Mathematical modeling in bone biology: from intracellular signaling to tissue mechanics

A Mathematical Model for Fibrous Dysplasia: The Role of the Flow of Mutant Cells

Fibrous dysplasia (FD) is a mosaic non-inheritable genetic disorder of the skeleton in which normal bone is replaced by structurally unsound fibro-osseous tissue. There is no curative treatment for FD, partly because its pathophysiology is not yet fully known. We present a simple mathematical model of the disease incorporating its basic known biology, to gain insight on the dynamics of the involved bone-cell populations, and shed light on its pathophysiology. We develop an analytical study of the model and study its basic properties. The existence and stability of steady states are studied, an analysis of sensitivity on the model parameters is done, and different numerical simulations provide findings in agreement with the analytical results. We discuss the model dynamics match with known facts on the disease, and how some open questions could be addressed using the model.

Phase diagrams of bone remodeling using a 3D stochastic cellular automaton

We propose a 3D stochastic cellular automaton model, governed by evolutionary game theory, to simulate bone remodeling dynamics. The model includes four voxel states: Formation, Quiescence, Resorption, and Environment. We simulate the Resorption and Formation processes on separate time scales to explore the parameter space and derive a phase diagram that illustrates the sensitivity of these processes to parameter changes. Combining these results, we simulate a full bone remodeling cycle. Furthermore, we show the importance of modeling small neighborhoods for studying local bone microenvironment controls. This model can guide experimental design and, in combination with other models, it could assist to further explore external impacts on bone remodeling. Consequently, this model contributes to an improved understanding of complex dynamics in bone remodeling dynamics and exploring alterations due to disease or drug treatment.

Computational modeling of multiple myeloma interactions with resident bone marrow cells

The interaction of multiple myeloma with bone marrow resident cells plays a key role in tumor progression and the development of drug resistance. The tumor cell response involves contact-mediated and paracrine interactions. The heterogeneity of myeloma cells and bone marrow cells makes it difficult to reproduce this environment in in-vitro experiments. The use of in-silico established tools can help to understand these complex problems. In this article, we present a computational model based on the finite element method to define the interactions of multiple myeloma cells with resident bone marrow cells. This model includes cell migration, which is controlled by stress-strain equilibrium, and cell processes such as proliferation, differentiation, and apoptosis. A series of computational experiments were performed to validate the proposed model. Cell proliferation by the growth factor IGF-1 is studied for different concentrations ranging from 0-10 ng/mL. Cell motility is studied for different concentrations of VEGF and fibronectin in the range of 0-100 ng/mL. Finally, cells were simulated under a combination of IGF-1 and VEGF stimuli whose concentrations are considered to be dependent on the cancer-associated fibroblasts in the extracellular matrix. Results show a good agreement with previous in-vitro results. Multiple myeloma growth and migration are shown to correlate linearly to the IGF-1 stimuli. These stimuli are coupled with the mechanical environment, which also improves cell growth. Moreover, cell migration depends on the fiber and VEGF concentration in the extracellular matrix. Finally, our computational model shows myeloma cells trigger mesenchymal stem cells to differentiate into cancer-associated fibroblasts, in a dose-dependent manner.

Towards an agent-based model to simulate osseointegration in powder-bed 3D printed implant-like structures

The orthopedic industry is still searching for an efficient way to replace bone loss due to surgical procedures such as arthroplasty and limb-sparing surgery. Additive manufacturing (AM) presents an opportunity to manufacture affordable patient-specific implants. Optimization of the implant-bone interface to maximize osseointegration (bone ingrowth) has not been appropriately addressed. Mechanobiological models, suited to predict mechanical adaptation of bone, cannot be used to predict osseointegration inside implants as the implant is not exposed to any mechanical loading until it is fully accepted by the host body. Biological models relying on partial differential equations based on continuum approximation are not well-suited to predict the discrete phenomenon of osseointegration. This study proposes an agent-based modeling (ABM) approach for representing the osseointegration process for orthopedic implants produced by powder-bed additive manufacturing processes. Agent-Based Modeling (ABM) is a cellular automata based discrete computing technique that uses rule-based mathematics derived from experimental studies to simulate evolutionary phenomena. In this paper, osseointegration inside a hexagonal closed packing of AM powder particles is modeled using ABM. Cellular agents such as pre-osteoblasts and osteoblasts are realistically modeled as cubic cells. The proposed model underpredicts osseointegration at early stages but predicts osseointegration at around 21 days with sufficient accuracy when compared to the in vitro test conducted by Xue et al. in 2007.

