Investigation of cancellous bone architecture using structural optimisation

The optimality of bone's internal architecture and its remodelling is investigated by applying topological optimisation. The os-calcis (heel bone) which is subject to a relatively simple loading environment was used as the test problem. The solution is compared with X-ray and CT images of the os-calcis and is demonstrated to agree favourably with the bone's typical trabecular orientation in the coronal plane. The optimality of bone remodelling is further investigated by applying a perturbation during optimisation which leads to a different optimum topology. This is compared with in-vivo experimental results from a model in which the same perturbation was applied. The findings indicate that there may exist more than one optimum state for the same mechanical condition and that structural optimisation is able to find these multiple states. It is therefore suggested that structural optimisation can be used to investigate the mechanisms of bone remodelling.     

On the role of bone damage in calcium homeostasis

Bone serves as the reservoir of some minerals including calcium. If calcium is needed anywhere in the body, it can be removed from the bone matrix by resorption and put back into the blood flow. During bone remodelling the resorbed tissue is replaced by osteoid which gets mineralized very slowly. Then, calcium homeostasis is controlled by bone remodelling, among other processes: the more intense is the remodelling activity, the lower is the mineral content of bone matrix. Bone remodelling is initiated by the presence of microstructural damage. Some experimental evidences show that the fatigue properties of bone are degraded and more microdamage is accumulated due to the external load as the mineral content increases. That damage initiates bone remodelling and the mineral content is so reduced. Therefore, this process prevents the mineral content of bone matrix to reach very high (non-physiological) values. A bone remodelling model has been used to simulate this regulatory process. In this model, damage is an initiation factor for bone remodelling and is estimated through a fatigue algorithm, depending on the macroscopic strain level. Mineral content depends on bone remodelling and mineralization rate. Finally, the bone fatigue properties are defined as dependent on the mineral content, closing the interconnection between damage and mineral content. The remodelling model was applied to a simplified example consisting of a bar under tension with an initially heterogeneous mineral distribution. Considering the fatigue properties as dependent on the mineral content, the mineral distribution tends to be homogeneous with an ash fraction within the physiological range. If such dependance is not considered and fatigue properties are assumed constant, the homogenization is not always achieved and the mineral content may rise up to high non-physiological values. Thus, the interconnection between mineral content and fatigue properties is essential for the maintenance of bone's structural integrity as well as for the calcium homeostasis.     

Quantifying the strain history of bone: spatial uniformity and self-similarity of low-magnitude strains

We hypothesize that when a broad spectrum of bone strain is considered, strain history is similar for different bones in different species. Using a data collection protocol with a fine resolution, mid-diaphyseal strains were measured in vivo for both weightbearing and non-weightbearing bones in three species: dog, sheep, and turkey, with strain information collected continuously while the animals performed their natural daily activities. The daily strain history was quantified by both counting cyclic strain events (to quantify the distribution of strains of different magnitudes) and by estimating the average spectral characteristics of the strain (to quantify the frequency content of the strain signals). Counting of the daily (12-24 h) strain events show that large strains (> 1000 microstrain) occur relatively few times a day, while very small strains (< 10 microstrain) occur thousands of times a day. The lower magnitude strains (< approximately 200 microstrain) are found to be more uniform around the bone cross-section than the higher magnitude, peak strains. Strain dynamics are found to be well described by a power-law relationship and exhibit self-similar characteristics. These data lead to the suggestion that the organization of bone tissue is driven by the continual barrage of activity spanning a wide but consistent range of frequency and amplitude, and until the mechanism of bone's mechanosensory system is fully understood, all portions of bone's strain history should be considered to possibly play a role in bone adaptation.     

Morpho-functional correlations of the structure of bone cells and adjoining bone matrix in the developing bone

By means of scanning and transmissive electron microscopy methods structure of the developing bone has been studied. Interconnection of the cell structure and spatial organization of the adjoining matrix has been demonstrated. On the surface of the growing bone not only forming areas have been revealed, where under osteoblasts at various functional states, osteoid layer is determined, but also areas of resorption and completed osteogenesis. This demonstrates an interrupted character of osteogenesis at modelling. At the same time for the remodelling process presence of erosive lacunae is specific; they are filled with a newly deposited collagenous matrix. Therefore, it is possible to suppose that formation of the bone as an organ during the postnatal development includes in itself both mechanisms supporting its form at outgrowth of the osseous matrix volume (modeling) and its continuous rearrangement and adaptation to real conditions of functioning (remodelling).     

