Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling

Computational models of bone fracture healing and applications: a review

Fracture healing is a very complex physiological process involving multiple events at different temporal and spatial scales, such as cell migration and tissue differentiation, in which mechanical stimuli and biochemical factors assume key roles. With the continuous improvement of computer technology in recent years, computer models have provided excellent solutions for studying the complex process of bone healing. These models not only provide profound insights into the mechanisms of fracture healing, but also have important implications for clinical treatment strategies. In this review, we first provide an overview of research in the field of computational models of fracture healing based on CiteSpace software, followed by a summary of recent advances, and a discussion of the limitations of these models and future directions for improvement. Finally, we provide a systematic summary of the application of computational models of fracture healing in three areas: bone tissue engineering, fixator optimization and clinical treatment strategies. The application of computational models of bone healing in clinical treatment is immature, but an inevitable trend, and as these models become more refined, their role in guiding clinical treatment will become more prominent.

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.

Advances in mechanobiological pharmacokinetic-pharmacodynamic models of osteoporosis treatment - Pathways to optimise and exploit existing therapies

Osteoporosis (OP) is a chronic progressive bone disease which is characterized by reduction of bone matrix volume and changes in the bone matrix properties which can ultimately lead to bone fracture. The two major forms of OP are related to aging and/or menopause. With the worldwide increase of the elderly population, particularly age-related OP poses a serious health issue which puts large pressure on health care systems. A major challenge for development of new drug treatments for OP and comparison of drug efficacy with existing treatments is due to current regulatory requirements which demand testing of drugs based on bone mineral density (BMD) in phase 2 trials and fracture risk in phase 3 trials. This requires large clinical trials to be conducted and to be run for long time periods, which is very costly. This, together with the fact that there are already many drugs available for treatment of OP, makes the development of new drugs inhibitive. Furthermore, an increased trend of the use of different sequential drug therapies has been observed in OP management, such as sequential anabolic-anticatabolic drug treatment or switching from one anticatabolic drug to another. Running clinical trials for concurrent and sequential therapies is neither feasible nor practical due to large number of combinatorial possibilities. In silico mechanobiological pharmacokinetic-pharmacodynamic (PK-PD) models of OP treatments allow predictions beyond BMD, i.e. bone microdamage and degree of mineralisation can also be monitored. This will help to inform clinical drug usage and development by identifying the most promising scenarios to be tested clinically (confirmatory trials rather than exploratory only trials), optimise trial design and identify subgroups of the population that show benefit-risk profiles (both good and bad) that are different from the average patient. In this review, we provide examples of the predictive capabilities of mechanobiological PK-PD models. These include simulation results of PMO treatment with denosumab, implications of denosumab drug holidays and coupling of bone remodelling models with calcium and phosphate systems models that allows to investigate the effects of co-morbidities such as hyperparathyroidism and chronic kidney disease together with calcium and vitamin D status on drug efficacy.

Mechanobiochemical bone remodelling around an uncemented acetabular component: influence of bone orthotropy

Mechanical loosening of an implant is often caused by bone resorption, owing to stress/strain shielding. Adaptive bone remodelling elucidates the response of bone tissue to alterations in mechanical and biochemical environments. This study aims to propose a novel framework of bone remodelling based on the combined effects of bone orthotropy and mechanobiochemical stimulus. The proposed remodelling framework was employed in the finite element model of an implanted hemipelvis to predict evolutionary changes in bone density and associated orthotropic bone material properties. In order to account for variations in load transfer during common daily activities, several musculoskeletal loading conditions of hip joint corresponding to sitting down/up, stairs ascend/descend and normal walking were considered. The bone remodelling predictions were compared with those of isotropic strain energy density (SED)-based, isotropic mechanobiochemical and orthotropic strain-based bone remodelling formulations. Although similar trends of bone resorption were predicted by orthotropic mechanobiochemical (MBC) and orthotropic strain-based models across implanted acetabulum, more volume (10-20%) of bone elements was subjected to bone resorption for the orthotropic MBC model. Higher bone resorption (75-85%) was predicted by the orthotropic strain-based and orthotropic MBC models compared to the isotropic MBC and SED-based models. Higher bone apposition (35-160%) across the implanted acetabulum was predicted by the isotropic MBC model, compared to the SED-based model. The remodelling predictions indicated that a reduction in estrogen level might lead to an increase in bone resorption. The study highlighted the importance of including mechanobiochemical stimulus and bone anisotropy to predict bone remodelling adequately.