Dynamic modeling of bone remodeling, osteolytic metastasis and PK/PD therapy: introducing variable order derivatives as a simplification technique

Bone is constantly being renewed: in the adult skeleton, bone resorption and formation are in a tightly coupled balance, allowing for a constant bone density to be maintained. Yet this micro-environment provides the necessary conditions for the growth and proliferation of tumor cells, and thus bone is a common site for the development of metastases, mainly from primary breast and prostate cancer. Mathematical and computational models with differential equations can replicate this bone remodeling process. These models have been extended to include the effects of disruptive tumor pathologies in the bone dynamics, as metastases contribute to the decoupling between bone resorption and formation and to the self-perpetuating tumor growth cycle. Such models may also contemplate the counteraction effects of currently used therapies, and, in the case of treatments with drugs, their pharmocokinetics and pharmacodynamics. We present a thorough overview of biochemical models for bone remodeling, in the presence of a tumour together with anti-cancer and anti-resorptive therapy, formulated as systems of first-order differential equations, or simplified using variable order derivatives. The latter models, of which some are new to this paper, result in equations with fewer parameters, and allow accounting for anomalous diffusion processes. In this way, more compact and parsimonious models, that promptly highlight tumorous bone interactions, are achieved, providing an effective framework to counteract the loss of bone integrity on the affected areas.

Load adaptation through bone remodeling: a mechanobiological model coupled with the finite element method

This work proposes a novel tissue-scale mechanobiological model of bone remodeling to study bone's adaptation to distinct loading conditions. The devised algorithm describes the mechanosensitivity of bone and its impact on bone cells' functioning through distinct signaling factors. In this study, remodeling is mechanically ruled by variations of the strain energy density (SED) of bone, which is determined by performing a linear elastostatic analysis combined with the finite element method. Depending on the SED levels and on a set of biological signaling factors ([Formula: see text] parameters), osteoclasts and osteoblasts can be mechanically triggered. To reproduce this phenomenon, this work proposes a new set of [Formula: see text] parameters. The combined response of osteoclasts and osteoblasts will then affect bone's apparent density, which is correlated with other mechanical properties of bone, through a phenomenological law. Thus, this novel model proposes a constant interplay between the mechanical and biological components of the process. The spatiotemporal simulation used to validate this new approach is a benchmark example composed by two distinct phases: (1) pre-orientation and (2) load adaptation. On both of them, bone is able to adapt its morphology according to the loading condition, achieving the required trabecular distribution to withstand the applied loads. Moreover, the equilibrium morphology reflects the orientation of the load. These preliminary results support the new approach proposed in this study.

Toward a Mathematical Modeling of Diseases' Impact on Bone Remodeling: Technical Review

A wide variety of bone diseases have hitherto been discovered, such as osteoporosis, Paget’s disease, osteopetrosis, and metastatic bone disease, which are not well defined in terms of changes in biochemical and mechanobiological regulatory factors. Some of these diseases are secondary to other pathologies, including cancer, or to some clinical treatments. To better understand bone behavior and prevent its deterioration, bone biomechanics have been the subject of mathematical modeling that exponentially increased over the last years. These models are becoming increasingly complex. The current paper provides a timely and critical analysis of previously developed bone remodeling mathematical models, particularly those addressing bone diseases. Besides, mechanistic pharmacokinetic/pharmacodynamic (PK/PD) models, which englobe bone disease and its treatment’s effect on bone health. Therefore, the review starts by presenting bone remodeling cycle and mathematical models describing this process, followed by introducing some bone diseases and discussing models of pathological mechanisms affecting bone, and concludes with exhibiting the available bone treatment procedures considered in the PK/PD models.

Dynamics of Bone Cell Interactions and Differential Responses to PTH and Antibody-Based Therapies

We propose a mathematical model describing the dynamics of osteoblasts and osteoclasts in bone remodeling. The goal of this work is to develop an integrated modeling framework for bone remodeling and bone cell signaling dynamics that could be used to explore qualitatively combination treatments for osteoporosis in humans. The model has been calibrated using 57 checks from the literature. Specific global optimization methods based on qualitative objectives have been developed to perform the model calibration. We also added pharmacokinetics representations of three drugs to the model, which are teriparatide (PTH(1-34)), denosumab (a RANKL antibody) and romosozumab (a sclerostin antibody), achieving excellent goodness-of-fit of human clinical data. The model reproduces the paradoxical effects of PTH on the bone mass, where continuous administration of PTH results in bone loss but intermittent administration of PTH leads to bone gain, thus proposing an explanation of this phenomenon. We used the model to simulate different categories of osteoporosis. The main attributes of each disease are qualitatively well captured by the model, for example changes in bone turnover in the disease states. We explored dosing regimens for each disease based on the combination of denosumab and romosozumab, identifying adequate ratios and doses of both drugs for subpopulations of patients in function of categories of osteoporosis and the degree of severity of the disease.

