innovations in micro-computed tomography (microCT) in the medical field have resulted in the development of techniques that allow the precise quantification of bone density and fabric related parameters of trabecular bone. For the purpose of this study, the technique was applied to a small sample of Perodicticus potto and Galago senegalensis femora to see if differences in loading environment elicit the predicted effects on trabecular structure. While the overall bone volume was approximately three times larger in the potto, there was no significant difference in the apparent volume density in the two taxa. When regional differences in the proximal femur were examined, the cancellous bone of the femoral head of Perodicticus potto and Galago senegalensis, while not differing in volume density, showed differences in trabecular orientation, with the potto having more randomly oriented trabeculae than the bushbaby. This was as hypothesized, given that the bushbaby submits its femora to more stereotypical loading environments than the potto. In the femoral neck, the cancellous bone was not only more randomly oriented, it was also denser in the potto compared with the bushbaby. This suggests that trabecular morphology may be extremely sensitive to certain differences in the loading environment and that this information, combined with information on cortical bone structure and external geometry, will result in a more complete understanding of how bone shape and composition correspond to loading and locomotor patterns. Ultimately, a synthesis of these different lines of evidence may have considerable applications in paleontological studies that attempt to reconstruct bone use from morphology.

Modeling and remodeling in a developing artiodactyl calcaneus: a model for evaluating Frost's Mechanostat hypothesis and its corollaries

The artiodactyl (mule deer) calcaneus was examined for structural and material features that represent regional differences in cortical bone modeling and remodeling activities. Cortical thickness, resorption and formation surfaces, mineral content (percent ash), and microstructure were quantified between and within skeletally immature and mature bones. These features were examined to see if they are consistent with predictions of Frost's Mechanostat paradigm of mechanically induced bone adaptation in a maturing "tension/compression" bone (Frost, 1990a,b, Anat Rec 226:403-413, 414-422). Consistent with Frost's hypothesis that surface modeling activities differ between the "compression" (cranial) and "tension" (caudal) cortices, the elliptical cross-section of the calcaneal diaphysis becomes more elongated in the direction of bending as a result of preferential (> 95%) increase in thickness of the compression cortex. Regional differences in mineral content and population densities of new remodeling events (NREs = resorption spaces plus newly forming secondary osteons) support Frost's hypothesis that intracortical remodeling activities differ between the opposing cortices: 1.) in immature and mature bones, the compression cortex had attained a level of mineralization averaging 8.9 and 6.8% greater (P < 0.001), respectively, than that of the tension cortex, and 2.) there are on average 350 to 400% greater population densities of NREs in the tension cortices of both age groups (P < 0.0003). No significant differences in cortical thickness, mineral content, porosity, or NREs were found between medial and lateral cortices of the skeletally mature bones, suggesting that no modeling or remodeling differences exist along a theoretical neutral axis. However, in mature bones these cortices differed considerably in secondary osteon cross-sectional area and population density. Consistent with Frost's hypothesis, remodeling in the compression cortex produced bone with microstructural organization that differs from the tension cortex. However, the increased remodeling activity of the tension cortex does not appear to be related to a postulated low-strain environment. Although most findings are consistent with predictions of Frost's Mechanostat paradigm, there are several notable inconsistencies. Additional studies are needed to elucidate the nature of the mechanisms that govern the modeling and remodeling activities that produce and maintain normal bone. It is proposed that the artiodactyl calcaneus will provide a useful experimental model for these studies.

Bone Load Estimation for the Proximal Femur Using Single Energy Quantitative CT Data

A density-based load estimation method was applied to determine femoral load patterns. Two-dimensional finite element models were constructed using single energy quantitative computed tomography (QCT) data from two femora. Basic load cases included parabolic pressure joint loads and constant tractions on the greater trochanter. An optimization procedure adjusted magnitudes of the basic load cases, such that the applied mechanical stimulus approached the ideal stimulus throughout each model. Dominant estimated load directions were generally consistent with published experimental data for gait. Other estimated loads suggested that loads at extreme joint orientations may be important to maintenance of bone structure. Remodeling simulations with the estimated loads produced density distributions qualitatively similar to the QCT data sets. Average nodal density errors between QCT data and predictions were 0.24 g/cm(3) and 0.28 g/cm(3). The results indicate that density-based load estimation could improve understanding of loading patterns on bones.

