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

SUMMARY 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. © 2014 The Authors. International Journal for Numerical Methods in Biomedical Engineering published by John Wiley & Sons Ltd

Computational Modeling of Interactions between Multiple Myeloma and the Bone Microenvironment

Multiple Myeloma (MM) is a B-cell malignancy that is characterized by osteolytic bone lesions. It has been postulated that positive feedback loops in the interactions between MM cells and the bone microenvironment form reinforcing ‘vicious cycles’, resulting in more bone resorption and MM cell population growth in the bone microenvironment. Despite many identified MM-bone interactions, the combined effect of these interactions and their relative importance are unknown. In this paper, we develop a computational model of MM-bone interactions and clarify whether the intercellular signaling mechanisms implemented in this model appropriately drive MM disease progression. This new computational model is based on the previous bone remodeling model of Pivonka et al. [1], and explicitly considers IL-6 and MM-BMSC (bone marrow stromal cell) adhesion related pathways, leading to formation of two positive feedback cycles in this model. The progression of MM disease is simulated numerically, from normal bone physiology to a well established MM disease state. Our simulations are consistent with known behaviors and data reported for both normal bone physiology and for MM disease. The model results suggest that the two positive feedback cycles identified for this model are sufficient to jointly drive the MM disease progression. Furthermore, quantitative analysis performed on the two positive feedback cycles clarifies the relative importance of the two positive feedback cycles, and identifies the dominant processes that govern the behavior of the two positive feedback cycles. Using our proposed quantitative criteria, we identify which of the positive feedback cycles in this model may be considered to be ‘vicious cycles’. Finally, key points at which to block the positive feedback cycles in MM-bone interactions are identified, suggesting potential drug targets.

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.

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.

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.

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.

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, Patget’s disease, osteopetrosis, and metastatic bone disease, which is not well defined. 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 more and more involving the impact of bone diseases on its quality. The current paper provides a meaningful and critical analysis of previously developed bone remodeling mathematical models, particularly those addressing bone diseases. Besides, mechanistic pharmacokinetic/pharmacodynamic (PK/PD) models, which englobe either bone disease and its treatments’ 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 ends by exhibiting the available bone treatment procedures and the PK/PD models considering these lasts.

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.

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.

The Cellular Dynamics of Bone Remodeling: A Mathematical Model

The mechanical properties of vertebrate bone are largely determined by a process which involves the complex interplay of three different cell types. This process is called bone remodeling and occurs asynchronously at multiple sites in the mature skeleton. The cells involved are bone resorbing osteoclasts, bone matrix producing osteoblasts, and mechanosensing osteocytes. These cells communicate with each other by means of autocrine and paracrine signaling factors and operate in complex entities, the so-called bone multicellular units (BMUs). To investigate the BMU dynamics in silico, we develop a novel mathematical model resulting in a system of nonlinear partial differential equations (PDEs) with time delays. The model describes the osteoblast and osteoclast populations together with the dynamics of the key messenger molecule RANKL and its decoy receptor OPG. Scaling theory is used to address parameter sensitivity and predict the emergence of pathological remodeling regimes. The model is studied numerically in one and two space dimensions using finite difference schemes in space and explicit delay equation solvers in time. The computational results are in agreement with in vivo observations and provide new insights into the role of the RANKL/OPG pathway in the spatial regulation of bone remodeling.

A novel mathematical model of bone remodelling cycles for trabecular bone at the cellular level

