General analysis of mathematical models for bone remodeling

Tumor growth for remodeling process: A 2D approach

This paper aims to present a comprehensive framework for coupling tumor-bone remodeling processes in a 2-dimensional geometry. This is achieved by introducing a bio-inspired damage that represents the growing tumor, which subsequently affects the main populations involved in the remodeling process, namely, osteoclasts, osteoblasts, and bone tissue. The model is constructed using a set of differential equations based on the Komarova's and Ayati's models, modified to incorporate the bio-inspired damage that may result in tumor mass formation. Three distinct models were developed. The first two models are based on the Komarova's governing equations, with one demonstrating an osteolytic behavior and the second one an osteoblastic model. The third model is a variation of Ayati's model, where the bio-inspired damage is induced through the paracrine and autocrine parameters, exhibiting an osteolytic behavior. The obtained results are consistent with existing literature, leading us to believe that our in-silico experiments will serve as a cornerstone for paving the way towards targeted interventions and personalized treatment strategies, ultimately improving the quality of life for those affected by these conditions.

A coupled mathematical model between bone remodeling and tumors: a study of different scenarios using Komarova's model

This paper aims to construct a general framework of coupling tumor-bone remodeling processes in order to produce plausible outcomes of the effects of tumors on the number of osteoclasts, osteoblasts, and the frequency of the bone turnover cycle. In this document, Komarova's model has been extended to include the effect of tumors on the bone remodeling processes. Thus, we explored three alternatives for coupling tumor presence into Komarova's model: first, using a "damage" parameter that depends on the tumor cell concentration. A second model follows the original structure of Komarova, including the tumor presence in those equations powered up to a new parameter, called the paracrine effect of the tumor on osteoclasts and osteoblasts; the last model is replicated from Ayati and collaborators in which the impact of the tumor is included into the paracrine parameters. Through the models, we studied their stability and considered some examples that can reproduce the tumor effects seen in clinic and experimentally. Therefore, this paper has three parts: the exposition of the three models, the results and discussion (where we explore some aspects and examples of the solution of the models), and the conclusion.

Modeling osteoporosis to design and optimize pharmacological therapies comprising multiple drug types

For the treatment of postmenopausal osteoporosis, several drug classes with different mechanisms of action are available. Since only a limited set of dosing regimens and drug combinations can be tested in clinical trials, it is currently unclear whether common medication strategies achieve optimal bone mineral density gains or are outperformed by alternative dosing schemes and combination therapies that have not been explored so far. Here, we develop a mathematical framework of drug interventions for postmenopausal osteoporosis that unifies fundamental mechanisms of bone remodeling and the mechanisms of action of four drug classes: bisphosphonates, parathyroid hormone analogs, sclerostin inhibitors, and receptor activator of NF-κB ligand inhibitors. Using data from several clinical trials, we calibrate and validate the model, demonstrating its predictive capacity for complex medication scenarios, including sequential and parallel drug combinations. Via simulations, we reveal that there is a large potential to improve gains in bone mineral density by exploiting synergistic interactions between different drug classes, without increasing the total amount of drug administered.

Generalized Structural Kinetic Modeling: A Survey and Guide

Many current challenges involve understanding the complex dynamical interplay between the constituents of systems. Typically, the number of such constituents is high, but only limited data sources on them are available. Conventional dynamical models of complex systems are rarely mathematically tractable and their numerical exploration suffers both from computational and data limitations. Here we review generalized modeling, an alternative approach for formulating dynamical models to gain insights into dynamics and bifurcations of uncertain systems. We argue that this approach deals elegantly with the uncertainties that exist in real world data and enables analytical insight or highly efficient numerical investigation. We provide a survey of recent successes of generalized modeling and a guide to the application of this modeling approach in future studies such as complex integrative ecological models.

Antigen receptor therapy in bone metastasis via optimal control for different human life stages

In this work we propose a bone metastasis model using power law growth functions in order to describe the biochemical interactions between bone cells and cancer cells. Experimental studies indicate that bone remodeling cycles are different for human life stages: childhood, young adulthood, and adulthood. In order to include such differences in our study, we estimate the model parameter values for each human life stage via bifurcation analysis. Results reveal an intrinsic relationship between the active period of remodeling cycles and the proliferation of cancer cells. Subsequently, using optimal control theory we analyze a possible antigen receptor therapy as a new treatment for bone metastasis. Theoretical results such as existence of optimal solutions are proved. Numerical simulations for late stages of bone metastasis are presented and a discussion of our results is carried out.