Osteoporosis induced by cellular senescence: A mathematical model

Osteoporosis is a disease characterized by loss of bone mass, where bones become fragile and more likely to fracture. Bone density begins to decrease at age 50, and a state of osteoporosis is defined by loss of more than 25%. Cellular senescence is a permanent arrest of normal cell cycle, while maintaining cell viability. The number of senescent cells increase with age. Since osteoporosis is an aging disease, it is natural to consider the question to what extend senescent cells induce bone density loss and osteoporosis. In this paper we use a mathematical model to address this question. We determine the percent of bone loss for men and women during age 50 to 100 years, and the results depend on the rate η of net formation of senescent cell, with η = 1 being the average rate. In the case η = 1, the model simulations are in agreement with empirical data. We also consider senolytic drugs, like fisetin and quercetin, that selectively eliminate senescent cells, and assess their efficacy in terms of reducing bone loss. For example, at η = 1, with estrogen hormonal therapy and early treatment with fisetin, bone density loss for women by age 75 is 23.4% (below osteoporosis), while with no treatment with fisetin it is 25.8% (osteoporosis); without even a treatment with estrogen hormonal therapy, bone loss of 25.3% occurs already at age 65.

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.

Modelling the Effects of Growth and Remodelling on the Density and Structure of Cancellous Bone

A two-stage model is proposed for investigating remodelling characteristics in bone over time and distance to the growth plate. The first stage comprises a partial differential equation (PDE) for bone density as a function of time and distance from the growth plate. This stage clarifies the contributions to changes in bone density due to remodelling and growth processes and tracks the rate at which new bone emanates from the growth plate. The second stage consists of simulating the remodelling process to determine remodelling characteristics. Implementing the second stage requires the rate at which bone moves away from the growth plate computed during the first stage. The second stage is also needed to confirm that remodelling characteristics predicted by the first stage may be explained by a realistic model for remodelling and to compute activation frequency. The model is demonstrated on microCT scans of tibia of juvenile female rats in three experimental groups: sham-operated control, oestrogen deprived, and oestrogen deprived followed by treatment. Model predictions for changes in bone density and remodelling characteristics agree with the literature. In addition, the model provides new insight into the role of treatment on the density of new bone emanating from the growth plate and provides quantitative descriptions of changes in remodelling characteristics beyond what has been possible to ascertain by experimentation alone.

Studying the mechanical properties of the mandible and injury prediction under the effect of ossification factors

BACKGROUND AND OBJECTIVE

It is essential to know the quantitative interactions between biological tissues and external mechanical and chemical stimuli. This assists the physicians to better know the quantitative behavior of the tissue and plan more effective therapy. In the literature, the effect of the chemical and mechanical loading was investigated on the bone biological cell activities and some mechanical features, but a lack of prediction of bone injury under the chemical and mechanical factors was sensed. Therefore, the present study aims to investigate the effects of the application of major chemical factors involved in ossification, including RANKL1 (Receptor Activator of Nuclear Factor Kappa Beta Ligand), PTH2 (Parathyroid Hormone), and OPG3 (Osteoprotegerin) on the mandibular bone biological osteoblast and osteoclast activities and mechanical properties. Moreover, the study assesses the bone injury possibility under uniform mastication pressure applied on the premolar tooth in terms of the mechanostat theory undergoing the effects of the chemical factors.

METHODS

A 3D geometry of the mandible-tooth assembly was generated from the CT image dataset. The geometry was next purified, solidified, and exported to FEM4 (Finite Element Method) software to be meshed, where boundary conditions and loading were applied. Moreover, the mathematical system of differential equations to model the chemical factor effects on osteoblast and osteoclast activities as well as bone matrix volume fraction and elastic stiffness relations were applied. Next, the values of the equivalent strain were calculated to predict the injury states of the bone.

RESULTS

The results complied with the literature data. The results showed that RANKL and PTH increased the values of the equivalent strain from 450 με to 11500 με, while OPG reduced that from 450 με to 300 με.

CONCLUSIONS

Therefore, RANKL and PTH doses of this study put the bone at risk of injury compared to the baseline of no dose applied, while OPG secured the bone from injury.