After an initial phase of growth and development, bone undergoes a continuous cycle of repair, renewal and optimisation by a process called remodelling. This paper describes a novel mathematical model of the trabecular bone remodelling cycle. It is essentially formulated to simulate a remodelling event at a fixed position in the bone, integrating bone removal by osteoclasts and formation by osteoblasts. The model is developed to construct the variation in bone thickness at a particular point during the remodelling event, derived from standard bone histomorphometric analyses. The novelties of the approach are the adoption of a predator–prey model to describe the dynamic interaction between osteoclasts and osteoblasts, using a genetic algorithm–based solution; quantitative reconstruction of the bone remodelling cycle; and the introduction of a feedback mechanism in the bone formation activity to co-regulate bone thickness. The application of the model is first demonstrated by using experimental data recorded for normal (healthy) bone remodelling to predict the temporal variation in the number of osteoblasts and osteoclasts. The simulated histomorphometric data and remodelling cycle characteristics compare well with the specified input data. Sensitivity studies then reveal how variations in the model’s parameters affect its output; it is hoped that these parameters can be linked to specific biochemical factors in the future. Two sample pathological conditions, hypothyroidism and primary hyperparathyroidism, are examined to demonstrate how the model could be applied more broadly, and, for the first time, the osteoblast and osteoclast populations are predicted for these conditions. Further data are required to fully validate the model’s predictive capacity, but this work shows it has potential, especially in the modelling of pathological conditions and the optimisation of the treatment of those conditions.

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.

Optimizing the dynamics of bone turnover with genetic algorithm

Control systems and the modeling strategies are not only limited to engineering problems. These approaches can be used in the field of bio-mathematics as well and modern studies have promoted this approach to a great extent. The computational modeling and simulation of bone metastasis is painful yet critical after cancer invades the body. This vicious cycle is complex, and several research centers worldwide are devoted to understanding the dynamics and setting up a treatment strategy for this life-threatening behavior of cancer. Cancerous cells activation and the corresponding process of metastasis is reported to boost during the periodic waves of COVID-19, due to the inflammatory nature of the infection associated with SARS-2 and its variants. The bone cells are comprised of two types of cells responsible for bone formation and resorption. The computational framework of such cells, in spatial form, can help the researchers forecast the bone dynamics in a robust manner where the impact of cancer is incorporated into the computational model as a source of perturbation. A series of computational models are presented to explore the complex behavior of bone metastasis with COVID-19 induced infection. The finite difference algorithm is used to simulate the nonlinear computational model. The results obtained are in close agreement with the experimental findings. The computational results can help explore the vicious cycle’s fate and help set up control strategies through drug therapies.

A mathematical model for describing the metastasis of cancer in bone tissue

Metastasis is the rapid proliferation of cancer cells (secondary tumour) at a specific place, generally leading to death. This occurs at anatomical parts providing the necessary environment for vas...

Bone Dynamics in Metabolic Bone Disease:

Bone remodeling: A tissue-level process emerging from cell-level molecular algorithms

The human skeleton undergoes constant remodeling throughout the lifetime. Processes occurring on microscopic and molecular scales degrade bone and replace it with new, fully functional tissue. Multiple bone remodeling events occur simultaneously, continuously and independently throughout the body, so that the entire skeleton is completely renewed about every ten years.Bone remodeling is performed by groups of cells called Bone Multicellular Units (BMU). BMUs consist of different cell types, some specialized in the resorption of old bone, others encharged with producing new bone to replace the former. These processes are tightly regulated so that the amount of new bone produced is in perfect equilibrium with that of old bone removed, thus maintaining bone microscopic structure.To date, many regulatory molecules involved in bone remodeling have been identified, but the precise mechanism of BMU operation remains to be fully elucidated. Given the complexity of the signaling pathways already known, one may question whether such complexity is an inherent requirement of the process or whether some subset of the multiple constituents could fulfill the essential role, leaving functional redundancy to serve an alternative safety role. We propose in this work a minimal model of BMU function that involves a limited number of signals able to account for fully functional BMU operation. Our main assumptions were i) at any given time, any cell within a BMU can select only one among a limited choice of decisions, i.e. divide, die, migrate or differentiate, ii) this decision is irreversibly determined by depletion of an appropriate internal inhibitor and iii) the dynamics of any such inhibitor are coupled to that of specific external mediators, such as hormones, cytokines, growth factors. It was thus shown that efficient BMU operation manifests as an emergent process, which results from the individual and collective decisions taken by cells within the BMU unit in the absence of any external planning.