Measurement of microstructural strain in cortical bone

It is well known that mechanical factors affect bone remodeling such that increased mechanical demand results in net bone formation, whereas decreased demand results in net bone resorption. Current theories suggest that bone modeling and remodeling is controlled at the cellular level through signals mediated by osteocytes. The objective of this study was to investigate how macroscopically applied bone strains similar in magnitude to those that occur in vivo are manifest at the microscopic level in the bone matrix. Using a digital image correlation strain measurement technique, experimentally determined bone matrix strains around osteocyte lacuna resulting from macroscopic strains of approximately 2,000 microstrain (0.2%) reach levels of over 30,000 microstrain (3%) over fifteen times greater than the applied macroscopic strain. Strain patterns were highly heterogeneous and in some locations similar to observed microdamage around osteocyte lacuna indicating the resulting strains may represent the precursors to microdamage. This information may lead to a better understanding of how bone cells are affected by whole bone functional loading.     

Simulation of vertebral trabecular bone loss using voxel finite element analysis

Trabecular bone loss in human vertebral bone is characterised by thinning and eventual perforation of the horizontal trabeculae. Concurrently, vertical trabeculae are completely lost with no histological evidence of significant thinning. Such bone loss results in deterioration in apparent modulus and strength of the trabecular core. In this study, a voxel-based finite element program was used to model bone loss in three specimens of human vertebral trabecular bone. Three sets of analyses were completed. In Set 1, strain adaptive resorption was modelled, whereby elements which were subject to the lowest mechanical stimulus (principal strain) were removed. In Set 2, both strain adaptive and microdamage mechanisms of bone resorption were included. Perforation of vertical trabeculae occurred due to microdamage resorption of elements with strains that exceeded a damage threshold. This resulted in collapse of the trabecular network under compression loading for two of the specimens tested. In Set 3, the damage threshold strain was gradually increased as bone loss progressed, resulting in reduced levels of microdamage resorption. This mechanism resulted in trabecular architectures in which vertical trabeculae had been perforated and which exhibited similar apparent modulus properties compared to experimental values reported in the literature. Our results indicate that strain adaptive remodelling alone does not explain the deterioration in mechanical properties that have been observed experimentally. Our results also support the hypothesis that horizontal trabeculae are lost principally by strain adaptive resorption, while vertical trabeculae may be lost due to perforation from microdamage resorption followed by rapid strain adaptive resorption of the remaining unloaded trabeculae.     

Static vs dynamic loads as an influence on bone remodelling

Remodelling activity in the avian ulna was assessed under conditions of disuse alone, disuse with a superimposed continuous compressive load, and disuse interrupted by a short daily period of intermittent loading. The ulnar preparation consisted of the 110 mm section of the bone shaft between two submetaphyseal osteotomies. Each end of the preparation was transfixed by a stainless steel pin and the shaft either protected from normal functional loading with the pins joined by external fixators, loaded continuously in compression by joining the pins with springs, or loaded intermittently in compression for a single 100 s period per day by engaging the pins in an Instron machine. Similar loads (525 N) were used in both static and dynamic cases. The strains engendered were determined by strain gauges, and at their maximum around the bone's midshaft were − 0.002. The intermittent load was applied at a frequency of 1 Hz as a ramped square wave, with a rate of change of strain during the ramp of 0.01 s⁻¹. Peak strain at the midshaft of the ulna during wing flapping in the intact bone was recorded from bone bonded strain gauges in vivo as −0.0033 with a maximum rate of change of strain of 0.056 s⁻¹.

Examination of bone sections from the midpoint of the preparation after an 8 week period indicated that in both non-loaded and statically loaded bones there was an increase in both endosteal diameter and intra cortical porosity. These changes produced a decrease in cross sectional area which was similar in the two groups (− 13%). Intermittently loaded bones however showed a 24% increase in cross sectional area resulting from new bone deposited predominantly, but not exclusively, on the periosteal surface. It appears that in this preparation, a static load sufficient to produce strains in the functional dynamic strain range has no effect on bone remodelling, whereas a similar load applied intermittently for a short daily period may be associated with a substantial increase in bone mass.     