Mathematical models of cystic fibrosis as a systemic disease

Cystic fibrosis (CF) is widely known as a disease of the lung, even though it is in truth a systemic disease, whose symptoms typically manifest in gastrointestinal dysfunction first. CF ultimately impairs not only the pancreas and intestine but also the lungs, gonads, liver, kidneys, bones, and the cardiovascular system. It is caused by one of several mutations in the gene of the epithelial ion channel protein CFTR. Intense research and improved antimicrobial treatments during the past eight decades have steadily increased the predicted life expectancy of a person with CF (pwCF) from a few weeks to over 50 years. Moreover, several drugs ameliorating the sequelae of the disease have become available in recent years, and notable treatments of the root cause of the disease have recently generated substantial improvements in health for some but not all pwCF. Yet, numerous fundamental questions remain unanswered. Complicating CF, for instance in the lung, is the fact that the associated insufficient chloride secretion typically perturbs the electrochemical balance across epithelia and, in the airways, leads to the accumulation of thick, viscous mucus and mucus plaques that cannot be cleared effectively and provide a rich breeding ground for a spectrum of bacterial and fungal communities. The subsequent infections often become chronic and respond poorly to antibiotic treatments, with outcomes sometimes only weakly correlated with the drug susceptibility of the target pathogen. Furthermore, in contrast to rapidly resolved acute infections with a single target pathogen, chronic infections commonly involve multi-species bacterial communities, called "infection microbiomes," that develop their own ecological and evolutionary dynamics. It is presently impossible to devise mathematical models of CF in its entirety, but it is feasible to design models for many of the distinct drivers of the disease. Building upon these growing yet isolated modeling efforts, we discuss in the following the feasibility of a multi-scale modeling framework, known as template-and-anchor modeling, that allows the gradual integration of refined sub-models with different granularity. The article first reviews the most important biomedical aspects of CF and subsequently describes mathematical modeling approaches that already exist or have the potential to deepen our understanding of the multitude aspects of the disease and their interrelationships. The conceptual ideas behind the approaches proposed here do not only pertain to CF but are translatable to other systemic diseases. This article is categorized under: Congenital Diseases > Computational Models.

Individual hematotoxicity prediction of further chemotherapy cycles by dynamic mathematical models in patients with gastrointestinal tumors

PURPOSE

Hematotoxicity is a common side-effect of cytotoxic gastrointestinal (GI) cancer therapies. An unsolved problem is to predict the individual risk therefore to decide on treatment adaptions. We applied an established biomathematical prediction model and primarily evaluated its predictive value in patients undergoing chemotherapy for GI cancers in curative intent.

METHODS

In a prospective, observational multicenter study on patients with gastro-esophageal or pancreatic cancer (n = 28) receiving myelosuppressive adjuvant or neoadjuvant chemotherapy (FLO(T) or FOLFIRINOX), individual model parameters were learned based on patients' observed laboratory values during the first chemotherapy cycle and further external data resources. Grades of hematotoxicity of subsequent cycles were predicted by model simulation and compared with observed data.

RESULTS

The most common high-grade hematological toxicity was neutropenia [19/28 patients (68%)]. For the FLO(T) regimen, individual grades of thrombocytopenia and leukopenia could be well predicted for cycles 2-4, as well as grades of neutropenia for cycle 2. Prediction accuracy for neutropenia in the third and fourth cycle differed by one toxicity grade on average. For the FOLFIRINOX-regimen, thrombocytopenia predictions showed a maximum deviation of one toxicity grade up to the end of therapy (8 cycles). Deviations of predictions were less than one degree on average up to cycle 4 for neutropenia, and up to cycle 6 for leukopenia.

CONCLUSION

The biomathematical model showed excellent short-term and decent long-term prediction performance for all relevant hematological side effects associated with FLO(T)/FOLFIRINOX. Clinical utility of this precision-medicine approach needs to be further investigated in a larger cohort.

Modeling and simulation of bone cells dynamic behavior under the late effect of breast cancer treatments

Breast Cancer (BC) treatments have been proven to interfere with the health of bones. Chemotherapy and endocrinal treatment regimens such as tamoxifen and aromatase inhibitors are frequently prescribed for women with BC. However, these drugs increase bone resorption and reduce the Bone Mineral Density (BMD), thus increasing the risk of bone fracture. In the current study, a mechanobiological bone remodeling model has been developed by coupling cellular activities, mechanical stimuli, and the effect of breast cancer treatments (chemotherapy, tamoxifen, and aromatase inhibitors). This model algorithm has been programmed and implemented on MATLAB software to simulate different treatment scenarios and their effects on bone remodeling and also predict the evolution of Bone Volume fraction (BV/TV) and the associated Bone Density Loss (BDL) over a period of time. The simulation results, achieved from different combinations of Breast Cancer treatments, allow the researchers to predict the intensity of each combination treatment on BV/TV and BMD. The combination of chemotherapy, tamoxifen, and aromatase inhibitors, followed by the combination of chemotherapy and tamoxifen remain the most harmful regimen. This is because they have a strong ability to induce the bone degradation which is represented by a decrease of 13.55% and 11.55% of the BV/TV value, respectively. These results were compared with the experimental studies and clinical observations which showed good agreement. The proposed model can be used by clinicians and physicians to choose the most appropriate combination of treatments, according to the patient's case.

