[Bone Cell Biology Assessed by Microscopic Approach. A mathematical approach to understand bone remodeling].

It is well known that bone tissue can change its outer shape and internal structure by remodeling according to a changing mechanical environment. However, the mechanism of bone functional adaptation induced by the collaborative metabolic activities of bone cells in response to mechanical stimuli remains elusive. In this article, we focus on the hierarchy of bone structure and function from the microscopic cellular level to the macroscopic tissue level. We provide an overview of a mathematical approach to understand the adaptive changes in trabecular morphology under the application of mechanical stress.

Aging effects on osteoclast progenitor dynamics affect variability in bone turnover via feedback regulation.

Bone turnover markers (BTMs) are commonly used in osteoporosis treatment as indicators of cell activities of bone-resorbing osteoclasts and bone-forming osteoblasts. The wide variability in their values due to multiple factors, such as aging and diseases, makes it difficult for physicians to utilize them for clinical decision-making. The progenitors of osteoclasts and osteoblasts are indispensable for a comprehensive interpretation of the variability in BTM values because these upstream progenitors strongly regulate the downstream cell activities of bone turnover. However, understanding the complex interactions among the multiple populations of bone cells is challenging. In this study, we aimed to gain a fundamental understanding of the mechanism by which the progenitor dynamics affect the variability in bone turnover through in silico experiments by exploring the cell dynamics with aging effects on osteoporosis. Negative feedback control driven by the consumptive loss of progenitors prevents rapid bone loss due to excessive bone turnover, and through feedback regulation, aging effects on osteoclast differentiation and osteoclast progenitor proliferation cause variability in the osteoclast and osteoblast activity balance and its temporal transition. By expressing the variability in the bone turnover status, our model describes the individualities of patients based on their clinical backgrounds. Therefore, our model could play a powerful role in assisting tailored treatment and has the potential to resolve the various health problems associated with osteoporosis worldwide.

Chondrocyte hypertrophy in the growth plate promotes stress anisotropy affecting long bone development through chondrocyte column formation.

The length of long bones is determined by column formation of proliferative chondrocytes and subsequent chondrocyte hypertrophy in the growth plate during bone development. Despite the importance of mechanical loading in long bone development, the mechanical conditions of the cells within the growth plate, such as the stress field, remain unclear owing to the difficulty in investigating spatiotemporal changes within dynamically growing tissues. In this study, the mechanisms of longitudinal bone growth were investigated from a mechanical perspective through column formation of proliferative chondrocytes within the growth plate before secondary ossification center formation using continuum-based particle models (CbPMs). A one-factor model, which simply describes essential aspects of a biological signaling cascade regulating cell activities within the growth plate, was developed and incorporated into CbPM. Subsequently, the developmental process and maintenance of the growth plate structure and resulting bone morphogenesis were simulated. Thus, stress anisotropy in the proliferative zone that affects bone elongation through chondrocyte column formation was identified and found to be promoted by chondrocyte hypertrophy. These results provide further insights into the mechanical regulation of multicellular dynamics during bone development.

Development of continuum-based particle models of cell growth and proliferation for simulating tissue morphogenesis.

Biological tissues acquire various characteristic shapes through morphogenesis. Tissue shapes result from the spatiotemporally heterogeneous cellular activities influenced by mechanical and biochemical environments. To investigate multicellular tissue morphogenesis, this study aimed to develop a novel multiscale method that can connect each cellular activity to the mechanical behaviors of the whole tissue by constructing continuum-based particle models of cellular activities. This study proposed mechanical models of cell growth and proliferation that are expressed as volume expansion and cell division by extending the material point method. By simulating cell hypertrophy and proliferation under both free and constraint conditions, the proposed models demonstrated potential for evaluating the mechanical state and tracing cells throughout tissue morphogenesis. Moreover, the effect of a cell size checkpoint was incorporated into the cell proliferation model to investigate the mechanical behaviors of the whole tissue depending on the condition of cellular activities. Consequently, the accumulation of strain energy density was suppressed because of the influence of the checkpoint. In addition, the whole tissues acquired different shapes depending on the influence of the checkpoint. Thus, the models constructed herein enabled us to investigate the change in the mechanical behaviors of the whole tissue according to each cellular activity depending on the mechanical state of the cells during morphogenesis.

Computational framework for analyzing flow-induced strain on osteocyte as modulated by microenvironment.

