Bone remodelling is carried out by ‘bone multicellular units’ ($$\text{ BMU }$$s) in which active osteoclasts and active osteoblasts are spatially and temporally coupled. The refilling of new bone by osteoblasts towards the back of the $$\text{ BMU }$$occurs at a rate that depends both on the number of osteoblasts and on their secretory activity. In cortical bone, a linear phenomenological relationship between matrix apposition rate and $$\text{ BMU }$$cavity radius is found experimentally. How this relationship emerges from the combination of complex, nonlinear regulations of osteoblast number and secretory activity is unknown. Here, we extend our previous mathematical model of cell development within a single cortical $$\text{ BMU }$$to investigate how osteoblast number and osteoblast secretory activity vary along the $$\text{ BMU }$$’s closing cone. The mathematical model is based on biochemical coupling between osteoclasts and osteoblasts of various maturity and includes the differentiation of osteoblasts into osteocytes and bone lining cells, as well as the influence of $$\text{ BMU }$$cavity shrinkage on osteoblast development and activity. Matrix apposition rates predicted by the model are compared with data from tetracycline double labelling experiments. We find that the linear phenomenological relationship observed in these experiments between matrix apposition rate and $$\text{ BMU }$$cavity radius holds for most of the refilling phase simulated by our model, but not near the start and end of refilling. This suggests that at a particular bone site undergoing remodelling, bone formation starts and ends rapidly, supporting the hypothesis that osteoblasts behave synchronously. Our model also suggests that part of the observed cross-sectional variability in tetracycline data may be due to different bone sites being refilled by $$\text{ BMU }$$s at different stages of their lifetime. The different stages of a $$\text{ BMU }$$’s lifetime (such as initiation stage, progression stage, and termination stage) depend on whether the cell populations within the $$\text{ BMU }$$are still developing or have reached a quasi-steady state whilst travelling through bone. We find that due to their longer lifespan, active osteoblasts reach a quasi-steady distribution more slowly than active osteoclasts. We suggest that this fact may locally enlarge the Haversian canal diameter (due to a local lack of osteoblasts compared to osteoclasts) near the $$\text{ BMU }$$’s point of origin.
Mathematical Modeling of Spatio-Temporal Dynamics of a Single Bone Multicellular Unit
Abstract During bone remodeling, bone-resorbing osteoclasts and bone-forming osteoblasts are organized in bone multicellular units (BMUs), which travel at a rate of 20–40 μm/d for 6–12 mo, maintaining a cylindrical structure. However, the interplay of local BMU geometry with biochemical regulation is poorly understood. We developed a mathematical model of BMU describing changes in time and space of the concentrations of proresorptive cytokine RANKL and its inhibitor osteoprotegerin (OPG), in osteoclast and osteoblast numbers, and in bone mass. We assumed that osteocytes surrounding a microfracture produce RANKL, which attracted osteoclasts. OPG and RANKL were produced by osteoblasts and diffused through bone, RANKL was eliminated by binding to OPG and RANK. Osteoblasts were coupled to osteoclasts through paracrine factors. The evolution of the BMU arising from this model was studied using numerical simulations. Our model recapitulated the spatio-temporal dynamics observed in vivo in a cross-section of bone. In response to a RANKL field, osteoclasts moved as a well-confined cutting cone. The coupling of osteoclasts to osteoblasts allowed for sufficient recruitment of osteoblasts to the resorbed surfaces. The RANKL field was the highest at the microfracture in front of the BMU, whereas the OPG field peaked at the back of the BMU, resulting in the formation of a RANKL/OPG gradient, which strongly affected the rate of BMU progression and its size. Thus, the spatial organization of a BMU provides important constraints on the roles of RANKL and OPG as well as possibly other regulators in determining the outcome of remodeling in the BMU.