Proximal Femoral Density Patterns are Consistent with Bicentric Joint Loads

We developed an alternate method for density-based load estimation and applied it to estimate hip joint load distributions for two femora. Two-dimensional finite element models were constructed from single energy quantitative computed tomography (QCT) data. Load estimation was performed using five loading regions on the femoral head. Within each loading region, individual nodal loads, normal to the local surface, were supplied as input to the load estimation. An optimization procedure independently adjusted individual nodal load magnitudes in each region, and the magnitudes of muscle forces on the greater trochanter, such that the applied tissue stimulus approached the reference stimulus throughout the model. Dominant estimated load resultant directions were generally consistent with published experimental data for loads during gait. The estimated loads also suggested that loads near the extremes of the articulating surface may be important (even required) for development and maintenance of normal bone architecture. Estimated load distributions within nearly all regions predicted bicentric loading patterns, which are consistent with observations of hip joint incongruity. Remodeling simulations with the estimated loads predicted density distributions with features qualitatively similar to the QCT data sets. This study illustrates how applications of density-based bone load estimation can improve understanding of dominant loading patterns in other bones and joints. The prediction of bicentric loading suggests a very fine level of local adaptation to details of joint loading.

An Analysis and Comparison of Convergence and Uniqueness of Time-Independent Bone Adaptation Models

Several stimuli are proposed in the bone remodeling theory. It is not clear, if a unique solution exists and if the result is convergent using a certain stimulus. In this study, the strain stimulus, strain energy stimulus and the von Mises stress stimulus for bone remodeling are compared and applied to a square plate model using the finite element method. In the plane stress state, the remodeling equilibrium equations are transformed into functions of only the principal strains and the graphs of these functions are drawn in a diagram using the principal strains as the variables of two coordinate axes. The equation of the sum of principal strain squared equal to a constant is a circle in the diagram. The remodeling equilibrium equation of the strain stimulus is a quadrangle fitting into the circle, the remodeling equilibrium equation of the strain energy stimulus is an ellipse and the remodeling equilibrium equation of the von Mises stress stimulus is also an ellipse close to the principal strains circle when we take the same constants in the above equations. Using the finite element method, two models are performed with the uniform initial elastic properties and with the semi-random initial distribution of the elastic properties. The principal strains as the final finite element results converge within 2% of the objective constant for all the different stimuli. The obtained Young's moduli of two models as the adaptation object are different but in equilibrium, i.e. the equilibrium solution of adaptation model is not unique. The principal strains can not be used to examine the uniqueness of solution, since two different solutions can have the same results of principal strains. Using a certain stimulus, certain initial properties and a certain iterative equation, the solution is unique in equilibrium. The results using the model in this study show also that the same results can be obtained using any of the three stimuli when a proper constant in each remodeling equilibrium equation is chosen.

Development of epiphyseal structure and function in Didelphis virginiana (Marsupiala, Didelphidae)

This study addressed the question of how the epiphyses of growing mammals change their external shape and internal architecture during postnatal development. Ontogenetic transformations in the external form and internal structure of the fore- and hindlimb epiphyses were examined in a mixed cross-sectional sample of Didelphis virginiana using two methods: morphometric analysis of linear epiphyseal dimensions and histological staining of serially sectioned epiphyses. Metric data indicate that Virginia opossums are born with relatively short hindlimbs and long forelimbs, but by the time they are weaned their hindlimbs are longer than their forelimbs. Functional integration of the locomotor system in D. virginiana involves a decoupling of fore- and hindlimb growth rates so that between birth and weaning, femoral length, diaphyseal cross-sectional area, and articular surface area increase at a significantly faster rate than the corresponding humeral dimensions. Histological results demonstrate that these differences in growth rate are reflected in morphology of the humeral and femoral growth plate and epiphyseal cartilages. The humeral cartilages exhibit a level of cellular organization characteristic of more mature limb elements at earlier developmental stages compared to the femoral cartilages, which assume this anisotropic structure relatively later in postnatal development. Results presented here also reveal that the formation of articular cartilage and the initiation of epiphyseal ossification in D. virginiana are both correlated with the development of independent positional behaviors prior to weaning. These histological data, therefore, suggest that mechanical loading associated with the postnatal onset of locomotor and postural development may provide an important stimulus for the progression of ossification and the formation of articular cartilage in the epiphyses of growing mammals. J. Morphol. 239:283-296, 1999. © 1999 Wiley-Liss, Inc.