Influence of growth plate morphology on bone trabecular groups, a framework computational approach

The morpholog of the growth plate undergoes various transformations during each stage of development, affecting its shape, width, density, and other characteristics. This significantly impacts the distribution of stress in the epiphysis of long bones. To the best of our knowledge, this study represents the first attempt to examine the relationship between growth plate morphology and trabecular bone patterns. Our analysis was conducted using a finite element model and we analyzed two medical cases: trabecular patterns in the femoral epiphysis and the calcaneus bone. Our findings revealed a correlation between the formation of main trabecular groups and growth plate morphology. We investigated how an increased density in high-shear stress zones, which are typically located at the periphery of the growth plate, may occur to prevent failure by shear. This is evident in cases such as slipped capital femoral epiphysis or sever's disease, different simulations align with the clinical data available in the literature from a qualitative and quantitative point of view. Our results suggest that further research should focus on understanding the impact of growth plate morphology on bone remodeling and exploring potential preventive measures for different bone disorders.

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.

Both network architecture and micro cracks effects on lacuno-canalicular liquid flow efficiency within the context of multiphysics approach for bone remodeling

When physical forces are applied to bone, its mechanical adaptive behaviors change according to the microarchitecture configuration. This leads to changes in biological and physical thresholds in the remodeling cell population, involving sensor cells (osteocytes) interacting with each other and changes in osteocyte shape due to variation in lacunar shape. The resulting alterations in fluid flow leads to changes in the membrane electrical potential and shear stress. Eventual creation of microcracks, may lead in turn to modify cell activity. In contrast, the redundancy in the lacuno canalicular network (LCN) interconnectivity maintains partial flow. Our goal was to investigate the role of fluid flow in LCN by proposing a model of electro-mechanical energy spread through inhomogeneous microarchitectures. We focused on mechano-sensitivity to changes in load-induced flow impacted by neighboring micro cracks and quantifying its critical role in changing, velocity, shear stress and orientation of liquid mass transportation from one cell to another. To enhance the concept of intricacy LCN micro-structure to fluid flow, we provide a new combined effects factor considered as osteocytes sensor efficiency. We customized an influence function for each osteocyte, coupling: in one hand, the spatial distribution within remodeling influence areas, conducting a significant fluid spread, leading hydro-dynamic behavior and impacted further by presence of micro cracks and; in other hand, the fluid electro kinetic behavior. As an attempt to fill the limitations stated by many of the recent studies, we reveal in numerical simulation, some results which cannot be measured in vitro/in vivo studies. Numerical calculations were performed in order to evaluate, among many others, how liquid flow conditions changes between lacunas, how the orientation and the magnitude of the governing flow in LCN can regulate osteocytes efficiency. In addition to be regulated by osteocytes, a direct effects of fluid flow are also acting on osteoblast activity. In summary, this new approach considers mechano-sensitivity in relation to liquid flow dynamic and suggests additional pathway for Osseo integration via osteoblast regulation. However, this novel modeling approach may help improve the mapping and design bone scaffolds and/or selection of scaffold implantation regions.

A unified framework of cell population dynamics and mechanical stimulus using a discrete approach in bone remodelling

Multiphysics models have become a key tool in understanding the way different phenomenon are related in bone remodeling and various approaches have been proposed, yet, to the best of the author's knowledge there is no model able to link a cell population model with a mechanical stimulus model using a discrete approach, which allows for an easy implementation. This article couples two classical models, the cell population model from Komarova and the Nackenhorst model in a 2D domain, where correlations between the mechanical loading and the cell population dynamics can be established, furthermore the effect of different paracrine and autocrine regulators is seen on the overall density of a portion of trabecular bone. A discretization is performed using frame 1D finite elements, representing the trabecular structure. The Nackenhorst model is implemented by using the finite element method to calculate the strain energy as the main mechanical stimulus that determines the bone mass density evolution in time. This density is normalized to be added to the bone mass percentage proposed by the Komarova model, where coupling terms have been added as well that guarantee a stable response. In the simulations, the equations were solved employing the finite element method with a user subroutine implemented in ABAQUS (2017) and by applying a direct formulation. The methodology presented can model the cell dynamics occurring in bone remodelling in accordance with the asynchronous nature of this process, yet allowing to differentiate zones with higher density, the main trabecular groups are obtained for the proximal femur. Finally, the model is tested in pathological cases, such as osteoporosis and osteopetrosis, yielding results similar to the pathology behavior. Furthermore, the discrete modelling technique is shown to be of use in this particular application.

