The Role of Osteocytes in Targeted Bone Remodeling: A Mathematical Model

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.

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.

BIOMEDICAL ENGINEERING OF SCLEROSTIN ACTION IN THE BONE REMODELING

Mathematical modeling of biological processes has bridged the fields of experimental as well as theoretical research and has carried forward remarkable innovation. Sclerostin is a fundamental communication element for bone remodeling and its activity regulates the reabsorption and deposition of new bone tissue. During this research, we have presented several studies, which illustrate the function of sclerostin in communication with the Wnt signaling pathways. This article features the sclerostin-based pathological patterns related to diseases such as bone cancer. To have a good remodeling process, the osteocytes must recruit the pre-osteoblast cells from the mesenchymal stem cells with the help of the signal mechanism given by the Wnt pathway. The Wnt signal pathway is a complex transduction of a pool of well-conserved genes whose expression regulates various activities like gene translation, cell adhesion, cell differentiation, mitogenic stimulation and polarity cell. The complexity of the interaction of the Wnt pathway is due to the ligands of Wnt itself, to the proteins R-spondin and norrin. The receptors on the surface of the cell, then, activate a process of transduction of the intracellular signal that initiates gene transduction. The hypothesis of a sort of “steady state” has therefore proved indispensable to establish a sort of common base on which the two phases. This paper seeks to give a qualitative view of the action of sclerostin through a simple mathematical model. We use a logic related to stimulation and inhibition signals of new tissue production and illustrate the role of sclerostin in the mechanical and biochemical interaction during the bone remodeling process.

Connecting the Dots: Linking Osteocyte Activity and Therapeutic Modulation of Sclerostin by Extending a Multiscale Systems Model

The goal of this work was to extend a mathematical, multiscale systems model of bone function, remodeling, and health in order to explore hypotheses related to therapeutic modulation of sclerostin an...

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.

Mechanobiological osteocyte feedback drives mechanostat regulation of bone in a multiscale computational model

Significant progress has been made to identify the cells and signaling molecules involved in the mechanobiological regulation of bone remodeling. It is now well accepted that osteocytes act as mechanosensory cells in bone expressing several signaling molecules such as nitric oxide (NO) and sclerostin (Scl) which are able to control bone remodeling responses. In this paper, we present a comprehensive multiscale computational model of bone remodeling which incorporates biochemical osteocyte feedback. The mechanostat theory is quantitatively incorporated into the model using mechanical feedback to control expression levels of NO and Scl. The catabolic signaling pathway RANK–RANKL–OPG is co-regulated via (continuous) PTH and NO, while the anabolic Wnt signaling pathway is described via competitive binding reactions between Wnt, Scl and the Wnt receptors LRP5/6. Using this novel model of bone remodeling, we investigate the effects of changes in the mechanical loading and hormonal environment on bone balance. Our numerical simulations show that we can calibrate the mechanostat anabolic and catabolic regulatory mechanisms so that they are mutually exclusive. This is consistent with previous models that use a Wolff-type law to regulate bone resorption and formation separately. Furthermore, mechanical feedback provides an effective mechanism to obtain physiological bone loss responses due to mechanical disuse and/or osteoporosis.

Biology of Bone Tissue: Structure, Function, and Factors That Influence Bone Cells

Bone tissue is continuously remodeled through the concerted actions of bone cells, which include bone resorption by osteoclasts and bone formation by osteoblasts, whereas osteocytes act as mechanosensors and orchestrators of the bone remodeling process. This process is under the control of local (e.g., growth factors and cytokines) and systemic (e.g., calcitonin and estrogens) factors that all together contribute for bone homeostasis. An imbalance between bone resorption and formation can result in bone diseases including osteoporosis. Recently, it has been recognized that, during bone remodeling, there are an intricate communication among bone cells. For instance, the coupling from bone resorption to bone formation is achieved by interaction between osteoclasts and osteoblasts. Moreover, osteocytes produce factors that influence osteoblast and osteoclast activities, whereas osteocyte apoptosis is followed by osteoclastic bone resorption. The increasing knowledge about the structure and functions of bone cells contributed to a better understanding of bone biology. It has been suggested that there is a complex communication between bone cells and other organs, indicating the dynamic nature of bone tissue. In this review, we discuss the current data about the structure and functions of bone cells and the factors that influence bone remodeling.