The Bidirectional Interactions Between Inflammation and Coagulation in Fracture Hematoma

IMPACT STATEMENT

The review leads to better understanding of the interrelation between inflammation mediators and coagulation factors in the early fracture hematoma, and their influences on hematoma formation in the beginning of fracture healing. Furthermore, development of therapies aimed at simultaneous modulation of both coagulation factors and inflammation factors that affect hematoma structure, rather than specific factors, may be most promising.

A new conceptual framework for the therapy by optimized multidimensional pulses of therapeutic activity. The case of multiple myeloma model

We developed simulation methodology to assess eventual therapeutic efficiency of exogenous multiparametric changes in a four-component cellular system described by the system of ordinary differential equations. The method is numerically implemented to simulate the temporal behavior of a cellular system of multiple myeloma cells. The problem is conceived as an inverse optimization task where the alternative temporal changes of selected parameters of the ordinary differential equations represent candidate solutions and the objective function quantifies the goals of the therapy. The system under study consists of two main cellular components, tumor cells and their cellular environment, respectively. The subset of model parameters closely related to the environment is substituted by exogenous time dependencies - therapeutic pulses combining continuous functions and discrete parameters subordinated thereafter to the optimization. Synergistic interaction of temporal parametric changes has been observed and quantified whereby two or more dynamic parameters show effects that absent if either parameter is stimulated alone. We expect that the theoretical insight into unstable tumor growth provided by the sensitivity and optimization studies could, eventually, help in designing combination therapies.

Fluctuating periodic solutions and moment boundedness of a stochastic model for the bone remodeling process

In this work, we model osteoclast-osteoblast population dynamics with random environmental fluctuations in order to understand the random variations of the bone remodeling process in real life. For this purpose, we construct a stochastic differential model for the interactions between the osteoclast and osteoblast cell populations using the parameter perturbation technique. We prove the existence of a globally attractive positive unique solution for the stochastically perturbed system. Also, the stochastic boundedness of the solution is demonstrated using its p-th order moments for p ≥ 1. Finally, we show that the introduction of noise in the deterministic model provides a fluctuating periodic solution. Numerical evidence supports our theoretical results and a discussion of the results is carried out.

A mathematical multiscale model of bone remodeling, accounting for pore space-specific mechanosensation

While bone tissue is a hierarchically organized material, mathematical formulations of bone remodeling are often defined on the level of a millimeter-sized representative volume element (RVE), "smeared" over all types of bone microstructures seen at lower observation scales. Thus, there is no explicit consideration of the fact that the biological cells and biochemical factors driving bone remodeling are actually located in differently sized pore spaces: active osteoblasts and osteoclasts can be found in the vascular pores, whereas the lacunar pores host osteocytes - bone cells originating from former osteoblasts which were then "buried" in newly deposited extracellular bone matrix. We here propose a mathematical description which considers size and shape of the pore spaces where the biological and biochemical events take place. In particular, a previously published systems biology formulation, accounting for biochemical regulatory mechanisms such as the rank-rankl-opg pathway, is cast into a multiscale framework coupled to a poromicromechanical model. The latter gives access to the vascular and lacunar pore pressures arising from macroscopic loading. Extensive experimental data on the biological consequences of this loading strongly suggest that the aforementioned pore pressures, together with the loading frequency, are essential drivers of bone remodeling. The novel approach presented here allows for satisfactory simulation of the evolution of bone tissue under various loading conditions, and for different species; including scenarios such as mechanical dis- and overuse of murine and human bone, or in osteocyte-free bone.

Mean-field approach to evolving spatial networks, with an application to osteocyte network formation

We consider evolving networks in which each node can have various associated properties (a state) in addition to those that arise from network structure. For example, each node can have a spatial location and a velocity, or it can have some more abstract internal property that describes something like a social trait. Edges between nodes are created and destroyed, and new nodes enter the system. We introduce a "local state degree distribution" (LSDD) as the degree distribution at a particular point in state space. We then make a mean-field assumption and thereby derive an integro-partial differential equation that is satisfied by the LSDD. We perform numerical experiments and find good agreement between solutions of the integro-differential equation and the LSDD from stochastic simulations of the full model. To illustrate our theory, we apply it to a simple model for osteocyte network formation within bones, with a view to understanding changes that may take place during cancer. Our results suggest that increased rates of differentiation lead to higher densities of osteocytes, but with a smaller number of dendrites. To help provide biological context, we also include an introduction to osteocytes, the formation of osteocyte networks, and the role of osteocytes in bone metastasis.