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.

Parameter reduction, sensitivity studies, and correlation analyses applied to a mechanobiologically regulated bone cell population model of the bone metabolism

When striving for reconstructing and predicting bone remodeling processes by means of mathematical models, cell population models have become a popular option. From a conceptual point of view, these models are able to take into account an arbitrary amount of regulatory mechanisms driving the development of bone cells and their activities. However, in most cases, the models include a large number of parameters; and most of those parameters cannot be measured, which certainly compromises the credibility of cell population models. Here, new insights are presented as to the potential improvement of this unsatisfactory situation. In particular, a previously published bone remodeling model was considered, and based on combination and merging of the original parameters, the total number of parameters could be reduced from 28 to 18, without impairing the model's versatility and significance. Furthermore, a comprehensive number of one- and two-variable sensitivity studies were performed, pointing out which parameters (alone and in combination with other parameters) influence the model predictions significantly - for that purpose, the mean squared relative error (MSRE) between simulations based on the original parameters and based on varied parameters was considered as failure measure. It has turned out that the model is significantly more sensitive to parameters which can be considered as phenomenological (such as differentiation, proliferation, and apoptosis rates) than to parameters which are directly related to specific processes (such as dissociation rate constants, and maximum concentrations of the involved factors). Using common correlation measures (such as Pearson, Spearman, and partial ranked correlation coefficients), correlation studies revealed that the correlations between most parameters and the MSRE are weak, while a few parameters exhibited moderate correlations. In conclusion, the results shown in this paper provide valuable insights concerning the design of new experiments allowing for measurement of the parameters which are most influential in the context of bone remodeling simulation.

Strange attractors in discrete slow power-law models of bone remodeling

Recently, a family of nonlinear mathematical discrete systems to describe biological interactions was considered. Such interactions are modeled by power-law functions where the exponents involve regulation processes. Considering exponent values giving rise to hyperbolic equilibria, we show that the systems exhibit irregular behavior characterized by strange attractors. The systems are numerically analyzed for different parameter values. Depending on the initial conditions, the orbits of each system either diverge to infinity or approach a periodic orbit or a strange attractor. Such dynamical behavior is identified by their Lyapunov exponents and local dimension. Finally, an application to the biochemical process of bone remodeling is presented. The existence of deterministic chaos in this process reveals a possible explanation of reproducibility failure and variation of effects in clinical experiments.

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.

Revealing instabilities in a generalized triadic supply network: A bifurcation analysis

Supply networks are exposed to instabilities and thus a high level of risk. To mitigate this risk, it is necessary to understand how instabilities are formed in supply networks. In this paper, we focus on instabilities in inventory dynamics that develop due to the topology of the supply network. To be able to capture these topology-induced instabilities, we use a method called generalized modeling, a minimally specified modeling approach adopted from ecology. This method maps the functional dependencies of production rates on the inventory levels of different parts and products, which are imposed by the network topology, to a set of elasticity parameters. We perform a bifurcation analysis to investigate how these elasticities affect the stability. First, we show that dyads and serial supply chains are immune to topology-induced instabilities. In contrast, in a simple triadic network, where a supplier acts as both a first and a second tier supplier, we can identify instabilities that emerge from saddle-node, Hopf, and global homoclinic bifurcations. These bifurcations lead to different types of dynamical behavior, including exponential convergence to and divergence from a steady state, temporary oscillations around a steady state, and co-existence of different types of dynamics, depending on initial conditions. Finally, we discuss managerial implications of the results.