Breast Cancer Exosomal microRNAs Facilitate Pre-Metastatic Niche Formation in the Bone: A Mathematical Model

Pre-metastatic niche is a location where cancer cells, separating from a primary tumor, find "fertile soil" for growth and proliferation, ensuring successful metastasis. Exosomal miRNAs of breast cancer are known to enter the bone and degrade it, which facilitates cancer cells invasion into the bone interior and ensures its successful colonization. In this paper, we use a mathematical model to first describe, in health, the continuous remodeling of the bone by bone-forming osteoblasts, bone-resorbing osteoclasts and the RANKL-OPG-RANK signaling system, which keeps the balance between bone formation and bone resorption. We next demonstrate how breast cancer exosomal miRNAs disrupt this balance, either by increasing or by decreasing the ratio of osteoclasts/osteoblasts, which results in abnormal high bone resorption or abnormal high bone forming, respectively, and in bone weakening in both cases. Finally we consider the case of abnormally high resorption and evaluate the effect of drugs, which may increase bone density to normal level, thus protecting the bone from invasion by cancer cells.

Spatio-temporal simulations of bone remodelling using a bone cell population model based on cell availability

Here we developed a spatio-temporal bone remodeling model to simulate the action of Basic Multicelluar Units (BMUs). This model is based on two major extensions of a temporal-only bone cell population model (BCPM). First, the differentiation into mature resorbing osteoclasts and mature forming osteoblasts from their respective precursor cells was modelled as an intermittent process based on precursor cells availability. Second, the interaction between neighbouring BMUs was considered based on a "metabolic cost" argument which warrants that no new BMU will be activated in the neighbourhood of an existing BMU. With the proposed model we have simulated the phases of the remodelling process obtaining average periods similar to those found in the literature: resorption ( ∼ 22 days)-reversal (∼8 days)-formation (∼65 days)-quiescence (560-600 days) and an average BMU activation frequency of ∼1.6 BMUs/year/mm³. We further show here that the resorption and formation phases of the BMU become coordinated only by the presence of TGF-β (transforming growth factor β), i.e., a major coupling factor stored in the bone matrix. TGF-β is released through resorption so upregulating osteoclast apoptosis and accumulation of osteoblast precursors, i.e., facilitating the transition from the resorption to the formation phase at a given remodelling site. Finally, we demonstrate that this model can explain targeted bone remodelling as the BMUs are steered towards damaged bone areas in order to commence bone matrix repair.

Mathematical modeling of the effects of Wnt-10b on bone metabolism

Bone health is determined by factors including bone metabolism or remodeling. Wnt-10b alters osteoblastogenesis through pre-osteoblast proliferation and differentiation and osteoblast apoptosis rate, which collectively lead to the increase of bone density. To model this, we adapted a previously published model of bone remodeling. The resulting model for the bone compartment includes differential equations for active osteoclasts, pre-osteoblasts, osteoblasts, osteocytes, and the amount of bone present at the remodeling site. Our alterations to the original model consist of extending it past a single remodeling cycle and implementing a direct relationship to Wnt-10b. Four new parameters were estimated and validated using normalized data from mice. The model connects Wnt-10b to bone metabolism and predicts the change in trabecular bone volume caused by a change in Wnt-10b input. We find that this model predicts the expected increase in pre-osteoblasts and osteoblasts while also pointing to a decrease in osteoclasts when Wnt-10b is increased.

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 multi-model approach to predict efficacious clinical dose for an anti-TGF-β antibody (GC2008) in the treatment of osteogenesis imperfecta

Osteogenesis imperfecta (OI) is a heterogeneous group of inherited bone dysplasias characterized by reduced skeletal mass and bone fragility. Although the primary manifestation of the disease involves the skeleton, OI is a generalized connective tissue disorder that requires a multidisciplinary treatment approach. Recent studies indicate that application of a transforming growth factor beta (TGF-β) neutralizing antibody increased bone volume fraction (BVF) and strength in an OI mouse model and improved bone mineral density (BMD) in a small cohort of patients with OI. In this work, we have developed a multitiered quantitative pharmacology approach to predict human efficacious dose of a new anti-TGF-β antibody drug candidate (GC2008). This method aims to translate GC2008 pharmacokinetic/pharmacodynamic (PK/PD) relationship in patients, using a number of appropriate mathematical models and available preclinical and clinical data. Compartmental PK linked with an indirect PD effect model was used to characterize both pre-clinical and clinical PK/PD data and predict a GC2008 dose that would significantly increase BMD or BVF in patients with OI. Furthermore, a physiologically-based pharmacokinetic model incorporating GC2008 and the body's physiological properties was developed and used to predict a GC2008 dose that would decrease the TGF-β level in bone to that of healthy individuals. By using multiple models, we aim to reveal information for different aspects of OI disease that will ultimately lead to a more informed dose projection of GC2008 in humans. The different modeling efforts predicted a similar range of pharmacologically relevant doses in patients with OI providing an informed approach for an early clinical dose setting.