Bone fracture healing in mechanobiological modeling: A review of principles and methods

Bone fracture is a very common body injury. The healing process is physiologically complex, involving both biological and mechanical aspects. Following a fracture, cell migration, cell/tissue differentiation, tissue synthesis, and cytokine and growth factor release occur, regulated by the mechanical environment. Over the past decade, bone healing simulation and modeling has been employed to understand its details and mechanisms, to investigate specific clinical questions, and to design healing strategies. The goal of this effort is to review the history and the most recent work in bone healing simulations with an emphasis on both biological and mechanical properties. Therefore, we provide a brief review of the biology of bone fracture repair, followed by an outline of the key growth factors and mechanical factors influencing it. We then compare different methodologies of bone healing simulation, including conceptual modeling (qualitative modeling of bone healing to understand the general mechanisms), biological modeling (considering only the biological factors and processes), and mechanobiological modeling (considering both biological aspects and mechanical environment). Finally we evaluate different components and clinical applications of bone healing simulation such as mechanical stimuli, phases of bone healing, and angiogenesis.

Bone remodeling as a spatial evolutionary game

Bone remodeling is a complex process involving cell-cell interactions, biochemical signaling and mechanical stimuli. Early models of the biological aspects of remodeling were non-spatial and focused on the local dynamics at a fixed location in the bone. Several spatial extensions of these models have been proposed, but they generally suffer from two limitations: first, they are not amenable to analysis and are computationally expensive, and second, they neglect the role played by bone-embedded osteocytes. To address these issues, we developed a novel model of spatial remodeling based on the principles of evolutionary game theory. The analytically tractable framework describes the spatial interactions between zones of bone resorption, bone formation and quiescent bone, and explicitly accounts for regulation of remodeling by bone-embedded, mechanotransducing osteocytes. Using tools from the theory of interacting particle systems we systematically classified the different dynamic regimes of the spatial model and identified regions of parameter space that allow for global coexistence of resorption, formation and quiescence, as observed in physiological remodeling. In coexistence scenarios, three-dimensional simulations revealed the emergence of sponge-like bone clusters. Comparison between spatial and non-spatial dynamics revealed substantial differences and suggested a stabilizing role of space. Our findings emphasize the importance of accounting for spatial structure and bone-embedded osteocytes when modeling the process of bone remodeling. Thanks to the lattice-based framework, the proposed model can easily be coupled to a mechanical model of bone loading.

An intramembranous ossification model for the in silico analysis of bone tissue formation in tooth extraction sites

The accurate modeling of biological processes allows us to predict the spatiotemporal behavior of living tissues by computer-aided (in silico) testing, a useful tool for the development of medical strategies, avoiding the expenses and potential ethical implications of in vivo experimentation. A model for bone healing in mouth would be useful for selecting proper surgical techniques in dental procedures. In this paper, the formulation and implementation of a model for Intramembranous Ossification is presented aiming to describe the complex process of bone tissue formation in tooth extraction sites. The model consists in a mathematical description of the mechanisms in which different types of cells interact, synthesize and degrade extracellular matrices under the influence of biochemical factors. Special attention is given to angiogenesis, oxygen-dependent effects and growth factor-induced apoptosis of fibroblasts. Furthermore, considering the depth-dependent vascularization of mandibular bone and its influence on bone healing, a functional description of the cell distribution on the severed periodontal ligament (PDL) is proposed. The developed model was implemented using the finite element method (FEM) and successfully validated by simulating an animal in vivo experiment on dogs reported in the literature. A good fit between model outcome and experimental data was obtained with a mean absolute error of 3.04%. The mathematical framework presented here may represent an important tool for the design of future in vitro and in vivo tests, as well as a precedent for future in silico studies on osseointegration and mechanobiology.

A mathematical model of cell equilibrium and joint cell formation in multiple myeloma

In Multiple Myeloma Bone Disease healthy bone remodelling is affected by tumour cells by means of paracrine cytokinetic signalling in such a way that osteoclast formation is enhanced and the growth of osteoblast cells inhibited. The participating cytokines are described in the literature. Osteoclast-induced myeloma cell growth is also reported. Based on existing mathematical models for healthy bone remodelling a three-way equilibrium model is presented for osteoclasts, osteoblasts and myeloma cell populations to describe the progress of the illness in a scenario in which there is a secular increase in the cytokinetic interactive effectiveness of paracrine processes. The equilibrium state for the system is obtained. The paracrine interactive effectiveness is explored by parameter variation and the stable region in the parameter space is identified. Then recently-discovered joint myeloma-osteoclast cells are added to the model to describe the populations inside lytic lesions. It transpires that their presence expands the available parameter space for stable equilibrium, thus permitting a detrimental, larger population of osteoclasts and myeloma cells. A possible relapse mechanism for the illness is explored by letting joint cells dissociate. The mathematics then permits the evaluation of the evolution of the cell populations as a function of time during relapse.