Spatio-temporal simulations of bone remodelling using a bone cell population model based on cell availability
Here we developed a spatio-temporal bone remodeling model to simulate the action of Basic Multicelluar Units (BMUs). This model is based on two major extensions of a temporal-only bone cell population model (BCPM). First, the differentiation into mature resorbing osteoclasts and mature forming osteoblasts from their respective precursor cells was modelled as an intermittent process based on precursor cells availability. Second, the interaction between neighbouring BMUs was considered based on a ``metabolic cost'' argument which warrants that no new BMU will be activated in the neighbourhood of an existing BMU. With the proposed model we have simulated the phases of the remodelling process obtaining average periods similar to those found in the literature: resorption ($\sim22$ days) - reversal ($\sim$8 days) - formation ($\sim$ 65 days) - quiescence (560 - 600 days) and an average BMU activation frequency of $\sim$1.6 BMUs/year/mm$^3$. We further show here that the resorption and formation phases of the BMU become coordinated only by the presence of TGF-$\upbeta$ (transforming growth factor $\upbeta$, i.e. a major coupling factor stored in the bone matrix. TGF-$\upbeta$ is released through resorption so upregulating osteoclast apoptosis and accumulation of osteoblast precursors, i.e. facilitating the transition from the resorption to the formation phase at a given remodelling site. Finally, we demonstrate that this model can explain targeted bone remodelling as the BMUs are steered towards damaged bone areas in order to commence bone matrix repair.
A review of mathematical modeling of bone remodeling from a systems biology perspective
Bone remodeling is an essential, delicately balanced physiological process of coordinated activity of bone cells that remove and deposit new bone tissue in the adult skeleton. Due to the complex nature of this process, many mathematical models of bone remodeling have been developed. Each of these models has unique features, but they have underlying patterns. In this review, the authors highlight the important aspects frequently found in mathematical models for bone remodeling and discuss how and why these aspects are included when considering the physiology of the bone basic multicellular unit, which is the term used for the collection of cells responsible for bone remodeling. The review also emphasizes the view of bone remodeling from a systems biology perspective. Understanding the systemic mechanisms involved in remodeling will help provide information on bone pathology associated with aging, endocrine disorders, cancers, and inflammatory conditions and enhance systems pharmacology. Furthermore, some features of the bone remodeling cycle and interactions with other organ systems that have not yet been modeled mathematically are discussed as promising future directions in the field.
A review of mathematical modeling of bone remodeling from a systems biology perspective
Bone remodeling is an essential physiological process in the adult skeleton. Due to the complex nature of this process, many mathematical models of bone remodeling have been developed. Each of these models has unique features, but they have underlying patterns. In this review, the authors highlight the important aspects frequently found in mathematical models for bone remodeling and discuss how and why these aspects are included when considering the physiology of the bone basic multicellular unit, which is the term used for the collection of cells responsible for bone remodeling. The review also emphasizes the view of bone remodeling from a systems biology perspective. Understanding the systemic mechanisms involved in remodeling will help provide information on bone pathology associated with aging, endocrine disorders, cancers, and inflammatory conditions and enhance systems pharmacology. Furthermore, some features of the bone remodeling cycle and interactions with other organ systems that have not yet been modeled mathematically are discussed as promising future directions in the field.
The Role of Osteocytes in Targeted Bone Remodeling: A Mathematical Model
Until recently many studies of bone remodeling at the cellular level have focused on the behavior of mature osteoblasts and osteoclasts, and their respective precursor cells, with the role of osteocytes and bone lining cells left largely unexplored. This is particularly true with respect to the mathematical modeling of bone remodeling. However, there is increasing evidence that osteocytes play important roles in the cycle of targeted bone remodeling, in serving as a significant source of RANKL to support osteoclastogenesis, and in secreting the bone formation inhibitor sclerostin. Moreover, there is also increasing interest in sclerostin, an osteocyte-secreted bone formation inhibitor, and its role in regulating local response to changes in the bone microenvironment. Here we develop a cell population model of bone remodeling that includes the role of osteocytes, sclerostin, and allows for the possibility of RANKL expression by osteocyte cell populations. We have aimed to give a simple, yet still tractable, model that remains faithful to the underlying system based on the known literature. This model extends and complements many of the existing mathematical models for bone remodeling, but can be used to explore aspects of the process of bone remodeling that were previously beyond the scope of prior modeling work. Through numerical simulations we demonstrate that our model can be used to explore theoretically many of the qualitative features of the role of osteocytes in bone biology as presented in recent literature.
