Metastatic disease is a lethal stage of cancer progression. It is characterized by the spread of aberrant cells from a primary tumor to distant tissues like the bone. Several treatments are used to deal with bone metastases formation, but they are palliative since the disease is considered incurable. Computational and mathematical models are used to understand the underlying mechanisms of how bone metastasis evolves. In this way, new therapies aiming to reduce or eliminate the metastatic burden in the bone tissue may be proposed. We present an optimal control approach to analyze some common treatments for bone metastasis. In particular, we focus on denosumab treatment, an anti-resorptive therapy, and radiotherapy treatment which has a cell killing action. We base our work in a variant of an existing model introduced by Komarova. The new model incorporates a logistic equation in order to describe the bone metastasis evolution. We provide proofs of existence and uniqueness of solutions to the corresponding optimal control problems for each treatment. Moreover, we present some numerical simulations to analyze the effectiveness of both treatments when different interactions between cancer and bone cells occur. A discussion of the obtained results is provided.
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...
Studying Bone Remodelling and Tumour Growth for Therapy Predictive Control
Bone remodelling consists of cycles of bone resorption and formation executed mainly by osteoclasts and osteoblasts. Healthy bone remodelling is disrupted by diseases such as Multiple Myeloma and bone metastatic diseases. In this paper, a simple mathematical model with differential equations, which takes into account the evolution of osteoclasts, osteoblasts, bone mass and bone metastasis growth, is improved with a pharmacokinetic and pharmacodynamic (PK/PD) scheme of the drugs denosumab, bisphosphonates, proteasome inhibitors and paclitaxel. The major novelty is the inclusion of drug resistance phenomena, which resulted in two variations of the model, corresponding to different paradigms of the origin and development of the tumourous cell resistance condition. These models are then used as basis for an optimization of the drug dose applied, paving the way for personalized medicine. A Nonlinear Model Predictive Control scheme is used, which takes advantage of the convenient properties of a suggested adaptive and democratic variant of Particle Swarm Optimization. Drug prescriptions obtained in this way provide useful insights into dose administration strategies. They also show how results may change depending on which of the two very different paradigms of drug resistance is used to model the behaviour of the tumour.
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
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.
Modeling Osteocyte Network Formation: Healthy and Cancerous Environments
Advanced cancers, such as prostate and breast cancers, commonly metastasize to bone. In the bone matrix, dendritic osteocytes form a spatial network allowing communication between osteocytes and the osteoblasts located on the bone surface. This communication network facilitates coordinated bone remodelling. In the presence of a cancerous microenvironment, the morphology of this network changes. Commonly osteocytes appear to be either overdifferentiated (i.e., there are more dendrites than healthy bone) or underdeveloped(i.e., dendrites do not fully form). In addition to structural changes, histological sections from metastatic breast cancer xenografted mice show that number of osteocytes per unit area is different between healthy bone and cancerous bone. We present a stochastic agent-based model for bone formation incorporating osteoblasts and osteocytes that allows us to probe both network structure and density of osteocytes in bone. Our model both allows for the simulation of our spatial network model and analysis of mean-field equations in the form of integro-partial differential equations. We considered variations of our model to study specific physiological hypotheses related to osteoblast differentiation; for example predicting how changing biological parameters, such as rates of bone secretion, rates of cancer formation and rates of osteoblast differentiation can allow for qualitatively different network morphologies. We then used our model to explore how commonly applied therapies such as bisphosphonates (e.g. zoledronic acid) impact osteocyte network formation.
Osteoprotegerin in Bone Metastases: Mathematical Solution to the Puzzle
Bone is a common site for cancer metastasis. To create space for their growth, cancer cells stimulate bone resorbing osteoclasts. Cytokine RANKL is a key osteoclast activator, while osteoprotegerin (OPG) is a RANKL decoy receptor and an inhibitor of osteoclastogenesis. Consistently, systemic application of OPG decreases metastatic tumor burden in bone. However, OPG produced locally by cancer cells was shown to enhance osteolysis and tumor growth. We propose that OPG produced by cancer cells causes a local reduction in RANKL levels, inducing a steeper RANKL gradient away from the tumor and towards the bone tissue, resulting in faster resorption and tumor expansion. We tested this hypothesis using a mathematical model of nonlinear partial differential equations describing the spatial dynamics of OPG, RANKL, PTHrP, osteoclasts, tumor and bone mass. We demonstrate that at lower expression rates, tumor-derived OPG enhances the chemotactic RANKL gradient and osteolysis, whereas at higher expression rates OPG broadly inhibits RANKL and decreases osteolysis and tumor burden. Moreover, tumor expression of a soluble mediator inducing RANKL in the host tissue, such as PTHrP, is important for correct orientation of the RANKL gradient. A meta-analysis of OPG, RANKL and PTHrP expression in normal prostate, carcinoma and metastatic tissues demonstrated an increase in expression of OPG, but not RANKL, in metastatic prostate cancer, and positive correlation between OPG and PTHrP in metastatic prostate cancer. The proposed mechanism highlights the importance of the spatial distribution of receptors, decoys and ligands, and can be applied to other systems involving regulation of spatially anisotropic processes.