Integrating new discoveries into the "vicious cycle" paradigm of prostate to bone metastases

In prostate to bone metastases, the "vicious cycle" paradigm has been traditionally used to illustrate how metastases manipulate the bone forming osteoblasts and resorbing osteoclasts in order to yield factors that facilitate growth and establishment. However, recent advances have illustrated that the cycle is far more complex than this simple interpretation. In this review, we will discuss the role of exosomes and hematopoietic/mesenchymal stem/stromal cells (MSC) that facilitate the establishment and activation of prostate metastases and how cells including myeloid-derived suppressor cells, macrophages, T cells, and nerve cells contribute to the momentum of the vicious cycle. The increased complexity of the tumor-bone microenvironment requires a system level approach. The evolution of computational models to interrogate the tumor-bone microenvironment is also discussed, and the application of this integrated approach should allow for the development of effective therapies to treat and cure prostate to bone metastases.

Regenerative orthopaedics: in vitro, in vivo...in silico

In silico, defined in analogy to in vitro and in vivo as those studies that are performed on a computer, is an essential step in problem-solving and product development in classical engineering fields. The use of in silico models is now slowly easing its way into medicine. In silico models are already used in orthopaedics for the planning of complicated surgeries, personalised implant design and the analysis of gait measurements. However, these in silico models often lack the simulation of the response of the biological system over time. In silico models focusing on the response of the biological systems are in full development. This review starts with an introduction into in silico models of orthopaedic processes. Special attention is paid to the classification of models according to their spatiotemporal scale (gene/protein to population) and the information they were built on (data vs hypotheses). Subsequently, the review focuses on the in silico models used in regenerative orthopaedics research. Contributions of in silico models to an enhanced understanding and optimisation of four key elements-cells, carriers, culture and clinics-are illustrated. Finally, a number of challenges are identified, related to the computational aspects but also to the integration of in silico tools into clinical practice.

Modeling TGF-β in early stages of cancer tissue dynamics

Recent works have highlighted a double role for the Transforming Growth Factor (-): it inhibits cancer in healthy cells and potentiates tumor progression during late stage of tumorigenicity, respectively; therefore it has been termed the “Jekyll and Hyde” of cancer or, alternatively, an “excellent servant but a bad master”. It remains unclear how this molecule could have the two opposite behaviours. In this work, we propose a - multi scale mathematical model at molecular, cellular and tissue scales. The multi scalar behaviours of the - are described by three coupled models built up together which can approximatively be related to distinct microscopic, mesoscopic, and macroscopic scales, respectively. We first model the dynamics of - at the single-cell level by taking into account the intracellular and extracellular balance and the autocrine and paracrine behaviour of -. Then we use the average estimates of the - from the first model to understand its dynamics in a model of duct breast tissue. Although the cellular model and the tissue model describe phenomena at different time scales, their cumulative dynamics explain the changes in the role of - in the progression from healthy to pre-tumoral to cancer. We estimate various parameters by using available gene expression datasets. Despite the fact that our model does not describe an explicit tissue geometry, it provides quantitative inference on the stage and progression of breast cancer tissue invasion that could be compared with epidemiological data in literature. Finally in the last model, we investigated the invasion of breast cancer cells in the bone niches and the subsequent disregulation of bone remodeling processes. The bone model provides an effective description of the bone dynamics in healthy and early stages cancer conditions and offers an evolutionary ecological perspective of the dynamics of the competition between cancer and healthy cells.

The role of osteoclasts in bone tissue engineering

The success of scaffold-based bone regeneration approaches strongly depends on the performance of the biomaterial utilized. Within the efforts of regenerative medicine towards a restitutio ad integrum (i.e. complete reconstruction of a diseased tissue), scaffolds should be completely degraded within an adequate period of time. The degradation of synthetic bone substitute materials involves both chemical dissolution (physicochemical degradation) and resorption (cellular degradation by osteoclasts). Responsible for bone resorption are osteoclasts, cells of haematopoietic origin. Osteoclasts play also a crucial role in bone remodelling, which is essential for the regeneration of bone defects. There is, however, surprisingly limited knowledge about the detailed effects of osteoclasts on biomaterials degradation behaviour. This review covers the relevant fundamental knowledge and progress made in the field of osteoclast activity related to biomaterials used for bone regeneration. In vitro studies with osteoclastic precursor cells on synthetic bone substitute materials show that there are specific parameters that inhibit or enhance resorption. Moreover, analyses of the bone-material interface reveal that biomaterials composition has a significant influence on their degradation in contact with osteoclasts. Crystallinity, grain size, surface bioactivity and density of the surface seem to have a less significant effect on osteoclastic activity. In addition, the topography of the scaffold surface can be tailored to affect the development and spreading of osteoclast cells. The present review also highlights possible areas on which future research is needed and which are relevant to enhance our understanding of the complex role of osteoclasts in bone tissue engineering.

