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.
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 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.
<i>In silico</i> biology of bone modelling and remodelling: adaptation
Modelling and remodelling are the processes by which bone adapts its shape and internal structure to external influences. However, the cellular mechanisms triggering osteoclastic resorption and osteoblastic formation are still unknown. In order to investigate current biological theories, in silico models can be applied. In the past, most of these models were based on the continuum assumption, but some questions related to bone adaptation can be addressed better by models incorporating the trabecular microstructure. In this paper, existing simulation models are reviewed and one of the microstructural models is extended to test the hypothesis that bone adaptation can be simulated without particular knowledge of the local strain distribution in the bone. Validation using an experimental murine loading model showed that this is possible. Furthermore, the experimental model revealed that bone formation cannot be attributed only to an increase in trabecular thickness but also to structural reorganization including the growth of new trabeculae. How these new trabeculae arise is still an unresolved issue and might be better addressed by incorporating other levels of hierarchy, especially the cellular level. The cellular level sheds light on the activity and interplay between the different cell types, leading to the effective change in the whole bone. For this reason, hierarchical multi-scale simulations might help in the future to better understand the biomathematical laws behind bone adaptation.
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.
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.
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.
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.
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.
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.
A new biological bone remodeling in silico model combined with advanced discretization methods
Abstract Bone remodeling remains a highly researched topic investigated by many strands of science. The main purpose of this work is formulating a new computational framework for biological simulation, extending the version of the bone remodeling model previously proposed by Komarova. Thus, considering only the biological aspect of the remodeling process, the action of osteoclasts and osteoblasts is taken into account as well as its impact on bone mass. It is conducted a spatiotemporal analysis of a remodeling cycle obtaining a dynamic behavior of bone cells very similar to the biological process already described in the literature. The numerical example used is based on bone images obtained with scanning electron microscopy. During simulation, it is possible to observe the variation of bone's architecture through isomaps. These maps are obtained through the combination of biological bone remodeling models with three distinct numerical techniques—finite element method (FEM), radial point interpolation method (RPIM), and natural neighbor radial point interpolation method (NNRPIM). A study combining these numerical techniques allows to compare their performance. Ultimately, this work supports the inclusion of meshless methods due to their smoother results and its easiness to be combined with medical images from CT scans and MRI.
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.
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.
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.
MODEL-BASED ANALYSIS OF IGF-1 EFFECT ON OSTEOBLAST AND OSTEOCLAST REGULATION IN BONE TURNOVER
The main determinant of bone Ca accretion is a bimolecular regulatory network on osteoblast (OB) and osteoclast (OC). Even though IGF-1 is known as an important regulator in bone cell cycle, little has been done to model IGF-1 action in bone cell regulation. Thus, the objective is to develop a mathematical model that depicts the regulatory action of IGF-1 onto the OB and OC interaction, and to evaluate adolescent and adult bone Ca accretion in response to differences in IGF-1 levels. As a result, a dynamic model of OB and OC with two main regulatory systems, i.e., Receptor Activator for Nuclear Factor [Formula: see text]B (RANK)-RANK Ligand (RANKL)-osteoprogerin (OPG) system, and TGF-[Formula: see text], was augmented with the IGF-1, and incorporated into Ca kinetic data to predict exchangeable bone Ca. The developed model could predict a change in OB and OC levels in response to perturbations in regulators, producing results consistent with bone physiology and published experimental data. The model also estimated parametric difference in regulators between adults and adolescents, suggesting that RANKL/OPG in adolescents was about 4 times higher than in adults, while adolescent serum PTH and IGF-1 concentrations were 60% and 220% of those of adults, respectively. This study highlighted the influence of IGF-1 on the regulation of bone cells in positively modulating bone Ca, suggesting that IGF-1 may be an effective target for reducing bone loss by promoting mature OB.
The bone remodelling cycle
The bone remodelling cycle replaces old and damaged bone and is a highly regulated, lifelong process essential for preserving bone integrity and maintaining mineral homeostasis. During the bone remodelling cycle, osteoclastic resorption is tightly coupled to osteoblastic bone formation. The remodelling cycle occurs within the basic multicellular unit and comprises five co-ordinated steps; activation, resorption, reversal, formation and termination. These steps occur simultaneously but asynchronously at multiple different locations within the skeleton. Study of rare human bone disease and animal models have helped to elucidate the cellular and molecular mechanisms that regulate the bone remodelling cycle. The key signalling pathways controlling osteoclastic bone resorption and osteoblastic bone formation are receptor activator of nuclear factor-κB (RANK)/RANK ligand/osteoprotegerin and canonical Wnt signalling. Cytokines, growth factors and prostaglandins act as paracrine regulators of the cycle, whereas endocrine regulators include parathyroid hormone, vitamin D, calcitonin, growth hormone, glucocorticoids, sex hormones, and thyroid hormone. Disruption of the bone remodelling cycle and any resulting imbalance between bone resorption and formation leads to metabolic bone disease, most commonly osteoporosis. The advances in understanding the cellular and molecular mechanisms underlying bone remodelling have also provided targets for pharmacological interventions which include antiresorptive and anabolic therapies. This review will describe the remodelling process and its regulation, discuss osteoporosis and summarize the commonest pharmacological interventions used in its management.
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.