Osteocytes buried in bone matrix are major mechanosensory cells that regulate bone remodeling in response to interstitial fluid flow in a lacuno-canalicular porosity. To gain an understanding of the mechanism of osteocyte mechanosensing, it is important to be able to evaluate the local strain on the osteocyte process membrane induced by the interstitial fluid flow. The microenvironment of the osteocytes, including the pericellular matrix (PCM) and canalicular ultrastructure, is a key modulator of the flow-induced strain on the osteocyte process membrane because it produces heterogeneous flow patterns in the pericellular space. To investigate the effect of changes in the microenvironment of osteocytes on the flow-induced strain, we developed a novel computational framework for analyzing the fluid-structure interaction. Computer simulations based on the proposed framework enabled evaluation of the spatial distribution of flow-induced strain on the osteocyte process membrane according to changes in the PCM density and canalicular curvature. The simulation results reveal that a decrease in PCM density and an increase in canalicular curvature, each of which is associated with aging and bone disease, have the notable effect of enhancing local flow-induced strain on the osteocyte process membrane. We believe that the proposed computational framework is a promising framework for investigating cell-specific mechanical stimuli and that it has the potential to accelerate the mechanobiological study of osteocytes by providing a deeper understanding of their mechanical environment in living bone tissue.

High-resolution image-based simulation reveals membrane strain concentration on osteocyte processes caused by tethering elements.

Osteocytes are vital for regulating bone remodeling by sensing the flow-induced mechanical stimuli applied to their cell processes. In this mechanosensing mechanism, tethering elements (TEs) connecting the osteocyte process with the canalicular wall potentially amplify the strain on the osteocyte processes. The ultrastructure of the osteocyte processes and canaliculi can be visualized at a nanometer scale using high-resolution imaging via ultra-high voltage electron microscopy (UHVEM). Moreover, the irregular shapes of the osteocyte processes and the canaliculi, including the TEs in the canalicular space, should considerably influence the mechanical stimuli applied to the osteocytes. This study aims to characterize the roles of the ultrastructure of osteocyte processes and canaliculi in the mechanism of osteocyte mechanosensing. Thus, we constructed a high-resolution image-based model of an osteocyte process and a canaliculus using UHVEM tomography and investigated the distribution and magnitude of flow-induced local strain on the osteocyte process by performing fluid-structure interaction simulation. The analysis results reveal that local strain concentration in the osteocyte process was induced by a small number of TEs with high tension, which were inclined depending on the irregular shapes of osteocyte processes and canaliculi. Therefore, this study could provide meaningful insights into the effect of ultrastructure of osteocyte processes and canaliculi on the osteocyte mechanosensing mechanism.

Theoretical concept of cortical to cancellous bone transformation.

Microstructures of cortical and cancellous bones are altered continually by load-adaptive remodeling; in addition, their cellular mechanisms are similar despite the remarkably different porosities. The cortico-cancellous transitional zone is a site of vigorous remodeling, and intracortical remodeling cavitates the inner cortex to promote its trabecularization, which is considered the main cause of bone loss because of aging. Therefore, to prevent and treat age-related cortical bone loss effectively, it is indispensable to gain an integrated understanding of the cortical to the cancellous bone transformation via remodeling. We propose a novel theoretical concept to account for the transformation of dense cortical bone to porous cancellous bone. We develop a mathematical model of cortical and cancellous bone remodeling based on the concept that bone porosity is determined by the balance between the load-bearing function of mineralized bone and the material-transporting function of bone marrow. Remodeling simulations using this mathematical model enable the reproduction of the microstructures of cortical and cancellous bones simultaneously. Furthermore, current remodeling simulations have the potential to replicate cortical-to-cancellous bone transformation based on changes in the local balance between bone formation and resorption. We anticipate that the proposed mathematical model of cortical and cancellous bone remodeling will contribute to highlighting the essential features of cortical bone loss due to trabecularization of the cortex and help predict its spatial and temporal behavior during aging.

Continuum modeling for neuronal lamination during cerebral morphogenesis considering cell migration and tissue growth.

For neuronal lamination during cerebral morphogenesis, later-born neurons must migrate through already-accumulated neurons. This neuronal migration is biochemically regulated by signaling molecules and mechanically affected by tissue deformation. To understand the neuronal lamination mechanisms, we constructed a continuum model of neuronal migration in a growing deformable tissue. We performed numerical analyses considering the migration promoted by signaling molecules and the tissue growth induced by neuron accumulation. The results suggest that the promoted migration and the space ensured by tissue growth are essential for neuronal lamination. The proposed model can describe the coupling of mechanical and biochemical mechanisms for neuronal lamination.

