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

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

A theoretical framework for strain-related trabecular bone maintenance and adaptation

It is assumed that density and morphology of trabecular bone is partially controlled by mechanical forces. How these effects are expressed in the local metabolic functions of osteoclast resorption and osteoblast formation is not known. In order to investigate possible mechano-biological pathways for these mechanisms we have proposed a mathematical theory (Nature 405 (2000) 704). This theory is based on hypothetical osteocyte stimulation of osteoblast bone formation, as an effect of elevated strain in the bone matrix, and a role for microcracks and disuse in promoting osteoclast resorption. Applied in a 2-D Finite Element Analysis model, the theory explained the formation of trabecular patterns. In this article we present a 3-D FEA model based on the same theory and investigated its potential morphological predictability of metabolic reactions to mechanical loads. The computations simulated the development of trabecular morphological details during growth, relative to measurements in growing pigs, reasonably realistic. They confirmed that the proposed mechanisms also inherently lead to optimal stress transfer. Alternative loading directions produced new trabecular orientations. Reduction of load reduced trabecular thickness, connectivity and mass in the simulation, as is seen in disuse osteoporosis. Simulating the effects of estrogen deficiency through increased osteoclast resorption frequencies produced osteoporotic morphologies as well, as seen in post-menopausal osteoporosis. We conclude that the theory provides a suitable computational framework to investigate hypothetical relationships between bone loading and metabolic expressions.     

Analysis of microstructural and mechanical alterations of trabecular bone in a simulated three-dimensional remodeling process

Bone remodeling is a complex dynamic process, which modulates both bone mass and bone microstructure. In addition to bone mass, bone microstructure is an important contributor to bone quality in osteoporosis and fragility fractures. However, the quantitative knowledge of evolution of three-dimensional (3D) trabecular microstructure in adaptation to the external forces is currently limited. In this study, a new 3D simulation method of remodeling of human trabecular bone was developed to quantitatively study the dynamic evolution of bone mass and trabecular microstructure in response to different external loading conditions. The morphological features of trabecular plate and rod, such as thickness and number density in different orientations were monitored during the remodeling process using a novel imaging analysis technique, namely Individual Trabecula Segmentation (ITS). We showed that the volume fraction and microstructures of trabecular bone including, trabecular type and orientation, were determined by the applied mechanical load. Particularly, the morphological parameters of trabecular plates were more sensitive to the applied load, indicating that they played the major role in the mechanical properties of the trabecular bone. Reducing the applied load caused severe microstructural deteriorations of trabecular bone, such as trabecular plate perforation, rod breakage, and a conversion from plates to rods.     

A Computer-simulation Model Relating Bone-cell Metabolism to Mechanical Adaptation of Trabecular Architecture

The architecture of trabecular bone is thought to be an optimal mechanical structure in terms of maximal strength and stiffness, and minimal weight. The structural optimality seems to be maintained during growth and adulthood by adaptation of mass and structure through a relationship with actual mechanical usage. The formation and maintenance of the architecture is realized by bone-resorbing osteoclasts and bone-forming osteoblasts, the effector cells of bone metabolism. Hence, a feedback regulatory mechanism between external load and metabolism must exist. We have developed an FEA-based computer-simulation model to study explanations for the workings of such regulatory schemes (1: Huiskes et al. (2000), Nature, 404, 704-706). The model is based on a mechanosensory function of osteocytes, which are thought to react to the local strain-energy-density rate in the mineralized tissue, produced by dynamic external loading on the bone. As an effect of this signal, osteocytes are assumed to transfer an osteoblast recruitment stimulus to the surface, enhancing bone formation. Osteoclasts are assumed to resorb bone that is disused or damaged, in a spatially random manner. This model provides an explanation for the maintenance and adaptation of trabecular bone architecture as an optimal structure. In this article, the mathematical background of the model is specified.     