Experimental-based mechanobiological modeling of the anabolic and catabolic effects of breast cancer on bone remodeling

Bone is a biological tissue characterized by its hierarchical organization. This material has the ability to be continually renewed, which makes it highly adaptative to external loadings. Bone renewing is managed by a dynamic biological process called bone remodeling (BR), where continuous resorption of old bone and formation of new bone permits to change the bone composition and microstructure. Unfortunately, because of several factors, such as age, hormonal imbalance, and a variety of pathologies including cancer metastases, this process can be disturbed leading to various bone diseases. In this study, we have investigated the effect of breast cancer (BC) metastases causing osteolytic bone loss. BC has the ability to affect bone quantity in different ways in each of its primary and secondary stages. Based on a BR mathematical model, we modeled the BC cells' interaction with bone cells to assess their effect on bone volume fraction (BV/TV) evolution during the remodeling process. Some of the parameters used in our model have been determined experimentally using the enzyme-linked immune-sorbent assay (ELISA) and the MTT assay. Our numerical simulations show that primary BC plays a significant role in enhancing bone-forming cells' activity leading to a 6.22% increase in BV/TV over 1 year. On the other hand, secondary BC causes a noticeable decrease in BV/TV reaching 15.74% over 2 years.

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.

Clinical Data for Parametrization of In Silico Bone Models Incorporating Cell-Cytokine Dynamics: A Systematic Review of Literature

In silico simulations aim to provide fast, inexpensive, and ethical alternatives to years of costly experimentation on animals and humans for studying bone remodeling, its deregulation during osteoporosis and the effect of therapeutics. Within the varied spectrum of in silico modeling techniques, bone cell population dynamics and agent-based multiphysics simulations have recently emerged as useful tools to simulate the effect of specific signaling pathways. In these models, parameters for cell and cytokine behavior are set based on experimental values found in literature; however, their use is currently limited by the lack of clinical in vivo data on cell numbers and their behavior as well as cytokine concentrations, diffusion, decay and reaction rates. Further, the settings used for these parameters vary across research groups, prohibiting effective cross-comparisons. This review summarizes and evaluates the clinical trial literature that can serve as input or validation for in silico models of bone remodeling incorporating cells and cytokine dynamics in post-menopausal women in treatment, and control scenarios. The GRADE system was used to determine the level of confidence in the reported data, and areas lacking in reported measures such as binding site occupancy, reaction rates and cell proliferation, differentiation and apoptosis rates were highlighted as targets for further research. We propose a consensus for the range of values that can be used for the cell and cytokine settings related to the RANKL-RANK-OPG, TGF-β and sclerostin pathways and a Levels of Evidence-based method to estimate parameters missing from clinical trial literature.

Effects of assessing the bone remodeling process in biomechanical finite element stability evaluations of dental implants

BACKGROUND AND OBJECTIVE

While an accurate assessment of the biomechanical stability of implants is essential in dental prosthesis planning and associated treatment assurance, the bone remodeling process is often ignored in biomechanical studies using finite element (FE) analysis. In this study, we aimed to analyze the significance of assessing the bone remodeling process in FE analysis for evaluating the biomechanical stability of dental implants. We compared the FE results considering the bone remodeling process with FE results simulated using commonly used conditions, with no considerations of the bone remodeling process.

METHODS

The mathematical model proposed by Komarova et al. was used to calculate cell population dynamics and changes in bone density at a discrete site. The model was implemented in the FE software ABAQUS, using the UMAT subroutine. Three-dimensional FE models were constructed for two types of bone (III and IV) and three values of implant diameter (4.0, 4.5, and 5.0 mm). An average biting force of 50 N in the vertical direction was applied during the bone remodeling process for 150 days. Afterwards, the maximum biting force of 200 N in the 30° oblique direction was applied to evaluate the stability of the implant systems.

RESULTS

To understand the impact of bone remodeling on the resultant mechanical responses, we focused on peri-implant cancellous bone based on two parameters: apparent density change and microstrain distribution. The bone density decreased by an average of 5.3 % after implantation, and it was the lowest on the 6^th day. The average density increases of the peri-implant cancellous bone were 264.4 kgm³ (bone type III) and 220.0 kgm³ (bone type IV) over 150 days. For the bone stability analysis, the maximum principal strain in the peri-implant bone was used to evaluate the bone stability. If the bone remodeling process is ignored, then the bone volume within the fatigue failure range of the microstrain differs significantly from that if the bone remodeling process is considered, i.e., 60 % higher for bone type III and 33.4 % lower for bone type IV than when the bone remodeling process is considered.

CONCLUSIONS

The FE result without considering the bone remodeling process could be considered a conservative criterion for bone type III. However, in bone type IV, the FE result without considering the bone remodeling process tends to underestimate the risks. The bone remodeling process is more affected by the initial bone quality than the implant diameter.