A theoretical model of the effect of continuum damage on a bone adaptation model

Throughout life, bone is continuously turning over by the well-regulated processes of bone formation and resorption. Everyday activities damage bone, and this damage is normally repaired in a continuous remodelling process. When an imbalance in this remodelling process occurs, bones may become more susceptible to fracture. This paper is devoted to a theoretical modelling of the competition between damage and internal remodelling in bones. The general theory of adaptive damaged-elastic materials proposed here as a model for the physiological process of damaged-bone remodelling follows the general framework of continuum thermodynamics where new damaged-bone remodelling law and associated thermodynamical restrictions are stated, and specialized to the case of small strain in isothermal processes. An attempt is also made to derive: (a) the damage force (adaptive damage energy release rate ) which controls the microcracks propagation and arrest, and (b) the damage rule by introducing damage thresholds and loading/unloading conditions.     

An adaptation model for trabecular bone at different mechanical levels

Background  

Bone has the ability to adapt to mechanical usage or other biophysical stimuli in terms of its mass and architecture, indicating that a certain mechanism exists for monitoring mechanical usage and controlling the bone's adaptation behaviors. There are four zones describing different bone adaptation behaviors: the disuse, adaptation, overload, and pathologic overload zones. In different zones, the changes of bone mass, as calculated by the difference between the amount of bone formed and what is resorbed, should be different.     

A novel mathematical model of bone remodelling cycles for trabecular bone at the cellular level

After an initial phase of growth and development, bone undergoes a continuous cycle of repair, renewal and optimisation by a process called remodelling. This paper describes a novel mathematical model of the trabecular bone remodelling cycle. It is essentially formulated to simulate a remodelling event at a fixed position in the bone, integrating bone removal by osteoclasts and formation by osteoblasts. The model is developed to construct the variation in bone thickness at a particular point during the remodelling event, derived from standard bone histomorphometric analyses. The novelties of the approach are the adoption of a predator-prey model to describe the dynamic interaction between osteoclasts and osteoblasts, using a genetic algorithm-based solution; quantitative reconstruction of the bone remodelling cycle; and the introduction of a feedback mechanism in the bone formation activity to co-regulate bone thickness. The application of the model is first demonstrated by using experimental data recorded for normal (healthy) bone remodelling to predict the temporal variation in the number of osteoblasts and osteoclasts. The simulated histomorphometric data and remodelling cycle characteristics compare well with the specified input data. Sensitivity studies then reveal how variations in the model's parameters affect its output; it is hoped that these parameters can be linked to specific biochemical factors in the future. Two sample pathological conditions, hypothyroidism and primary hyperparathyroidism, are examined to demonstrate how the model could be applied more broadly, and, for the first time, the osteoblast and osteoclast populations are predicted for these conditions. Further data are required to fully validate the model's predictive capacity, but this work shows it has potential, especially in the modelling of pathological conditions and the optimisation of the treatment of those conditions.     

Thoughts on bone biomechanics

Simple simulations of stresses and strains in several types of localised anatomical features have been modelled using basic biomechanical thinking, experimental stress analysis employing photo-elastic methods and theoretical stress and strain analysis using a finite elements approach where the computational program utilises fast LaGrangian analysis of continua. The studies are providing results showing that, for given load proportions, the particular forms of some bones and bony architectural features seem related to stresses and strains that are relatively evenly distributed across surfaces (and therefore mechanically optimal). The results particularly show how apparently opposite (paradoxical) situations can readily occur within anatomical systems. They also provide strong biological information relevant to the assessment of such features of bone architecture in phylogenetic investigations. They even give information relevant to clinical problems.     

Understanding site-specific residual strain and architecture in bovine cortical bone

Living bone is considered as adaptive material to the mechanical functions, which continually undergoes change in its histological arrangement with respect to external prolonged loading. Such remodeling phenomena within bone depend on the degree of stimuli caused by the mechanical loading being experienced, and therefore, are specific to the sites. In the attempts of understanding strain adaptive phenomena within bones, different theoretical models have been proposed. Also, the existing literatures mostly follow the measurement of surface strains using strain gauges to experimentally quantify the strains experienced in the functional environment. In this work, we propose a novel idea of understanding site-specific functional adaptation to the prolonged load in bone on the basis of inherited residual strains and structural organization. We quantified the residual strains and amount of apatite crystals distribution, i.e., the degree of orientation, using X-ray diffraction procedures. The sites of naturally existing hole in bone, called foramen, are considered from bovine femur and metacarpal samples. Significant values of residual strains are found to exist in the specimens. Trends of residual strains noted in the specimens are mostly consistent with the degree of orientation of the crystallites. These features explain the response behavior of bone to the mechanical loading history near the foramen sites. Preferential orientation of crystals mapped around a femoral foramen specimen showed furnished tailored arrangement of the crystals around the hole. Effect of external loading at the femoral foramen site is also explained by the tensile loading experiment.     