Endocortical bone loss in osteoporosis: the role of bone surface availability

Age-related bone loss and postmenopausal osteoporosis are due to a dysregulation of bone remodelling in which less bone is reformed than resorbed. This dysregulation of bone remodelling does not occur with equal strength in all bone regions. Loss of bone is more pronounced near the endocortical surface. This leads to thinning of the cortical wall proceeding from the endosteum, a process sometimes called 'trabecularisation'. In this paper, we investigate the influence of the nonuniform distribution of bone surface within bone tissue for osteoporotic bone losses. We use a spatio-temporal computational model of bone remodelling in which microstructural changes of bone tissue are represented by a phenomenological relationship between bone specific surface and bone porosity. The simulation of an osteoporotic condition by our model shows that the evolution of bone porosity within a bone cross section is significantly influenced by the nonuniform availability of bone surface. Greater bone loss occurs near the endocortical wall, leading to cortical wall thinning and to an expansion of the medullary cavity similar to cross-sectional observations from human femur midshafts. Our model suggests that the rate of cortical wall thinning is fast/slow in the presence/absence of an adjacent trabecular or trabecularised bone compartment.

A negative feedback model for a mechanism based description of longitudinal observations. Application for bone turnover biomarkers

In modern medicine the diagnosis and prognosis of an abnormal metabolic condition is based on blood borne measurements involving one or more biomarker.

OBJECTIVE

This paper reports the development of a minimal negative feedback model for the description of longitudinal biomarkers concentrations for treatment of osteoporosis in postmenopausal women.

METHODS

Literature data were obtained from double-blind, placebo-controlled clinical trial over three years. There were four treatment groups: 1) Placebo, 2) Alendronate, 3) Conjugated Estrogen, and/or 4) Combination therapy. The negative feedback model consists of a biomarker and a companion controller. By considering the above basal biomarker values it is shown that the dynamics can be described by a second order differential equation without the involvement of biomarker production rate. The second order differential equation is also analogous to classical negative feedback servomechanism model with two parameters ω(n) and ξ. It was assumed that the rate constants defining the negative feedback model were equal which would set ξ to 0.707 with only ω(n) to be estimated.

RESULTS

ω(n) was estimated for both lumbar spine bone mineral density (BMD) and bone-specific alkaline phosphatase (BAP) in four treatments groups. The t(½) of BMD and BAP were estimated at 26.8 (0.30) and 9.4 (0.30) days respectively.

CONCLUSIONS

The negative feedback model of BMD supports the mechanism whereby Conjugated Estrogen and Alendronate decrease the clearance rate constant of BMD analogous to increased apoptosis of osteoclasts. The linked negative feedback models facilitate a mechanism based prediction of BMD using the concentrations of the bone turnover marker BAP.

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.

Role of mathematical modeling in bone fracture healing

Bone fracture healing is a complex physiological process commonly described by a four-phase model consisting of an inflammatory phase, two repair phases with soft callus formation followed by hard callus formation, and a remodeling phase, or more recently by an anabolic/catabolic model. Data from humans and animal models have demonstrated crucial environmental conditions for optimal fracture healing, including the mechanical environment, blood supply and availability of mesenchymal stem cells. Fracture healing spans multiple length and time scales, making it difficult to know precisely which factors and/or phases to manipulate in order to obtain optimal fracture-repair outcomes. Deformations resulting from physiological loading or fracture fixation at the organ scale are sensed at the cellular scale by cells inside the fracture callus. These deformations together with autocrine and paracrine signals determine cellular differentiation, proliferation and migration. The local repair activities lead to new bone formation and stabilization of the fracture. Although experimental data are available at different spatial and temporal scales, it is not clear how these data can be linked to provide a holistic view of fracture healing. Mathematical modeling is a powerful tool to quantify conceptual models and to establish the missing links between experimental data obtained at different scales. The objective of this review is to introduce mathematical modeling to readers who are not familiar with this methodology and to demonstrate that once validated, such models can be used for hypothesis testing and to assist in clinical treatment as will be shown for the example of atrophic nonunions.

