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

<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

A mechano-chemo-biological model for bone remodeling with a new mechano-chemo-transduction approach

Bone remodeling is a fundamental biological process that develops in bone tissue along its whole lifetime. It refers to a continuous bone transformation with new bone formation and old bone resorption that changes the internal microstructure and composition of the tissue. The main objectives of bone remodeling are: repair of the internal microcracks; adaptation of the macroscopic stiffness and strength to the actual changing mechanical demands; and control of the calcium homeostasis. Understanding this process and predicting its evolution is critical to reduce the effects of long-term disuse as happens during periods of reduced mobility. It is also important in the design of bone implants to avoid long-term stress shielding. Many mathematical models have been proposed from the earliest purely phenomenological to the latest that include biological knowledge. However, there still exists a lack of connection between the mechanical driving force and the biochemical and cell processes it triggers. Here, and following previous works that model independently the mechanobiological and biochemical processes in bone remodeling, we present a more complete model, useful for both cortical and trabecular bone, that uses a new mechanotransduction approach based on the effect of strains onto the bonding–unbonding rate of RANK/RANKL/OPG receptor–ligand reactions. We compare the results of this model with previous ones, showing a good agreement in similar conditions. We also apply it to realistic situations such as a femoral bone after implantation of a hip prosthesis, getting similar results to the clinical ones in the final bone density distribution. Finally, we extend this approach to the anisotropic case, getting not only the mean density, but also the directional homogenization of the microstructure. This biochemical approach permits, not only to predict the bone evolution under changes in the mechanical loads, but also, to consider anabolic and catabolic drugs to control bone density, such as those used in osteoporosis.

In silico experiments of bone remodeling explore metabolic diseases and their drug treatment

In silico experiments of bone remodeling enable observation and prediction of physiological and pathological bone conditions. , Bone structure and function are maintained by well-regulated bone metabolism and remodeling. Although the underlying molecular and cellular mechanisms are now being understood, physiological and pathological states of bone are still difficult to predict due to the complexity of intercellular signaling. We have now developed a novel in silico experimental platform, V-Bone, to integratively explore bone remodeling by linking complex microscopic molecular/cellular interactions to macroscopic tissue/organ adaptations. Mechano-biochemical couplings modeled in V-Bone relate bone adaptation to mechanical loading and reproduce metabolic bone diseases such as osteoporosis and osteopetrosis. V-Bone also enables in silico perturbation on a specific signaling molecule to observe bone metabolic dynamics over time. We also demonstrate that this platform provides a powerful way to predict in silico therapeutic effects of drugs against metabolic bone diseases. We anticipate that these in silico experiments will substantially accelerate research into bone metabolism and remodeling.

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.

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.

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.

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.

Mathematical modeling of postmenopausal osteoporosis and its treatment by the anti‐catabolic drug denosumab

SUMMARY Denosumab, a fully human monoclonal antibody, has been approved for the treatment of postmenopausal osteoporosis. The therapeutic effect of denosumab rests on its ability to inhibit osteoclast differentiation. Here, we present a computational approach on the basis of coupling a pharmacokinetics model of denosumab with a pharmacodynamics model for quantifying the effect of denosumab on bone remodeling. The pharmacodynamics model comprises an integrated systems biology‐continuum micromechanics approach, including a bone cell population model, considering the governing biochemical factors of bone remodeling (including the action of denosumab), and a multiscale micromechanics‐based bone mechanics model, for implementing the mechanobiology of bone remodeling in our model. Numerical studies of postmenopausal osteoporosis show that denosumab suppresses osteoclast differentiation, thus strongly curtailing bone resorption. Simulation results also suggest that denosumab may trigger a short‐term bone volume gain, which is, however, followed by constant or decreasing bone volume. This evolution is accompanied by a dramatic decrease of the bone turnover rate by more than one order of magnitude. The latter proposes dominant occurrence of secondary mineralization (which is not anymore impeded through cellular activity), leading to higher mineral concentration per bone volume. This explains the overall higher bone mineral density observed in denosumab‐related clinical studies. Copyright © 2013 John Wiley & Sons, Ltd.