Structural variation of the distal condyles of the third metacarpal and third metatarsal bones in the horse

This study examined 3-dimensional (3D) distribution of sectors with contrasting density in the equine third metacarpal (McIII) and third metatarsal (MtIII) bones with a view to explaining the aetiology of distal condylar fractures. Macroradiography and computed tomographic (CT) imaging were used in the nondestructive study of bones obtained from horses, most of which were Thoroughbreds in race training. Distal condylar regions of McIII and MtIII were also studied in microradiographs of 100 microm thick mediolateral sections cut perpendicular to the dorsal and palmar/plantar articular surfaces. Qualitative and quantitative results from all methods used (radiography, CT and microradiographic stereology) demonstrated a densification (sclerosis) of subchondral bone located in the palmar/plantar regions of the medial and lateral condyles of both McIII and MtIII. Substantial density gradients between the denser condyles and the subchondral bone of the sagittal groove were shown to equate with anatomical differences in loading intensity during locomotion. It is hypothesised that such differences in bone density results in stress concentration at the palmar/plantar aspect of the condylar grooves, which may predispose to fracture.

Mechanobiological adaptation of subchondral bone as a function of joint incongruity and loading

Computed tomography (CT) has been employed to determine non-invasively the distribution of subchondral bone density in joints and to evaluate their dominant loading pattern. The objective of this study was to investigate the relationship between subchondral bone adaptation, joint incongruity and loading, in order to determine to what extent the loading conditions and/or geometric configuration can be inferred from the distribution of subchondral density. Finite element models of joints with various degrees of incongruity were designed and a current remodeling theory implemented using the node-based approach. Appropriate combinations of joint incongruity and loading yielded subchondral bone density patterns consistent with experimental findings, specifically a bicentric distribution in the humero-ulnar joint and a monocentric distribution in the humero-radial joint. However, other combinations of incongruity and loading produced similar subchondral density patterns. Both the geometric joint configuration and the loading conditions influence the distribution of subchondral density in such a way that one of these factors must be known a priori to estimate the other. Since subchondral density can be assessed by CT and joint geometry by magnetic resonance imaging, the dominant loading pattern of joints may be potentially derived in the living using these non-invasive imaging methods.

Finite element models in tissue mechanics and orthopaedic implant design

This article attempts to review the literature on finite element modelling in three areas of biomechanics: (i) analysis of the skeleton, (ii) analysis and design of orthopaedic devices and (iii) analysis of tissue growth, remodelling and degeneration. It is shown that the method applied to bone and soft tissue has allowed researchers to predict the deformations of musculoskeletal structures and to explore biophysical stimuli within tissues at the cellular level. Next, the contribution of finite element modelling to the scientific understanding of joint replacement is reviewed. Finally, it is shown that, by incorporating finite element models into iterative computer procedures, adaptive biological processes can be simulated opening an exciting field of research by allowing scientists to test proposed 'rules' or 'algorithms' for tissue growth, adaptation and degeneration. These algorithms have been used to explore the mechanical basis of processes such as bone remodelling, fracture healing and osteoporosis. RELEVANCE: With faster computers and more reliable software, computer simulation is becoming an important tool of orthopaedic research. Future research programmes will use computer simulation to reduce the reliance on animal experimentation, and to complement clinical trials.