Modelling osteomyelitis

Background

This work focuses on the computational modelling of osteomyelitis, a bone pathology caused by bacteria infection (mostly Staphylococcus aureus). The infection alters the RANK/RANKL/OPG signalling dynamics that regulates osteoblasts and osteoclasts behaviour in bone remodelling, i.e. the resorption and mineralization activity. The infection rapidly leads to severe bone loss, necrosis of the affected portion, and it may even spread to other parts of the body. On the other hand, osteoporosis is not a bacterial infection but similarly is a defective bone pathology arising due to imbalances in the RANK/RANKL/OPG molecular pathway, and due to the progressive weakening of bone structure.

Results

Since both osteoporosis and osteomyelitis cause loss of bone mass, we focused on comparing the dynamics of these diseases by means of computational models. Firstly, we performed meta-analysis on a gene expression data of normal, osteoporotic and osteomyelitis bone conditions. We mainly focused on RANKL/OPG signalling, the TNF and TNF receptor superfamilies and the NF-kB pathway. Using information from the gene expression data we estimated parameters for a novel model of osteoporosis and of osteomyelitis. Our models could be seen as a hybrid ODE and probabilistic verification modelling framework which aims at investigating the dynamics of the effects of the infection in bone remodelling. Finally we discuss different diagnostic estimators defined by formal verification techniques, in order to assess different bone pathologies (osteopenia, osteoporosis and osteomyelitis) in an effective way.

Conclusions

We present a modeling framework able to reproduce aspects of the different bone remodeling defective dynamics of osteomyelitis and osteoporosis. We report that the verification-based estimators are meaningful in the light of a feed forward between computational medicine and clinical bioinformatics.

Multilevel computational modeling and quantitative analysis of bone remodeling

Our work focuses on bone remodeling with a multiscale breadth that ranges from modeling intracellular and intercellular RANK/RANKL signaling to tissue dynamics, by developing a multilevel modeling framework. Several important findings provide clear evidences of the multiscale properties of bone formation and of the links between RANK/RANKL and bone density in healthy and disease conditions. Recent studies indicate that the circulating levels of OPG and RANKL are inversely related to bone turnover and Bone Mineral Density (BMD) and contribute to the development of osteoporosis in postmenopausal women, and thalassemic patients. We make use of a spatial process algebra, the Shape Calculus, to control stochastic cell agents that are continuously remodeling the bone. We found that our description is effective for such a multiscale, multilevel process and that RANKL signaling small dynamic concentration defects are greatly amplified by the continuous alternation of absorption and formation resulting in large structural bone defects. This work contributes to the computational modeling of complex systems with a multilevel approach connecting formal languages and agent-based simulation tools.

A model for the change of cancellous bone volume and structure over time

A model is presented for characterizing the process by which cancellous bone changes in volume and structure over time. The model comprises simulations of local changes resulting from individual remodelling events, known as bone multicellular units (BMU), and an ordinary differential equation for connecting the number of remodelling events to real time. The model is validated on micro-CT scans of tibiae of normal rats, estrogen deprived rats and estrogen deprived rats treated with bisphosphonates. The model explains the asymptotic trends seen in changes of bone volume over time resulting from estrogen deprivation as well as trends seen subsequent to treatment. The model demonstrates that both bone volume and structure changes can be explained in terms of resetting remodelling parameters. The model also shows that either current understanding of the effects of bisphosphonates is not correct or that the simplest description of remodelling does not suffice to explain both the change in bone volume and structure of rats treated with bisphosphonates.

Bone mechanotransduction may require augmentation in order to strengthen the senescent skeleton

Physical exercise is thought to hold promise as a non-invasive countermeasure against skeletal fragility arising from post-menopausal and age-related osteoporosis. Importantly, mechanical loading and exercise are capable of increasing bone size via periosteal expansion, which by far, is the most effective means of strengthening the structure of a given bone. The focus of this review was to therefore explore whether exercise has the potential to increase periosteal modeling and bone size in the senescent skeleton. A survey of exercise trials in humans suggests that exercise interventions that enhance periosteal modeling in the young skeleton fail to do the same in the elderly skeleton. Underlying this ineffectiveness, in vitro studies indicate that aging lowers basal levels of cell function and degrades bone mechanotransduction at a variety of levels from altered second messenger signaling to gene expression driving proliferation and/or differentiation. Given these age-related alterations, the ultimate efficacy of an exercise intervention may depend upon concurrent supplementation that directly address deficits in signaling and/or cell function. In this context, in vivo animal models of mechanical loading that simulate the muted periosteal adaptation in the elderly hold potential to examine the efficacy of countermeasures. Preliminary in vivo experiments suggest that pharmacologically counteracting age-related deficits in cellular function can restore exercise induced periosteal modeling in the senescent skeleton to levels observed in young animals. If the safety and efficacy of this strategy were to be confirmed for human use, it would enable the utilization of exercise as a viable countermeasure against skeletal fragility at senescence.