An energy landscape approach to understanding variety and robustness in tissue morphogenesis.

During morphogenesis in development, multicellular tissues deform by mechanical forces induced by spatiotemporally regulated cellular activities, such as cell proliferation and constriction. Various morphologies are formed because of various spatiotemporal combinations and sequences of multicellular activities. Despite its potential to variations, morphogenesis is a surprisingly robust process, in which qualitatively similar morphologies are reproducibly formed even under spatiotemporal fluctuation of multicellular activities. To understand these essential characteristics of tissue morphogenesis, which involves the coexistence of various morphologies and robustness of the morphogenetic process, in this study, we propose a novel approach to capture the overall view of morphogenesis from mechanical viewpoints. This approach will enable visualization of the energy landscape, which includes morphogenetic processes induced by admissible histories of cellular activities. This approach was applied to investigate the morphogenesis of a sheet-like tissue with curvature, where it deformed to a concave or convex morphology depending on the history of growth and constriction. Qualitatively different morphologies were produced by bifurcation of the valley in the energy landscape. The depth and steepness of the valley near the stable states represented the degree of robustness to fluctuations of multicellular activities. Furthermore, as a realistic example, we showed an application of this approach to luminal folding observed in the initial stage of intestinal villus formation. This approach will be helpful to understand the mechanism of how various morphologies are formed and how tissues reproducibly achieve specific morphologies.

Mechano-Regulation of Trabecular Bone Adaptation Is Controlled by the Local in vivo Environment and Logarithmically Dependent on Loading Frequency.

It is well-established that cyclic, but not static, mechanical loading has anabolic effects on bone. However, the function describing the relationship between the loading frequency and the amount of bone adaptation remains unclear. Using a combined experimental and computational approach, this study aimed to investigate whether trabecular bone mechano-regulation is controlled by mechanical signals in the local in vivo environment and dependent on loading frequency. Specifically, by combining in vivo micro-computed tomography (micro-CT) imaging with micro-finite element (micro-FE) analysis, we monitored the changes in microstructural as well as the mechanical in vivo environment [strain energy density (SED) and SED gradient] of mouse caudal vertebrae over 4 weeks of either cyclic loading at varying frequencies of 2, 5, or 10 Hz, respectively, or static loading. Higher values of SED and SED gradient on the local tissue level led to an increased probability of trabecular bone formation and a decreased probability of trabecular bone resorption. In all loading groups, the SED gradient was superior in the determination of local bone formation and resorption events as compared to SED. Cyclic loading induced positive net (re)modeling rates when compared to sham and static loading, mainly due to an increase in mineralizing surface and a decrease in eroded surface. Consequently, bone volume fraction increased over time in 2, 5, and 10 Hz (+15%, +21% and +24%, p ≤ 0.0001), while static loading led to a decrease in bone volume fraction (-9%, p ≤ 0.001). Furthermore, regression analysis revealed a logarithmic relationship between loading frequency and the net change in bone volume fraction over the 4 week observation period (R 2 = 0.74). In conclusion, these results suggest that trabecular bone adaptation is regulated by mechanical signals in the local in vivo environment and furthermore, that mechano-regulation is logarithmically dependent on loading frequency with frequencies below a certain threshold having catabolic effects, and those above anabolic effects. This study thereby provides valuable insights toward a better understanding of the mechanical signals influencing trabecular bone formation and resorption in the local in vivo environment.

Intrauterine Pressures Adjusted by Reichert's Membrane Are Crucial for Early Mouse Morphogenesis.

Mammalian embryogenesis proceeds in utero with the support of nutrients and gases from maternal tissues. However, the contribution of the mechanical environment provided by the uterus to embryogenesis remains unaddressed. Notably, how intrauterine pressures are produced, accurately adjusted, and exerted on embryos are completely unknown. Here, we find that Reichert's membrane, a specialized basement membrane that wraps around the implanted mouse embryo, plays a crucial role as a shock absorber to protect embryos from intrauterine pressures. Notably, intrauterine pressures are produced by uterine smooth muscle contractions, showing the highest and most frequent periodic peaks just after implantation. Mechanistically, such pressures are adjusted within the sealed space between the embryo and uterus created by Reichert's membrane and are involved in egg-cylinder morphogenesis as an important biomechanical environment in utero. Thus, we propose the buffer space sealed by Reichert's membrane cushions and disperses intrauterine pressures exerted on embryos for egg-cylinder morphogenesis.