The Cellular Dynamics of Bone Remodeling: A Mathematical Model
The mechanical properties of vertebrate bone are largely determined by a process which involves the complex interplay of three different cell types. This process is called bone remodeling and occurs asynchronously at multiple sites in the mature skeleton. The cells involved are bone resorbing osteoclasts, bone matrix producing osteoblasts, and mechanosensing osteocytes. These cells communicate with each other by means of autocrine and paracrine signaling factors and operate in complex entities, the so-called bone multicellular units (BMUs). To investigate the BMU dynamics in silico, we develop a novel mathematical model resulting in a system of nonlinear partial differential equations (PDEs) with time delays. The model describes the osteoblast and osteoclast populations together with the dynamics of the key messenger molecule RANKL and its decoy receptor OPG. Scaling theory is used to address parameter sensitivity and predict the emergence of pathological remodeling regimes. The model is studied numerically in one and two space dimensions using finite difference schemes in space and explicit delay equation solvers in time. The computational results are in agreement with in vivo observations and provide new insights into the role of the RANKL/OPG pathway in the spatial regulation of bone remodeling.
A review of recent developments in mathematical modeling of bone remodeling
In this article, we summarize the developments in the mathematical modeling of the mechanics of bone and related biological phenomena. We will devote special attention to the results of the last 10–15 years, although we will cover some relevant classical work to better frame the more recent researches. We will propose a division of the literature based on the main aim of the model (mechanical/biomathematical) and the type of biological phenomena considered (stimulus, growth, cell population dynamics). Finally, we will suggest some possible directions for future investigations.
Analytic approach to explore dynamical osteoporotic bone turnover
The dynamics of the osteoporotic bone turnover is studied in this paper with the aid of stability analysis of the associated mathematical model. Osteoporosis, which is a common bone disorder, is studied in this papper in detail with an emphasis on the relative threshold values. We examine the expository signaling among the bone cells named osteoclast and osteoblast. Main functioning of osteoblasts is bone formation, whereas osteoclasts are bone removal cells.
Bone remodeling: A tissue-level process emerging from cell-level molecular algorithms
The human skeleton undergoes constant remodeling throughout the lifetime. Processes occurring on microscopic and molecular scales degrade bone and replace it with new, fully functional tissue. Multiple bone remodeling events occur simultaneously, continuously and independently throughout the body, so that the entire skeleton is completely renewed about every ten years.Bone remodeling is performed by groups of cells called Bone Multicellular Units (BMU). BMUs consist of different cell types, some specialized in the resorption of old bone, others encharged with producing new bone to replace the former. These processes are tightly regulated so that the amount of new bone produced is in perfect equilibrium with that of old bone removed, thus maintaining bone microscopic structure.To date, many regulatory molecules involved in bone remodeling have been identified, but the precise mechanism of BMU operation remains to be fully elucidated. Given the complexity of the signaling pathways already known, one may question whether such complexity is an inherent requirement of the process or whether some subset of the multiple constituents could fulfill the essential role, leaving functional redundancy to serve an alternative safety role. We propose in this work a minimal model of BMU function that involves a limited number of signals able to account for fully functional BMU operation. Our main assumptions were i) at any given time, any cell within a BMU can select only one among a limited choice of decisions, i.e. divide, die, migrate or differentiate, ii) this decision is irreversibly determined by depletion of an appropriate internal inhibitor and iii) the dynamics of any such inhibitor are coupled to that of specific external mediators, such as hormones, cytokines, growth factors. It was thus shown that efficient BMU operation manifests as an emergent process, which results from the individual and collective decisions taken by cells within the BMU unit in the absence of any external planning.