Mechanical loading of mouse caudal vertebrae increases trabecular and cortical bone mass-dependence on dose and genotype

Most in vivo studies addressing the skeletal responses of mice to mechanical loading have targeted cortical bone. To investigate trabecular bone responses also we have developed a caudal vertebral axial compression device (CVAD) that transmits mechanical loads to compress the fifth caudal vertebra via stainless steel pins inserted into the forth and sixth caudal vertebral bodies. Here, we used the CVAD in C57BL/6 (B6) and C3H/Hej (C3H) female mice (15 weeks of age) to investigate whether the effect of regular bouts of mechanical stimulation on bone modeling and bone mass was dependent on dose and genotype. A combined micro-computed tomographic and dynamic histomorphometric analysis was carried out at the end of a 4-week loading regimen (3,000 cycles, 10 Hz, 3 x week) for load amplitudes of 0N, 2N, 4N and 8N. Significant increases in trabecular bone mass of 9 and 21% for loads of 4N and 8N, respectively, were observed in B6 mice. A significant increase of 10% in trabecular bone mass occurred for a load of 8N in the C3H strain. For other loads, no significant increases were detected. Both mouse strains exhibited substantial increases in trabecular bone formation rates for all loads, B6: 111% (2N), 86% (4N), 164% (8N), C3H: 41% (2N), 38% (4N), 141% (8N). Significant decreases in osteoclast number of 146 and 93% for a load of 8N were detected in B6 and C3H mice, respectively. These findings demonstrate that the effect of loading on the structural and functional parameters of bone is dose and genotype dependent. The caudal vertebral loading model established here is proposed for further studies addressing the molecular processes involved in the skeletal responses to mechanical stimuli.     

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

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

Osteocyte-viability-based simulations of trabecular bone loss and recovery in disuse and reloading

Osteocyte apoptosis is known to trigger targeted bone resorption. In the present study, we developed an osteocyte-viability-based trabecular bone remodeling (OVBR) model. This novel remodeling model, combined with recent advanced simulation methods and analysis techniques, such as the element-by-element 3D finite element method and the ITS technique, was used to quantitatively study the dynamic evolution of bone mass and trabecular microstructure in response to various loading and unloading conditions. Different levels of unloading simulated the disuse condition of bed rest or microgravity in space. The amount of bone loss and microstructural deterioration correlated with the magnitude of unloading. The restoration of bone mass upon the reloading condition was achieved by thickening the remaining trabecular architecture, while the lost trabecular plates and rods could not be recovered by reloading. Compared to previous models, the predictions of bone resorption of the OVBR model are more consistent with physiological values reported from previous experiments. Whereas osteocytes suffer a lack of loading during disuse, they may suffer overloading during the reloading phase, which hampers recovery. The OVBR model is promising for quantitative studies of trabecular bone loss and microstructural deterioration of patients or astronauts during long-term bed rest or space flight and thereafter bone recovery.     

Adaptive diffuse domain approach for calculating mechanically induced deformation of trabecular bone

Remodelling of trabecular bone is essentially affected by the mechanical load of the trabeculae. Mathematical modelling and simulation of the remodelling process have to include time-consuming calculations of the displacement field within the complex trabecular structure under loading. We present an adaptive diffuse domain approach for calculating the elastic bone deformation based on micro computer tomogram data of real trabecular bone structures and compared it with a conventional voxel-based finite element method. In addition to allowing for higher computational efficiency, the adaptive approach is characterised by a very smooth representation of the bone surface, which suggests that this approach would be suitable as a basis for future simulations of bone resorption and formation processes within the trabecular structure.     

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.     

A sclerostin-based theory for strain-induced bone formation

Bone formation responds to mechanical loading, which is believed to be mediated by osteocytes. Previous theories assumed that loading stimulates osteocytes to secrete signals that stimulate bone formation. In computer simulations this 'stimulatory' theory successfully produced load-aligned trabecular structures. In recent years, however, it was discovered that osteocytes inhibit bone formation via the protein sclerostin. To reconcile this with strain-induced bone formation, one must assume that sclerostin secretion decreases with mechanical loading. This leads to a new 'inhibitory' theory in which loading inhibits osteocytes from inhibiting bone formation. Here we used computer simulations to show that a sclerostin-based model is able to produce a load-aligned trabecular architecture. An important difference appeared when we compared the response of the stimulatory and inhibitory models to loss of osteocytes, and found that the inhibitory pathway prevents the loss of trabeculae that is seen with the stimulatory model. Further, we demonstrated with combined stimulatory/inhibitory models that the two pathways can work side-by-side to achieve a load-adapted bone architecture.     