Mechanical implications of humero-ulnar incongruity--finite element analysis and experiment

Previous studies show that the humero-ulnar joint is physiologically incongruous [Eckstein et al. (1995a) Anat. Rec. 243, 318-326] and exhibits a bicentric (ventro-dorsal) distribution of subchondral mineralization [Eckstein et al. (1995b) J. Orthop. Res. 13, 286-278]. We therefore asked: (1) Does humero-ulnar incongruity bring about a bicentric distribution of contact pressure? (2) Do tensile stresses occur in the subchondral bone of the trochlear notch that are in the same order of magnitude as the compressive stresses? (3) Do ventral and dorsal maxima of subchondral bone density correlate with a bicentric distribution of strain energy density? To that end, a two-dimensional finite element model was designed. The shape and material properties of the bones were based on CT and the boundary conditions selected to agree with resisted elbow extension at 90 degrees of flexion. The incongruity and contact areas were determined experimentally from casts, and the pressure distribution with Fuji Prescale film. In the model and the experiment contact stresses above 2 MPa were recorded in the ventral and dorsal parts of the joint, and values below 0.5 MPa in the depth of the notch. In the model, tensile stresses of 2.9 MPa were observed in the subchondral bone of the ulna, but not in the humerus. The subchondral strain energy density yielded a bicentric pattern in a model with homogeneous subchondral bone properties. It is shown that humero-ulnar incongruity brings about a bicentric distribution of contact pressure, a tensile stress in the notch that is in the same order of magnitude as the compressive stress, and a distribution of strain energy density that correlates with subchondral density patterns.

Testing the daily stress stimulus theory of bone adaptation with natural and experimentally controlled strain histories

Theories of bone adaptation generally consider that a departure in some feature of the normal homeostatic mechanical stimulus governs mechanical adaptation. Specifically, the 'daily stress stimulus' theory commonly used in computational models of bone adaptation suggests that the mechanical stimulus arises from a synthesis of the peak magnitudes from each loading event during a day. In this study, the homeostatic daily strain history of the adult turkey ulna was established by categorizing and counting the natural wing activities of adult male turkeys over a full 24h period. Strain signals were recorded in vivo for each activity type at three mid-diaphysis sites using stacked rosette strain gages. Following surgical isolation and transverse metaphyseal pinning of the ulnae, additional strain signals were recorded during controlled axial and torsional loading regimens associated with documented maintenance, loss, or addition of bone mass. When the present data were incorporated into the daily stress stimulus formulation, the theory did not consistently discriminate maintenance versus formation regimens, i.e., some maintenance regimens were associated with a substantially higher daily stimulus than some regimens causing bone formation.

Different loads can produce similar bone density distributions

Finite element models of a generic long bone and the proximal femur were used to identify important load characteristics and to determine whether small changes in load affect bone adaptation simulations. We also examined the effect of implants on the sensitivity of bone adaptation simulations to changes in loads. For each model, a primary load set was selected and incorporated in a bone adaptation simulation to generate a primary density distribution. A density-based load estimation method was used to determine a secondary set of loading conditions for each model. Each secondary load set was incorporated in a bone adaptation simulation and the resulting density distribution was compared to the corresponding primary density distribution. Nearly identical density distributions were produced for the natural generic long bone model (average nodal density difference 0.02 g/cm3). For the natural proximal femur model, the density distributions were very similar, but differences were apparent (average nodal density difference 0.07 g/cm3). The same primary and secondary load sets were used for bone adaptation simulations with implant models. For the proximal femur model, density distribution differences with the implant were very slightly less than those of the natural model. For the generic long bone model, the implant amplified differences between density distributions (average nodal density difference 0.14 g/cm3). Thus, variations in loading conditions may partially explain variations in long-term total joint outcome. The total equivalent stimulus load magnitudes for the two load sets for the generic long bone model were within 1%, and the stimulus-weighted average load directions were within 1 degree. The similarity of these parameters and the natural generic long bone density distributions indicate that the overall magnitude and average load direction are key factors affecting bone adaptation.