Forceful mastication activates osteocytes and builds a stout jawbone.

Bone undergoes a constant reconstruction process of resorption and formation called bone remodeling, so that it can endure mechanical loading. During food ingestion, masticatory muscles generate the required masticatory force. The magnitude of applied masticatory force has long been believed to be closely correlated with the shape of the jawbone. However, both the mechanism underlying this correlation and evidence of causation remain largely to be determined. Here, we established a novel mouse model of increased mastication in which mice were fed with a hard diet (HD) to elicit greater masticatory force. A novel in silico computer simulation indicated that the masticatory load onto the jawbone leads to the typical bone profile seen in the individuals with strong masticatory force, which was confirmed by in vivo micro-computed tomography (micro-CT) analyses. Mechanistically, increased mastication induced Insulin-like growth factor (IGF)-1 and suppressed sclerostin in osteocytes. IGF-1 enhanced osteoblastogenesis of the cells derived from tendon. Together, these findings indicate that the osteocytes balance the cytokine expression upon the mechanical loading of increased mastication, in order to enhance bone formation. This bone formation leads to morphological change in the jawbone, so that the bone adapts to the mechanical environment to which it is exposed.

Capturing microscopic features of bone remodeling into a macroscopic model based on biological rationales of bone adaptation.

To understand Wolff's law, bone adaptation by remodeling at the cellular and tissue levels has been discussed extensively through experimental and simulation studies. For the clinical application of a bone remodeling simulation, it is significant to establish a macroscopic model that incorporates clarified microscopic mechanisms. In this study, we proposed novel macroscopic models based on the microscopic mechanism of osteocytic mechanosensing, in which the flow of fluid in the lacuno-canalicular porosity generated by fluid pressure gradients plays an important role, and theoretically evaluated the proposed models, taking biological rationales of bone adaptation into account. The proposed models were categorized into two groups according to whether the remodeling equilibrium state was defined globally or locally, i.e., the global or local uniformity models. Each remodeling stimulus in the proposed models was quantitatively evaluated through image-based finite element analyses of a swine cancellous bone, according to two introduced criteria associated with the trabecular volume and orientation at remodeling equilibrium based on biological rationales. The evaluation suggested that nonuniformity of the mean stress gradient in the local uniformity model, one of the proposed stimuli, has high validity. Furthermore, the adaptive potential of each stimulus was discussed based on spatial distribution of a remodeling stimulus on the trabecular surface. The theoretical consideration of a remodeling stimulus based on biological rationales of bone adaptation would contribute to the establishment of a clinically applicable and reliable simulation model of bone remodeling.

Poroelastic analysis of interstitial fluid flow in a single lamellar trabecula subjected to cyclic loading.

Trabecula, an anatomical unit of the cancellous bone, is a porous material that consists of a lamellar bone matrix and interstitial fluid in a lacuno-canalicular porosity. The flow of interstitial fluid caused by deformation of the bone matrix is believed to initiate a mechanical response in osteocytes for bone remodeling. In order to clarify the effect of the lamellar structure of the bone matrix--i.e., variations in material properties--on the fluid flow stimuli to osteocytes embedded in trabeculae, we investigated the mechanical behavior of an individual trabecula subjected to cyclic loading based on poroelasticity. We focused on variations in the trabecular permeability and developed an analytical solution containing both transient and steady-state responses for interstitial fluid pressure in a single trabecular model represented by a multilayered two-dimensional poroelastic slab. Based on the obtained solution, we calculated the pressure and seepage velocity of the interstitial fluid in lacuno-canalicular porosity, within the single trabecula, under various permeability distributions. Poroelastic analysis showed that a heterogeneous distribution of permeability produces remarkable variations in the fluid pressure and seepage velocity in the cross section of the individual trabecula, and suggests that fluid flow stimuli to osteocytes are mostly governed by the value of permeability in the neighborhood of the trabecular surfaces if there is no difference in the average permeability in a single trabecula.

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.

Microscale fluid flow analysis in a human osteocyte canaliculus using a realistic high-resolution image-based three-dimensional model.