Integrative modeling of drug therapy and the bone turnover
Denosumab has been successfully used for the postmenopausal osteoporosis treatment. This research is focused on the computational analysis of the effect of denosumab on bone remodeling.
An Integrated Computational Model of the Bone Microenvironment in Bone-Metastatic Prostate Cancer
Abstract Bone metastasis will impact most men with advanced prostate cancer. The vicious cycle of bone degradation and formation driven by metastatic prostate cells in bone yields factors that drive cancer growth. Mechanistic insights into this vicious cycle have suggested new therapeutic opportunities, but complex temporal and cellular interactions in the bone microenvironment make drug development challenging. We have integrated biologic and computational approaches to generate a hybrid cellular automata model of normal bone matrix homeostasis and the prostate cancer-bone microenvironment. The model accurately reproduces the basic multicellular unit bone coupling process, such that introduction of a single prostate cancer cell yields a vicious cycle similar in cellular composition and pathophysiology to models of prostate-to-bone metastasis. Notably, the model revealed distinct phases of osteolytic and osteogenic activity, a critical role for mesenchymal stromal cells in osteogenesis, and temporal changes in cellular composition. To evaluate the robustness of the model, we assessed the effect of established bisphosphonate and anti-RANKL therapies on bone metastases. At approximately 100% efficacy, bisphosphonates inhibited cancer progression while, in contrast with clinical observations in humans, anti-RANKL therapy fully eradicated metastases. Reducing anti-RANKL yielded clinically similar results, suggesting that better targeting or dosing could improve patient survival. Our work establishes a computational model that can be tailored for rapid assessment of experimental therapies and delivery of precision medicine to patients with prostate cancer with bone metastases. Cancer Res; 74(9); 2391–401. ©2014 AACR.
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.
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.
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.
Mathematical modeling of postmenopausal osteoporosis and its treatment by the anti‐catabolic drug denosumab
SUMMARY Denosumab, a fully human monoclonal antibody, has been approved for the treatment of postmenopausal osteoporosis. The therapeutic effect of denosumab rests on its ability to inhibit osteoclast differentiation. Here, we present a computational approach on the basis of coupling a pharmacokinetics model of denosumab with a pharmacodynamics model for quantifying the effect of denosumab on bone remodeling. The pharmacodynamics model comprises an integrated systems biology‐continuum micromechanics approach, including a bone cell population model, considering the governing biochemical factors of bone remodeling (including the action of denosumab), and a multiscale micromechanics‐based bone mechanics model, for implementing the mechanobiology of bone remodeling in our model. Numerical studies of postmenopausal osteoporosis show that denosumab suppresses osteoclast differentiation, thus strongly curtailing bone resorption. Simulation results also suggest that denosumab may trigger a short‐term bone volume gain, which is, however, followed by constant or decreasing bone volume. This evolution is accompanied by a dramatic decrease of the bone turnover rate by more than one order of magnitude. The latter proposes dominant occurrence of secondary mineralization (which is not anymore impeded through cellular activity), leading to higher mineral concentration per bone volume. This explains the overall higher bone mineral density observed in denosumab‐related clinical studies. Copyright © 2013 John Wiley & Sons, Ltd.
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 mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease
Multiple myeloma is a hematologic malignancy associated with the development of a destructive osteolytic bone disease.
Bone metastases
Bone is the most frequent site for metastasis for many cancers, notably for tumours originating in the breast and the prostate. Tumour cells can escape from the primary tumour site and colonize the bone microenvironment. Within the bone, these disseminated tumour cells, as well as those arising in the context of multiple myeloma, may assume a state of dormancy, remaining quiescent for years before resuming proliferation and causing overt metastasis, which causes bone destruction via activation of osteoclast-mediated osteolysis. This structural damage can lead to considerable morbidity, including pain, fractures and impaired quality of life. Although treatment of bone metastases and myeloma bone disease is rarely curative, disease control is often possible for many years through the use of systemic anticancer treatments on a background of multidisciplinary supportive care. This care should include bone-targeted agents to inhibit tumour-associated osteolysis and prevent skeletal morbidity as well as use of appropriate local treatments such as radiation therapy, orthopaedic surgery and specialist palliative care to minimize the impact of metastatic bone disease on physical functioning. In this Primer, we provide an overview of the clinical features, the pathophysiology and the specific treatment approaches to prevent and treat bone metastases from solid tumours as well as myeloma bone disease.