Osteocyte lacunae tissue strain in cortical bone

Current theories suggest that bone modeling and remodeling are controlled at the cellular level through signals mediated by osteocytes. However, the specific signals to which bone cells respond are still unknown. Two primary theories are: (1) osteocytes are stimulated via the mechanical deformation of the perilacunar bone matrix and (2) osteocytes are stimulated via fluid flow generated shear stresses acting on osteocyte cell processes within canaliculi. Recently, much focus has been placed on fluid flow theories since in vitro experiments have shown that bone cells are more responsive to analytically estimated levels of fluid shear stress than to direct mechanical stretching using macroscopic strain levels measured on bone in vivo. However, due to the complex microstructural organization of bone, local perilacunar bone tissue strains potentially acting on osteocytes cannot be reliably estimated from macroscopic bone strain measurements. Thus, the objective of this study was to quantify local perilacunar bone matrix strains due to macroscopically applied bone strains similar in magnitude to those that occur in vivo. Using a digital image correlation strain measurement technique, experimentally measured bone matrix strains around osteocyte lacunae resulting from macroscopic strains of approximately 2000 microstrain are significantly greater than macroscopic strain on average and can reach peak levels of over 30,000 microstrain locally. Average strain concentration factors ranged from 1.1 to 3.8, which is consistent with analytical and numerical estimates. This information should lead to a better understanding of how bone cells are affected by whole bone functional loading.     

The osteocyte as a wiring transmission system

The mechanism of transduction of mechanical strains into biological signals remains one of the more baffling problems of skeletal homeostasis. The updated literature ascribes to osteocytes the function of sensing the strains induced into the bone matrix by mechanical stresses. Whether the osteocytes perform such function by themselves or they are helped by other cells is also unknown. Indeed TEM investigations carried out in our laboratory pointed out the existence of a functional syncytium among all the cells of the osteogenic lineage (COL; stromal cells, osteoblasts or bone lining cells, osteocytes). On the basis of this finding, we suggested that COL may reciprocally modulate their function not only by volume transmission (paracrine and autocrine stimulation) but also by wiring transmission, namely in a neuronal like manner. Thanks to their location, osteocytes should theoretically be the first cells of COL functional syncytium to sense mechanical strains, whereas stromal cells should be the first to be activated by hormonal molecules diffusing across the endothelial lining. Since PTH and Estrogen receptors have also been localized on osteocytes, and considering that such hormones have been suggested to modulate the sensitivity to strain of the bone mechanosensor, we suggested that the osteocyte syncytium may constitute the microscopic bone structure that sense both mechanical strain and biochemical factors and, at any moment, after having combined the two types of stimuli, issues the appropriate signals to the other bone cells by volume and/or wiring-transmission. Stromal cells, on the other hand, besides transmitting signals from vascular endothelium to bone cells, may control the differentiation and then direct the course of the osteoblasts around the vascular framework.     

Relationships of loading history and structural and material characteristics of bone: development of the mule deer calcaneus

If a bone's morphologic organization exhibits the accumulated effects of its strain history, then the relative contributions of a given strain stimulus to a bone's development may be inferred from a bone's hierarchical organization. The artiodactyl calcaneus is a short cantilever, loaded habitually in bending, with prevalent compression in the cranial (Cr) cortex, tension in the caudal (Cd) cortex, and shear in the medial and lateral cortices (i.e., neutral axis). Artiodactyl calcanei demonstrate unusually heterogeneous structural and material organization between these cortices. This study examines potential relationships between developmental morphologic variations and the functional strain distribution of the deer calcaneus. One calcaneus was obtained from each of 36 (fetus to adult) wild deer. Predominant collagen fiber orientation (CFO), microstructural characteristics, mineral content (% ash), and geometric parameters were determined from transversely cut segments. Radiographs were examined for arched trabeculae, which may reflect tension/compression stress trajectories. Results showed that cross-sectional shape changes with age from quasi-circular to quasi-elliptical, with the long axis in the cranial-caudal direction of habitual bending. Cranial ("compression") cortical thickness increased at a greater rate than the Cd ("tension") cortex. Fetal bones exhibited arched trabeculae. Percent ash was not uniform (Cr > Cd), and this disparity increased with age (absolute differences: 2.5% fetuses, 4.3% adults). Subadult bones showed progressively more secondary osteons and osteocyte lacunae in the Cr cortex, but the Cd cortex tended to have more active remodeling in the subadult and adult bones. Nonuniform Cr:Cd CFO patterns first consistently appear in the subadults, and are correlated with secondary bone formation and habitual strain mode. Medial and lateral cortices in these groups exhibited elongated secondary osteons. These variations may represent "strain-mode-specific" (i.e., tension, compression, shear) adaptations. The heterogeneous organization may also be influenced by variations in longitudinal strain magnitude (highest in the Cr cortex) and principal strain direction-oblique in medial-lateral cortices (where shear strains also predominate). Other factors such as local reductions in longitudinal strain may influence the increased remodeling activity of the Cd cortex. Some structural variations, such as arched trabeculae, that are established early in ontogeny may be strongly influenced by genetic- or epigenetic-derived processes. Material variations, such as secondary osteon population densities and CFO, which appear later, may be products of extragenetic factors, including microdamage.     