Longitudinal Tibia Stress Fracture Risk During High-Volume Training: a Multi-Scale Modeling Pipeline Incorporating Bone Remodeling

Tibia stress fractures are prevalent during high-intensity training, yet a mechanistic model linking longitudinal training intensity, bone health, and long-term injury risk has yet to be demonstrated. The objective of this study was to develop and validate a multi-scale model of gross and tissue level loading on the tibia including bone remodeling on a timescale of week. Peak tensile tibial strain (3517 µstrain) during 4 m/s running was below injury thresholds, and the peak anteromedial tibial strain (1248 µstrain) was 0.17 standard deviations away from the mean of reported literature values. An initial study isolated the effects of cortical density and stiffness on tibial strain during a simulated eight week training period. Tibial strains and cortical microcracking correlated to initial cortical modulus, with all simulations presenting peak anteromedial tensile strains (1047-1600 µstrain) near day 11. Average cortical densities decreased by 7-8 percent of their nominal value by day 11, but the overall density change was <2% by the end of the simulated training period, in line with reported results. This study demonstrates the benefits of multi-scale models for investigating stress fracture risk and indicates that peak tibial strain, and thus injury risk, may increase early in a high intensity training program. Future studies could optimize training volume and recovery time to reduce injury risk during the most vulnerable training periods.

Remodeling Effects of the Combination of GGT Scaffolds, Percutaneous Electrical Stimulation, and Acupuncture on Large Bone Defects in Rats

The regeneration defect of bone is a long-term physiological process after bone injuries. To accelerate the bone remodeling process, the combination of chemical and physical stimulations provides an efficient strategy to allow maturation and to functionalize osteoclasts and osteoblasts. This study aims to investigate the dual effects of a tricalcium phosphate (TCP)-based gelatin scaffold (GGT) in combination with electroacupuncture stimulation on the activation of osteoclasts and osteoblasts, as well as new bone regrowth in vitro and in vivo. We demonstrated that electrical stimulation changes the pH of a culture medium and activates osteoblasts and osteoclasts in an in vitro co-culture system. Furthermore, we showed that electroacupuncture stimulation can enhance osteogenesis and new bone regrowth in vivo and can upregulate the mechanism among parathyroid hormone intact (PTH-i), calcium, osteoclasts, and osteoblasts in the bone-defected rats. Those results showed the potential interest to combine the electroacupuncture technique with GGT scaffolds to improve bone remodeling after injury.

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.

Mathematical modeling of the role of bone turnover in pH regulation in bone interstitial fluid

BACKGROUND AND AIMS

Bone turnover is strongly affected by pH of surrounding fluid, and in turn plays a role in maintaining systemic pH, however the quantitative contribution of bone processes to pH regulation is not known. Our goal was to develop a mathematical model describing pH regulation in the interstitial fluid and to examine the contribution of hydroxyapatite dissolution and precipitation to pH regulation.

MATERIALS AND METHODS

We modeled twelve reversible equilibrium reactions of sixteen calcium, phosphate, hydrogen and carbonate species in the interstitial fluid and examined the buffering capacity and range. The effect of hydroxyapatite dissolution and precipitation was modeled by assuming that the calcium, phosphate and hydroxide contained in the bone volume adjacent to the interstitial fluid is instantaneously added to or removed from the interstitial fluid.

RESULTS

The carbonate buffer was found to dominate electrochemical buffering system of the bone interstitial fluid. Nevertheless, the phosphate added during dissolution of bone hydroxyapatite significantly improved the interstitial fluid buffering capacity. In contrast, hydroxyapatite precipitation had limited effect on the interstitial fluid pH regulation.

CONCLUSION

This study provides mechanistic insights into the physicochemical processes underlying the known role of bone turnover processes in regulation of body pH homeostasis.

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.

A simple and effective 1D-element discrete-based method for computational bone remodeling

In-silico models applied to bone remodeling are widely used to investigate bone mechanics, bone diseases, bone-implant interactions, and also the effect of treatments of bone pathologies. This article proposes a new methodology to solve the bone remodeling problem using one-dimensional (1D) elements to discretize trabecular structures more efficiently for 2D and 3D domains. An Euler integration scheme is coupled with the momentum equations to obtain the evolution of material density at each step. For the simulations, the equations were solved by using the finite element method, and two benchmark tests were solved varying mesh parameters. Proximal femur and calcaneus bone were selected as study cases given the vast research available on the topology of these bones, and compared with the anatomical features of trabecular bone reported in the literature. The presented methodology has proven to be efficient in optimizing topologies of lattice structures; It can predict the trend of formation patterns of the main trabecular groups from two different cancellous bones (femur and calcaneus) using domains set up by discrete elements as a starting point. Preliminary results confirm that the proposed approach is suitable and useful in bone remodeling problems leading to a considerable computational cost reduction. Characteristics similar to those encountered in topological optimization algorithms were identified in the benchmark tests as well, showing the viability of the proposed approach in other applications such as bio-inspired design.