Complex Dynamics of Osteoclast Formation and Death in Long-Term Cultures

BackgroundOsteoclasts, cells responsible for bone resorption, contribute to the development of degenerative, metabolic and neoplastic bone diseases, which are often characterized by persistent changes in bone microenvironment. We aimed to investigate the dynamics of osteoclast formation and death in cultures that considerably exceeded the length of standard protocol and to design a mathematical model describing osteoclastogenesis.Methodology/Principal FindingsRAW 264.7 monocytic cells fuse to form multinucleated osteoclasts upon treatment with pro-resorptive cytokine RANKL. We have found that in long-term experiments (15–26 days), the dynamics of changes in osteoclast numbers was remarkably complex and qualitatively variable in different experiments. Whereas 19 of 46 experiments exhibited single peak of osteoclast formation, in 27 experiments we observed development of successive waves of osteoclast formation and death. Periodic changes in osteoclast numbers were confirmed in long-term cultures of mouse bone marrow cells treated with M-CSF and RANKL. Because the dynamics of changes in osteoclast numbers was found to be largely independent of monocytes, a two-species model of ordinary differential equations describing the changes in osteoclasts and monocytes was ineffective in recapitulating the oscillations in osteoclast numbers. Following experimental observation that medium collected from mature osteoclasts inhibited osteoclastogenesis in fresh cultures, we introduced a third variable, factor f, to describe osteoclast-derived inhibitor. This model allowed us to simulate the oscillatory changes in osteoclasts, which were coupled to oscillatory changes in the factor f, whereas monocytes changed exponentially. Importantly, to achieve the experimentally observed oscillations with increasing amplitude, we also had to assume that osteoclast presence stimulates osteoclast formation.Conclusions/SignificanceThis study identifies the critical role for osteoclast autocrine regulation in controlling long-term dynamic of osteoclast formation and death and describes the complementary roles for negative and positive feedback mediators in determining the sharp dynamics of activation and inactivation of osteoclasts.

The Expression of Osteoprotegerin and RANK Ligand and the Support of Osteoclast Formation by Stromal-Osteoblast Lineage Cells Is Developmentally Regulated**This work was supported by Grant AG-04875 from the National Institute on Aging.

Abstract. The one or more molecular mechanisms that determine the obligatory sequence of resorption followed by formation during bone remodeling is unclear

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.

Cellular mechanisms of bone remodeling

Bone remodeling is a tightly regulated process securing repair of microdamage (targeted remodeling) and replacement of old bone with new bone through sequential osteoclastic resorption and osteoblastic bone formation. The rate of remodeling is regulated by a wide variety of calcitropic hormones (PTH, thyroid hormone, sex steroids etc.). In recent years we have come to appreciate that bone remodeling proceeds in a specialized vascular structure,—the Bone Remodeling Compartment (BRC). The outer lining of this compartment is made up of flattened cells, displaying all the characteristics of lining cells in bone including expression of OPG and RANKL. Reduced bone turnover leads to a decrease in the number of BRCs, while increased turnover causes an increase in the number of BRCs. The secretion of regulatory factors inside a confined space separated from the bone marrow would facilitate local regulation of the remodeling process without interference from growth factors secreted by blood cells in the marrow space. The BRC also creates an environment where cells inside the structure are exposed to denuded bone, which may enable direct cellular interactions with integrins and other matrix factors known to regulate osteoclast/osteoblast activity. However, the denuded bone surface inside the BRC also constitutes an ideal environment for the seeding of bone metastases, known to have high affinity for bone matrix. Circulating osteoclast- and osteoblast precursor cells have been demonstrated in peripheral blood. The dominant pathway regulating osteoclast recruitment is the RANKL/OPG system, while many different factors (RUNX, Osterix) are involved in osteoblast differentiation. Both pathways are modulated by calcitropic hormones.

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.

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.

Osteocyte-Related Cytokines Regulate Osteoclast Formation and Bone Resorption

The process of bone remodeling is the result of the regulated balance between bone cell populations, namely bone-forming osteoblasts, bone-resorbing osteoclasts, and the osteocyte, the mechanosensory cell type. Osteoclasts derived from the hematopoietic stem cell lineage are the principal cells involved in bone resorption. In osteolytic diseases such as rheumatoid arthritis, periodontitis, and osteoporosis, the balance is lost and changes in favor of bone resorption. Therefore, it is vital to elucidate the mechanisms of osteoclast formation and bone resorption. It has been reported that osteocytes express Receptor activator of nuclear factor κΒ ligand (RANKL), an essential factor for osteoclast formation. RANKL secreted by osteocytes is the most important factor for physiologically supported osteoclast formation in the developing skeleton and in pathological bone resorption such as experimental periodontal bone loss. TNF-α directly enhances RANKL expression in osteocytes and promotes osteoclast formation. Moreover, TNF-α enhances sclerostin expression in osteocytes, which also increases osteoclast formation. These findings suggest that osteocyte-related cytokines act directly to enhance osteoclast formation and bone resorption. In this review, we outline the most recent knowledge concerning bone resorption-related cytokines and discuss the osteocyte as the master regulator of bone resorption and effector in osteoclast formation.