Estimation of bone permeability using accurate microstructural measurements

While interstitial fluid flow is necessary for the viability of osteocytes, it is also believed to play a role in bone's mechanosensory system by shearing bone cell membranes or causing cytoskeleton deformation and thus activating biochemical responses that lead to the process of bone adaptation. However, the fluid flow properties that regulate bone's adaptive response are poorly understood. In this paper, we present an analytical approach to determine the degree of anisotropy of the permeability of the lacunar-canalicular porosity in bone. First, we estimate the total number of canaliculi emanating from each osteocyte lacuna based on published measurements from parallel-fibered shaft bones of several species (chick, rabbit, bovine, horse, dog, and human). Next, we determine the local three-dimensional permeability of the lacunar-canalicular porosity for these species using recent microstructural measurements and adapting a previously developed model. Results demonstrated that the number of canaliculi per osteocyte lacuna ranged from 41 for human to 115 for horse. Permeability coefficients were found to be different in three local principal directions, indicating local orthotropic symmetry of bone permeability in parallel-fibered cortical bone for all species examined. For the range of parameters investigated, the local lacunar-canalicular permeability varied more than three orders of magnitude, with the osteocyte lacunar shape and size along with the 3-D canalicular distribution determining the degree of anisotropy of the local permeability. This two-step theoretical approach to determine the degree of anisotropy of the permeability of the lacunar-canalicular porosity will be useful for accurate quantification of interstitial fluid movement in bone.     

Multilevel finite element modeling for the prediction of local cellular deformation in bone

The underlying mechanisms by which bone cells respond to mechanical stimuli or how mechanical loads act on osteocytes housed in lacunae in bone are not well understood. In this study, a multilevel finite element (FE) approach is applied to predict local cell deformations in bone tissue. The local structure of the matrix dictates the local mechanical environment of an osteocyte. Cell deformations are predicted from detailed linear FE analysis of the microstructure, consisting of an arrangement of cells embedded in bone matrix material. This work has related the loads applied to a whole femur during the stance phase of the gait cycle to the strain of a single lacuna and of canaliculi. The predicted bone matrix strains around osteocyte lacunae and canaliculi were nonuniform and differed significantly from the macroscopically measured strains. Peak stresses and strains in the walls of the lacuna were up to six times those in the bulk extracellular matrix. Significant strain concentrations were observed at sites where the process meets the cell body.     

Bone remodelling algorithms incorporating both strain and microdamage stimuli

Biomechanical theories to predict bone remodelling have used either mechanical strain or microdamage as the stimulus driving cellular responses. Even though experimental data have implicated both stimuli in bone cell regulation, a mechano-regulatory system incorporating both stimuli has not yet been proposed. In this paper, we test the hypothesis that bone remodelling may be regulated by signals due to both strain and microdamage. Four mechano-regulation algorithms are studied where the stimulus is: strain, damage, combined strain/damage, and either strain or damage with damage-adaptive remodelling prioritised when damage is above a critical level. Each algorithm is implemented with both bone lining cell (surface) sensors and osteocyte cell (internal) sensors. Each algorithm is applied to prediction of a bone multicellular unit (BMU) remodelling on the surface of a bone trabecula. It is predicted that a regulatory system capable of responding to changes in either strain or microdamage but which prioritises removal of damaged bone when damage is above a critical level, is the only one that provides a plausible prediction of BMU behaviour. A mechanism for this may be that, below a certain damage threshold, osteocyte processes can sense changes in strain and fluid flow but above the threshold damage interferes with the signalling mechanism, or causes osteocyte apoptosis so that a remodelling response occurs to remove the dead osteocytes.