Testing two predictions for fracture load using computer models of trabecular bone

Aging induces several types of architectural changes in trabecular bone including thinning, increased levels of anisotropy, and perforation. It has been determined, on the basis of analysis of mathematical models, that reduction in fracture load caused by perforation is significantly higher than those due to equivalent levels of thinning or anisotropy. The analysis has also provided an expression which relates the fractional reduction of strength tau to the fraction of elements nu that have been removed from a network. Further, it was proposed that the ratio Gamma of the elastic constant of a sample and its linear response at resonance can be used as a surrogate for tau. Experimental validation of these predictions requires following architectural changes in a given sample of trabecular bone; techniques to study such changes using microcomputed tomography are only beginning to be available. In the present study, we use anatomically accurate computer models constructed from digitized images of bone samples for the purpose. Images of healthy bone are subjected to successive levels of synthetic degradation via surface erosion. Computer models constructed from these images are used to calculate their fracture load and other mechanical properties. Results from these computations are shown to be consistent with predictions derived from the analysis of mathematical models. Although the form of tau(nu) is known, parameters in the expression are expected to be sample-specific, and hence nu is not a reliable predictor of strength. We provide an example to demonstrate this. In contrast, analysis of model networks shows that the linear part of tau(Gamma) depends only on the structure of trabecular bone. Computations on models constructed from samples of iliac crest trabecular bone are shown to be in agreement with this assertion. Since Gamma can be computed from a vibrational assessment of bone, we argue that the latter can be used to introduce new surrogates for bone strength and hence diagnostic tools for osteoporosis.     

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.     

Load transfer analysis of the distal radius from in-vivo high-resolution CT-imaging.

Prevention of osteoporotic bone fractures requires accurate diagnostic methods to detect the increase in bone fragility at an early stage of osteoporosis. However, today's bone fracture risk prediction, primarily based on bone density measurement, is not sufficiently precise. There is increasing evidence that, in addition to bone density, also the bone microarchitecture and its mechanical loading conditions are important factors determining the fracture risk. Recently, it has been shown that new high-resolution imaging techniques in combination with new computer modeling techniques based on the finite-element (FE) method can account for these additional factors. These techniques might provide information that is more relevant for the prediction of bone fracture risk. So far, however, these new imaged-based FE techniques have not been feasible in-vivo. The objectives of this study were to quantify the load transfer through the trabecular network in a distal radius using a computer model based on in-vivo high-resolution images and to determine if common regions of fractures can be explained as a result of high tissue loading in these regions. The left distal radius and the two adjacent carpal bones of a healthy volunteer were imaged using a high-resolution three-dimensional CT system providing an isotropic resolution of 165 microm. The bone representation was converted into a FE-model that was used to calculate stresses and strains in the trabecular network. The two carpal bones were loaded using different load ratios (for each load case 1000 N in total) representing impact forces on the hand either in near-neutral position or ulnar/radial deviation. The load transfer through the trabecular network of the radius was characterized by the tissue strain energy density (SED) distribution for all load cases. It was found that the distribution of the tissue loading depends on the ratio of the forces acting on the carpal bones. For all load cases the higher SED values (on average: 0.02 +/- 0.08 (S.D.) N mm(-2)) are found in a 10 mm region adjacent to the articular surface which corresponds well with the region where Colles- or Chauffeur-fractures occur. We expect that, eventually, this new approach can lead to a better prediction of the fracture risk than methods based on bone density alone since it accounts for the bone microstructure as well as its loading conditions.     

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

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

Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy

The method of considering a single loading condition in the study of stress/morphology relationships in trabecular bone is expanded to include the multiple loading conditions experienced by bone in vivo. The bone daily loading histories are characterized in terms of stress magnitudes or cyclic strain energy density and the number of loading cycles. Relationships between local bone apparent density and loading history are developed which assume that bone mass is adjusted in response to strength or energy considerations. Three different bone maintenance criteria are described which are formulated based upon: (1) continuum model effective stress, (2) continuum model fatigue damage accumulation density, and (3) bone tissue strain energy density. These approaches can be applied to predict variations in apparent density within bone and among bones. We show that all three criteria have similar mathematical forms and may be related to the density (or concentration) of bone strain energy which is transferred (dissipated) in the mineralized tissue. The loading history and energy transfer concepts developed here can be applied to many different situations of growth, functional adaptation, injury, and aging of connective tissues.     

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

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

Differences in trabecular bone of leptin-deficient ob/ob mice in response to biomechanical loading

Objective: It is known that bone mineral density (BMD) and the strength of bone is predicted by body mass. Fat mass is a significant predictor of bone mineral density which correlates with body weight. This suggests that body fat regulates bone metabolism first by means of hormonal factors and second that the effects of muscle and loading are signaling factors in mechanotransduction. Leptin, a peptide hormone produced predominantly by white fat cells, is one of these hormonal factors. The aim of this study was to investigate and measure by micro-CT the different effects of weight-bearing on trabecular bone formation in mice without the stimulation of leptin.     