Modelling the anabolic response of bone using a cell population model

To maintain bone mass during bone remodelling, coupling is required between bone resorption and bone formation. This coordination is achieved by a network of autocrine and paracrine signalling molecules between cells of the osteoclastic lineage and cells of the osteoblastic lineage. Mathematical modelling of signalling between cells of both lineages can assist in the interpretation of experimental data, clarify signalling interactions and help develop a deeper understanding of complex bone diseases. Several mathematical models of bone cell interactions have been developed, some including RANK-RANKL-OPG signalling between cells and systemic parathyroid hormone PTH. However, to our knowledge these models do not currently include key aspects of some more recent biological evidence for anabolic responses. In this paper, we further develop a mathematical model of bone cell interactions by Pivonka et al. (2008) to include the proliferation of precursor osteoblasts into the model. This inclusion is important to be able to account for Wnt signalling, believed to play an important role in the anabolic responses of bone. We show that an increased rate of differentiation to precursor cells or an increased rate of proliferation of precursor osteoblasts themselves both result in increased bone mass. However, modelling these different processes separately enables the new model to represent recent experimental discoveries such as the role of Wnt signalling in bone biology and the recruitment of osteoblast progenitor cells by transforming growth factor β. Finally, we illustrate the power of the new model's capabilities by applying the model to prostate cancer metastasis to bone. In the bone microenvironment, prostate cancer cells are believed to release some of the same signalling molecules used to coordinate bone remodelling (i.e.,Wnt and PTHrP), enabling the cancer cells to disrupt normal signalling and coordination between bone cells. This disruption can lead to either bone gain or bone loss. We demonstrate that the new computational model developed here is capable of capturing some key observations made on the evolution of the bone mass due to metastasis of prostate cancer to the bone microenvironment.

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 spatial behavior 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.

Coping with time scales in disease systems analysis: application to bone remodeling

In this study we demonstrate the added value of mathematical model reduction for characterizing complex dynamic systems using bone remodeling as an example. We show that for the given parameter values, the mechanistic RANK-RANKL-OPG pathway model proposed by Lemaire et al. (J Theor Biol 229:293-309, 2004) can be reduced to a simpler model, which can describe the dynamics of the full Lemaire model to very good approximation. The response of both models to changes in the underlying physiology and therapeutic interventions was evaluated in four physiologically meaningful scenarios: (i) estrogen deficiency/estrogen replacement therapy, (ii) Vitamin D deficiency, (iii) ageing, and (iv) chronic glucocorticoid treatment and its cessation. It was found that on the time scale of disease progression and therapeutic intervention, the models showed negligible differences in their dynamic properties and were both suitable for characterizing the impact of estrogen deficiency and estrogen replacement therapy, Vitamin D deficiency, ageing, and chronic glucocorticoid treatment and its cessation on bone forming (osteoblasts) and bone resorbing (osteoclasts) cells. It was also demonstrated how the simpler model could help in elucidating qualitative properties of the observed dynamics, such as the absence of overshoot and rebound, and the different dynamics of onset and washout.

Trabecular bone remodelling simulated by a stochastic exchange of discrete bone packets from the surface

Human bone is constantly renewed through life via the process of bone remodelling, in which individual packets of bone are removed by osteoclasts and replaced by osteoblasts. Remodelling is mechanically controlled, where osteocytes embedded within the bone matrix are thought to act as mechanical sensors. In this computational work, a stochastic model for bone remodelling is used in which the renewal of bone material occurs by exchange of discrete bone packets. We tested different hypotheses of how the mechanical stimulus for bone remodelling is integrated by osteocytes and sent to actor cells on the bone's surface. A collective (summed) signal from multiple osteocytes as opposed to an individual (maximal) signal from a single osteocyte was found to lead to lower inner porosity and surface roughness of the simulated bone structure. This observation can be interpreted in that collective osteocyte signalling provides an effective surface tension to the remodelling process. Furthermore, the material heterogeneity due to remodelling was studied on a network of trabeculae. As the model is discrete, the age of individual bone packets can be monitored with time. The simulation results were compared with experimental data coming from quantitative back scattered electron imaging by transforming the information about the age of the bone packet into a mineral content. Discrepancies with experiments indicate that osteoclasts preferentially resorb low mineralized, i.e. young, bone at the bone's surface.