Osteocytes play a pivotal role in the regulation of skeletal mass. Osteocyte processes are thought to sense the flow of interstitial fluid that is driven through the osteocyte canaliculi by mechanical stimuli placed upon bone, but how this flow elicits a cellular response is virtually unknown. Modern theoretical models assume that osteocyte canaliculi contain ultrastructural features that amplify the fluid flow-derived mechanical signal. Unfortunately the calcified bone matrix has considerably hampered studies on the osteocyte process within its canaliculus. Using one of the few ultra high voltage electron microscopes (UHVEM) available worldwide, we applied UHVEM tomography at 2 MeV to reconstruct unique three-dimensional images of osteocyte canaliculi in 1 µm sections of human bone. A realistic three-dimensional image-based model of a single canaliculus was constructed, and the fluid dynamics of a Newtonian fluid flow within the canaliculus was analyzed. We created virtual 2.2 nm thick sections through a canaliculus and found that traditional TEM techniques create a false impression that osteocyte processes are directly attached to the canalicular wall. The canalicular wall had a highly irregular surface and contained protruding axisymmetric structures similar in size and shape to collagen fibrils. We also found that the microscopic surface roughness of the canalicular wall strongly influenced the fluid flow profiles, whereby highly inhomogeneous flow patterns emerged. These inhomogeneous flow patterns may induce deformation of cytoskeletal elements in the osteocyte process, thereby amplifying mechanical signals. Based on these observations, new and realistic models can be developed that will significantly enhance our understanding of the process of mechanotransduction in bone.

Effects of loading frequency on the functional adaptation of trabeculae predicted by bone remodeling simulation.

The process of bone remodeling is regulated by metabolic activities of many bone cells. While osteoclasts and osteoblasts are responsible for bone resorption and formation, respectively, activities of these cells are believed to be controlled by a mechanosensory system of osteocytes embedded in the extracellular bone matrix. Several experimental and theoretical studies have suggested that the strain-derived interstitial fluid flow in lacuno-canalicular porosity serves as the prime mover for bone remodeling. Previously, we constructed a mathematical model for trabecular bone remodeling that interconnects the microscopic cellular activities with the macroscopic morphological changes in trabeculae through the mechanical hierarchy. This model assumes that fluid-induced shear stress acting on osteocyte processes is a driving force for bone remodeling. The validity of this model has been demonstrated with a remodeling simulation using a two-dimensional trabecular model. In this study, to investigate the effects of loading frequency, which is thought to be a significant mechanical factor in bone remodeling, we simulated morphological changes of a three-dimensional single trabecula under cyclic uniaxial loading with various frequencies. The results of the simulation show the trabecula reoriented to the loading direction with the progress of bone remodeling. Furthermore, as the imposed loading frequency increased, the diameter of the trabecula in the equilibrium state was enlarged by remodeling. These results indicate that our simulation model can successfully evaluate the relationship between loading frequency and trabecular bone remodeling.

Trabecular bone remodelling simulation considering osteocytic response to fluid-induced shear stress.

In bone functional adaptation by remodelling, osteocytes in the lacuno-canalicular system are believed to play important roles in the mechanosensory system. Under dynamic loading, bone matrix deformation generates an interstitial fluid flow in the lacuno-canalicular system; this flow induces shear stress on the osteocytic process membrane that is known to stimulate the osteocytes. In this sense, the osteocytes behave as mechanosensors and deliver mechanical information to neighbouring cells through the intercellular communication network. In this study, bone remodelling is assumed to be regulated by the mechanical signals collected by the osteocytes. From the viewpoint of multi-scale biomechanics, we propose a mathematical model of trabecular bone remodelling that takes into account the osteocytic mechanosensory network system. Based on this model, a computational simulation of trabecular bone remodelling was conducted for a single trabecula under cyclic uniaxial loading, demonstrating functional adaptation to the applied mechanical loading as a load-bearing construct.

Estimation of bone permeability considering the morphology of lacuno-canalicular porosity.

Load-induced interstitial fluid flow in lacuno-canalicular porosity is believed to play an important role in cellular activities regulating adaptive bone remodeling. To investigate interstitial fluid behavior based on poroelasticity, it is important to determine the anisotropic permeability tensor reflecting the morphological features of the lacuno-canalicular porosity as fluid channels. In this study, we presented an estimation method of trabecular permeability by describing the analytical relationship between the volume orientation (VO) fabric tensor, which represents the canalicular orientation, and the permeability tensor. The relationship showed that the trabecular permeability tensor is proportional to the product of the volume fraction of the interstitial fluid and the VO fabric tensor of the canaliculi. We applied the proposed method to a two-dimensional fluorescent image of a trabecular cross section to quantify the canalicular anisotropy and the trabecular permeability tensor. The results indicated that the canaliculi are predominantly oriented in the radial direction of the trabecula, and the permeability depends strongly on the canalicular morphology.