The behavior of adaptive bone-remodeling simulation models

The process of adaptive bone remodeling can be described mathematically and simulated in a computer model, integrated with the finite element method. In the model discussed here, cortical and trabecular bone are described as continuous materials with variable density. The remodeling rule applied to simulate the remodeling process in each element individually is, in fact, an objective function for an optimization process, relative to the external load. Its purpose is to obtain a constant, preset value for the strain energy per unit bone mass, by adapting the density. If an element in the structure cannot achieve that, it either turns to its maximal density (cortical bone) or resorbs completely.

It is found that the solution obtained in generally a discontinuous patchwork. For a two-dimensional proximal femur model this patchwork shows a good resemblance with the density distribution of a real proximal femur.

It is shown that the discontinuous end configuration is dictated by the nature of the differential equations describing the remodeling process. This process can be considered as a nonlinear dynamical system with many degrees of freedom, which behaves divergent relative to the objective, leading to many possible solutions. The precise solution is dependent on the parameters in the remodeling rule, the load and the initial conditions. The feedback mechanism in the process is self-enhancing; denser bone attracts more strain energy, whereby the bone becomes even more dense. It is suggested that this positive feedback of the attractor state (the strain energy field) creates order in the end configuration. In addition, the process ensures that the discontinuous end configuration is a structure with a relatively low mass, perhaps a minimal-mass structure, although this is no explicit objective in the optimization process.

It is hypothesized that trabecular bone is a chaotically ordered structure which can be considered as a fractal with characteristics of optimal mechanical resistance and minimal mass, of which the actual morphology depends on the local (internal) loading characteristics, the sensor-cell density and the degree of mineralization.     

Analysis of bone architecture sensitivity for changes in mechanical loading, cellular activity, mechanotransduction, and tissue properties

Bone has an architecture which is optimized for its mechanical environment. In various conditions, this architecture is altered, and the underlying cause for this change is not always known. In the present paper, we investigated the sensitivity of the bone microarchitecture for four factors: changes in bone cellular activity, changes in mechanical loading, changes in mechanotransduction, and changes in mechanical tissue properties. The goal was to evaluate whether these factors can be the cause of typical bone structural changes seen in various pathologies. For this purpose, we used an established computational model for the simulation of bone adaptation. We performed two sensitivity analyses to evaluate the effect of the four factors on the trabecular structure, in both developing and adult bone. According to our simulations, alterations in mechanical load, bone cellular activities, mechanotransduction, and mechanical tissue properties may all result in bone structural changes similar to those observed in various pathologies. For example, our simulations confirmed that decreases in loading and increases in osteoclast number and activity may lead to osteoporotic changes. In addition, they showed that both increased loading and decreased bone matrix stiffness may lead to bone structural changes similar to those seen in osteoarthritis. Finally, we found that the model may help in gaining a better understanding of the contribution of individual disturbances to a complicated multi-factorial disease process, such as osteogenesis imperfecta.     

Fracture prediction for the proximal femur using finite element models: Part II--Nonlinear analysis.

In Part I we reported the results of linear finite element models of the proximal femur generated using geometric and constitutive data collected with quantitative computed tomography. These models demonstrated excellent agreement with in vitro studies when used to predict ultimate failure loads. In Part II, we report our extension of those finite element models to include nonlinear behavior of the trabecular and cortical bone. A highly nonlinear material law, originally designed for representing concrete, was used for trabecular bone, while a bilinear material law was used for cortical bone. We found excellent agreement between the model predictions and in vitro fracture data for both the onset of bone yielding and bone fracture. For bone yielding, the model predictions were within 2 percent for a load which simulated one-legged stance and 1 percent for a load which simulated a fall. For bone fracture, the model predictions were within 1 percent and 17 percent, respectively. The models also demonstrated different fracture mechanisms for the two different loading configurations. For one-legged stance, failure within the primary compressive trabeculae at the subcapital region occurred first, leading to load transfer and, ultimately, failure of the surrounding cortical shell. However, for a fall, failure of the cortical and trabecular bone occurred simultaneously within the intertrochanteric region. These results support our previous findings that the strength of the subcapital region is primarily due to trabecular bone whereas the strength of the intertrochanteric region is primarily due to cortical bone.