A multiscale mechanobiological model of bone remodelling predicts site-specific bone loss in the femur during osteoporosis and mechanical disuse

We propose a multiscale mechanobiological model of bone remodelling to investigate the site-specific evolution of bone volume fraction across the midshaft of a femur. The model includes hormonal regulation and biochemical coupling of bone cell populations, the influence of the microstructure on bone turnover rate, and mechanical adaptation of the tissue. Both microscopic and tissue-scale stress/strain states of the tissue are calculated from macroscopic loads by a combination of beam theory and micromechanical homogenisation. This model is applied to simulate the spatio-temporal evolution of a human midshaft femur scan subjected to two deregulating circumstances: (i) osteoporosis and (ii) mechanical disuse. Both simulated deregulations led to endocortical bone loss, cortical wall thinning and expansion of the medullary cavity, in accordance with experimental findings. Our model suggests that these observations are attributable to a large extent to the influence of the microstructure on bone turnover rate. Mechanical adaptation is found to help preserve intracortical bone matrix near the periosteum. Moreover, it leads to non-uniform cortical wall thickness due to the asymmetry of macroscopic loads introduced by the bending moment. The effect of mechanical adaptation near the endosteum can be greatly affected by whether the mechanical stimulus includes stress concentration effects or not.

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.

Mathematical Model of Bone Remodeling Captures the Antiresorptive and Anabolic Actions of Various Therapies

A better understanding of the molecular pathways regulating the bone remodeling process should help in the development of new antiresorptive regulators and anabolic regulators, that is, regulators of bone resorption and of bone formation. Understanding the mechanisms by which parathyroid hormone (PTH) influences bone formation and how it switches from anabolic to catabolic action is important for treating osteoporosis (Poole and Reeve in Curr Opin Pharmacol 5:612–617, 2005). In this paper we describe a mathematical model of bone remodeling that incorporates, extends, and integrates several models of particular aspects of this biochemical system (Cabal et al. in J Bone Miner Res 28(8):1830–1836, 2013; Lemaire et al. in J Theor Biol 229:293–309, 2004; Peterson and Riggs in Bone 46:49–63, 2010; Raposo et al. in J Clin Endocrinol Metab 87(9):4330–4340, 2002; Ross et al. in J Disc Cont Dyn Sys Series B 17(6):2185–2200, 2012). We plan to use this model as a bone homeostasis platform to develop anabolic and antiresorptive compounds. The model will allow us to test hypotheses about the dynamics of compounds and to test the potential benefits of combination therapies. At the core of the model is the idealized account of osteoclast and osteoblast signaling given by Lemaire et al. (J Theor Biol 229:293–309, 2004). We have relaxed some of their assumptions about the roles of osteoprotegerin, transforming growth factor $$\upbeta $$, and receptor activator of nuclear factor $$\upkappa $$B ligand; we have devised more detailed models of the interactions of these species. We have incorporated a model of the effect of calcium sensing receptor antagonists on remodeling (Cabal et al. in J Bone Miner Res 28(8):1830–1836, 2013). We have also incorporated a basic model of the effects of vitamin D on calcium homeostasis. We have included a simple model of the mechanism proposed by Bellido et al. (2003), Ross et al. (J Disc Cont Dyn Sys Series B 17(6):2185–2200, 2012), of the influence of PTH on osteoblast apoptosis, a mechanism that accounts for the anabolic response to pulsatile PTH administration. Finally, we have devised a simple model of the administration and effects of bisphosphonates. The biomarkers in the model are procollagen type 1 amino-terminal propeptide and C-terminal telopeptide. Bone mineral density is the model’s principal endpoint.

Dynamic Biochemical and Cellular Models of Bone Physiology: Integrating Remodeling Processes, Tumor Growth, and Therapy

Bone is an activeCoelho, Rui M.  Neto, Joana P. connective tissue composed of different types of cells. The dynamic behavior of bone remodeling processes is typically represented throughValério, Duarte differential equations, which represent the physiological phenomena occurring in this organ. These models takeVinga, Susana into account the tight biochemical regulation between osteoclasts and osteoblasts and have also been enriched with variables and parameters related to bone pathologies and treatment. This chapter reviews some of the more recent models describing bone physiology, focusing on those that include the main cellular processes, along the biochemical control, and also the pharmacokinetic/pharmacodynamic (PK/PD) of the most common treatments for diseases such as cancer. These models are then compared in terms of the simulations obtained and, finally, some highlights on integrating them with the biomechanical component of the system which will be given. These models are expected to provide a valuable insight into this complex system and to support the development of clinical decision systems for bone pathologies.

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.

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.

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.