Osteonal and hemi‐osteonal remodeling: The spatial and temporal framework for signal traffic in adult human bone
Abstract The bone replacement process in the adult skeleton is known as remodeling. When bone is removed by osteoclasts, new bone is laid down by osteoblasts in the same place, because the load bearing requirement is unchanged. Bone is usually replaced because it is too old to carry out its function, which is mainly mechanical in cortical bone and mainly support for homeostasis and hematopoiesis in cancellous bone. Remodeling always begins on a quiescent bone surface, separated from the marrow by flat lining cells that are one of the two modes of terminal differentiation of osteoblasts. Lining cells are gatekeepers, able to be informed of the need for remodeling, and to either execute or mediate all four components of its activation‐selection and preparation of the site, recruitment of mononuclear preosteoclasts, budding of new capillaries, and attraction of preosteoclasts to the chosen site where they fuse into multinucleated osteoclasts. In cortical bone, osteonal remodeling is carried out by a complex and unique structure, the basic multicellular unit (BMU) that comprises a cutting cone of osteoclasts in front, a closing cone lined by osteoblasts following behind, and connective tissue, blood vessels and nerves filling the cavity. The BMU maintains its size, shape and internal organization for many months as it travels through bone in a controlled direction. Individual osteoclast nuclei are short‐lived, turning over about 8% per d, replaced by new preosteoclasts that originated in the bone marrow and travel in the circulation to the site of resorption. Refilling of bone at each successive cross‐sectional location is accomplished by a team of osteoblasts, probably originating from precursors within the local connective tissue, all assembled within a narrow window of time, at the right location, and in the right orientation to the surface. Each osteoblast team forms bone most rapidly at its onset and slows down progressively. Some of the osteoblasts are buried as osteocytes, some die, and the remainder gradually assume the shape of lining cells. Cancellous bone is more accessible to study than cortical bone, but is geometrically complex. Although remodeling conforms to the same sequence of surface activation, resorption and formation, its three‐dimensional organization is difficult to visualize from two‐dimensional histologic sections. But the average sizes of resorption sites, formation sites, and completed structural units increase progressively, as they do in cortical bone, indicating that the cancellous BMU travels across the surface digging a trench rather than a tunnel, but maintaining its size, shape and individual identity by the continuous recruitment of new cells, just as in cortical bone, a process that can be visualized as hemiosteonal remodeling. The conclusion that all remodeling is carried out by individual BMUs has important implications for bone biology, since many questions about how BMUs operate cannot be answered by studying either intact organisms or isolated cell systems. Many different steps in remodeling and many factors that influence each step have been identified, but very little is known about how the process is regulated in vivo to achieve its biologic purposes; most factors studied to date are likely permissive rather than regulatory in nature. Based on the proposed conceptual model of the BMU, much in vitro experimentation is relevant to the growth, modeling and repair of bone, but not to its remodeling in the adult skeleton. Further progress in the understanding of in vivo physiology will require the characterization of gene expression in individual cells to be related to the spatial and temporal organization of the BMU. This is likely to be possible only for osteonal remodeling in cortical bone in which, because of its geometric simplicity, individual BMUs can consistently be observed in two‐dimensional, longitudinal sections. © 1994 Wiley‐Liss, Inc.
Towards a new spatial representation of bone remodeling
Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed. There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes. We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur in vitro and in vivo. By employing the level set approach it is possible to develop computational and mathematical representations of the spatialbehavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as in vitro and in vivo. This makes it possible to perform in silica experiments more closely resembling experimental observations.