Dynamic modeling of bone metastasis, microenvironment and therapy: Integrating parathyroid hormone (PTH) effect, anti-resorptive and anti-cancer therapy

Bone is a common site for the development of metastasis, as its microenvironment provides the necessary conditions for the growth and proliferation of cancer cells. Several mathematical models to describe the bone remodeling process and how osteoclasts and osteoblasts coupled action ensures bone homeostasis have been proposed and further extended to include the effect of cancer cells. The model proposed here includes the influence of the parathyroid hormone (PTH) as capable of triggering and regulating the bone remodeling cycle. It also considers the secretion of PTH-related protein (PTHrP) by cancer cells, which stimulates the production of receptor activator of nuclear factor kappa-B ligand (RANKL) by osteoblasts that activates osteoclasts, increasing bone resorption and the subsequent release of growth factors entrapped in the bone matrix, which induce tumor growth, giving rise to a self-perpetuating cycle known as the vicious cycle of bone metastases. The model additionally describes how the presence of metastases contributes to the decoupling between bone resorption and formation. Moreover, the effects of anti-cancer and anti-resorptive treatments, through chemotherapy and the administration of bisphosphonates or denosumab, are also included, along with their corresponding pharmacokinetics (PK) and pharmacodynamics (PD). The simulated models, available at http://sels.tecnico.ulisboa.pt/software/, are able to describe bone remodeling cycles, the growth of bone metastases and how treatment can effectively reduce tumor burden on bone and prevent loss of bone strength.

Stability analysis of a Komarova type model for the interactions of osteoblast and osteoclast cells during bone remodeling

In order to analyze theoretically the dynamics of osteoblast and osteoclast cells in the bone remodeling process we first consider a simplified Komarova model. The existence of periodic solutions, which is consistent with the biophysical phenomenon, has been observed only numerically for the general model. By a stability analysis of the simplified model we provide sufficient conditions to obtain existence and uniqueness of positive periodic solutions. Considering recent biological evidence about the participation of another cells like osteocytes in the regulation of bone remodeling, we incorporate to the simplified model a new term as a way to model the signaling of external agents in the remodeling process. Finally, we demonstrate that this new model has stable positive non-periodic solutions. All the theoretical results are accompanied by computational simulations.

Mathematical modelling of the pathogenesis of multiple myeloma-induced bone disease

Multiple myeloma (MM) is the second most common haematological malignancy and results in destructive bone lesions. The interaction between MM cells and the bone microenvironment plays an important role in the development of the tumour cells and MM-induced bone disease and forms a 'vicious cycle' of tumour development and bone destruction, intensified by suppression of osteoblast activity and promotion of osteoclast activity. In this paper, a mathematical model is proposed to simulate how the interaction between MM cells and the bone microenvironment facilitates the development of the tumour cells and the resultant bone destruction. It includes both the roles of inhibited osteoblast activity and stimulated osteoclast activity. The model is able to mimic the temporal variation of bone cell concentrations and resultant bone volume after the invasion and then removal of the tumour cells and explains why MM-induced bone lesions rarely heal even after the complete removal of MM cells. The behaviour of the model compares well with published experimental data. The model serves as a first step to understand the development of MM-induced bone disease and could be applied further to evaluate the current therapies against MM-induced bone disease and even suggests new potential therapeutic targets.

Effect of Rotation on Wave Propagation in Hollow Poroelastic Circular Cylinder

The objective of this paper is to study the effect of rotation on the wave propagation in an infinite poroelastic hollow circular cylinder. The frequency equation for poroelastic hollow circular cylinder is obtained when the boundaries are stress free and is examined numerically. The frequency, phase velocity, and attenuation coefficient are calculated for a pervious surface for various values of rotation, wave number, and thickness of the cylinder which are presented for nonaxial symmetric vibrations for a pervious surface. The dispersion curves are plotted for the poroelastic elastic behavior of the poroelastic material. Results are discussed for poroelastic material. The results indicate that the effect of rotation, wave number, and thickness on the wave propagation in the hollow poroelastic circular cylinder is very pronounced.

How to predict community responses to perturbations in the face of imperfect knowledge and network complexity

Recent attempts to predict the response of large food webs to perturbations have revealed that in larger systems increasingly precise information on the elements of the system is required. Thus, the effort needed for good predictions grows quickly with the system's complexity. Here, we show that not all elements need to be measured equally well, suggesting that a more efficient allocation of effort is possible. We develop an iterative technique for determining an efficient measurement strategy. In model food webs, we find that it is most important to precisely measure the mortality and predation rates of long-lived, generalist, top predators. Prioritizing the study of such species will make it easier to understand the response of complex food webs to perturbations.

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.

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.