Control of Bone Matrix Properties by Osteocytes
Osteocytes make up 90-95% of the cellular content of bone and form a rich dendritic network with a vastly greater surface area than either osteoblasts or osteoclasts. Osteocytes are well positioned to play a role in bone homeostasis by interacting directly with the matrix; however, the ability for these cells to modify bone matrix remains incompletely understood. With techniques for examining the nano- and microstructure of bone matrix components including hydroxyapatite and type I collagen becoming more widespread, there is great potential to uncover novel roles for the osteocyte in maintaining bone quality. In this review, we begin with an overview of osteocyte biology and the lacunar-canalicular system. Next, we describe recent findings from in vitro models of osteocytes, focusing on the transitions in cellular phenotype as they mature. Finally, we describe historical and current research on matrix alteration by osteocytes in vivo, focusing on the exciting potential for osteocytes to directly form, degrade, and modify the mineral and collagen in their surrounding matrix.
Osteoclasts Provide Coupling Signals to Osteoblast Lineage Cells Through Multiple Mechanisms
Bone remodeling is essential for the repair and replacement of damaged and old bone. The major principle underlying this process is that osteoclast-mediated resorption of a quantum of bone is followed by osteoblast precursor recruitment; these cells differentiate to matrix-producing osteoblasts, which form new bone to replace what was resorbed. Evidence from osteopetrotic syndromes indicate that osteoclasts not only resorb bone, but also provide signals to promote bone formation. Osteoclasts act upon osteoblast lineage cells throughout their differentiation by facilitating growth factor release from resorbed matrix, producing secreted proteins and microvesicles, and expressing membrane-bound factors. These multiple mechanisms mediate the coupling of bone formation to resorption in remodeling. Additional interactions of osteoclasts with osteoblast lineage cells, including interactions with canopy and reversal cells, are required to achieve coordination between bone formation and resorption during bone remodeling.
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.
<i>In silico</i> models of bone remodeling from macro to nano—from organ to cell
Abstract Computational modeling is a tool through which researchers can achieve a greater understanding of the mechanisms governing biological systems. In the field of bone biology, a plethora of models exist which attempt to replicate and investigate bone's dynamic behavior at different scales. At organ level, models are continuum based and describe the variation of bone's apparent density as a function of both biological and external mechanical stimuli. At tissue level, models include bone microarchitecture and more descriptive parameters such as trabecular thickness, osteoclast resorption depth, and activation frequency. Finally, at cell level, models employ partial differential equations to describe complex cellular interactions in the temporal domain. Although informative, these models exist in isolation. Consequently, their interpretation is limited. In this review, we present an overview of the organ‐, tissue‐, and cell‐level models and assess their ability to reflect bone's metabolic processes reliably. Existing interscale synergies are then presented along with a computational framework which could be exploited to achieve a fully integrated, multiscale modeling approach. WIREs Syst Biol Med 2011 3 241–251 DOI: 10.1002/wsbm.115 This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models
Simulated Interventions to Ameliorate Age-Related Bone Loss Indicate the Importance of Timing
Bone remodeling is the continuous process of bone resorption by osteoclasts and bone formation by osteoblasts, in order to maintain homeostasis. The activity of osteoclasts and osteoblasts is regulated by a network of signaling pathways, including Wnt, parathyroid hormone (PTH), RANKL/OPG and TGF-β, in response to stimuli such as mechanical loading. During aging there is a gradual loss of bone mass due to dysregulation of signaling pathways. This may be due to a decline in physical activity with age and/or changes in hormones and other signaling molecules. In particular, hormones such as PTH have a circadian rhythm which may be disrupted in aging. Due to the complexity of the molecular and cellular networks involved in bone remodeling, several mathematical models have been proposed to aid understanding of the processes involved. However, to date there are no models which explicitly consider the effects of mechanical loading, the circadian rhythm of PTH and the dynamics of signaling molecules on bone remodeling. Therefore, we have constructed a network model of the system using a modular approach which will allow further modifications as required in future research. The model was used to simulate the effects of mechanical loading and also the effects of different interventions such as continuous or intermittent administration of PTH. Our model predicts that the absence of regular mechanical loading and/or an impaired PTH circadian rhythm leads to a gradual decrease in bone mass over time which can be restored by simulated interventions and that the effectiveness of some interventions may depend on their timing.