A General Mechano-Pharmaco-Biological Model for Bone Remodeling Including Cortisol Variation

The process of bone remodeling requires a strict coordination of bone resorption and formation in time and space in order to maintain consistent bone quality and quantity. Bone-resorbing osteoclasts and bone-forming osteoblasts are the two major players in the remodeling process. Their coordination is achieved by generating the appropriate number of osteoblasts since osteoblastic-lineage cells govern the bone mass variation and regulate a corresponding number of osteoclasts. Furthermore, diverse hormones, cytokines and growth factors that strongly link osteoblasts to osteoclasts coordinated these two cell populations. The understanding of this complex remodeling process and predicting its evolution is crucial to manage bone strength under physiologic and pathologic conditions. Several mathematical models have been suggested to clarify this remodeling process, from the earliest purely phenomenological to the latest biomechanical and mechanobiological models. In this current article, a general mathematical model is proposed to fill the gaps identified in former bone remodeling models. The proposed model is the result of combining existing bone remodeling models to present an updated model, which also incorporates several important parameters affecting bone remodeling under various physiologic and pathologic conditions. Furthermore, the proposed model can be extended to include additional parameters in the future. These parameters are divided into four groups according to their origin, whether endogenous or exogenous, and the cell population they affect, whether osteoclasts or osteoblasts. The model also enables easy coupling of biological models to pharmacological and/or mechanical models in the future.

Integration of mechanics and biology in computer simulation of bone remodeling

Bone remodeling is a complex physiological process that spans across multiple spatial and temporal scales and is regulated by both mechanical and hormonal cues. An imbalance between bone resorption and bone formation in the process of bone remodeling may lead to various bone pathologies. One powerful and non-invasive approach to gain new insights into mechano-adaptive bone remodeling is computer modeling and simulation. Recent findings in bone physiology and advances in computer modeling have provided a unique opportunity to study the integration of mechanics and biology in bone remodeling. Our objective in this review is to critically appraise recent advances and developments and discuss future research opportunities in computational bone remodeling approaches that enable integration of mechanics and cellular and molecular pathways. Based on the critical appraisal of the relevant recent published literature, we conclude that multiscale in silico integration of personalized bone mechanics and mechanobiology combined with data science and analytics techniques offer the potential to deepen our knowledge of bone remodeling and provide ample opportunities for future research.

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.

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.

Spatial extension of a bone remodeling dynamics model and its finite element analysis

There are many works dealing with the dynamics of bone remodeling, proposing increasingly complex and complete models. In the recent years, the efforts started to focus on developing models that not only reproduce the temporal evolution, but also include the spatial aspects of this phenomenon. In this work, we propose the spatial extension of an existing model that includes the dynamics of osteocytes. The spatial dependence is modeled in terms of a linear diffusion, as proposed in previous works dealing with related problems. The resulting model is then written in its variational form, and fully discretized using the well-known finite element method and a combination of the implicit and explicit Euler schemes. The numerical algorithm is then analyzed, proving some a priori error estimates and its linear convergence. Finally, we extend the examples already published for the temporal model to one and two dimensions, showing the dynamics of the solution in the spatial domain.

Individual prediction of thrombocytopenia at next chemotherapy cycle: Evaluation of dynamic model performances

AIMS

Thrombocytopenia is a common major side-effect of cytotoxic cancer therapies. A clinically relevant problem is to predict an individual's thrombotoxicity in the next planned chemotherapy cycle in order to decide on treatment adaptation. To support this task, 2 dynamic mathematical models of thrombopoiesis under chemotherapy were proposed, a simple semimechanistic model and a comprehensive mechanistic model. In this study, we assess the performance of these models with respect to existing thrombocytopenia grading schemes.

METHODS

We consider close-meshed individual time series data of 135 non-Hodgkin's lymphoma patients treated with 6 cycles of CHOP/CHOEP chemotherapies. Individual parameter estimates were derived on the basis of these data considering a varying number of cycles per patient. Parsimony assumptions were applied to optimize parameter identifiability. Models' predictability are assessed by determining deviations of predicted and observed degrees of thrombocytopenia in the next cycles.

RESULTS

The mechanistic model results in better agreement of model prediction and individual time series data. Prediction accuracy of future cycle toxicities by the mechanistic model is higher even if the semimechanistic model is provided with data of more cycles for calibration.

CONCLUSION

We successfully established a quantitative and clinically relevant method for assessing prediction performances of biomathematical models of thrombopoiesis under chemotherapy. We showed that the more comprehensive mechanistic model outperforms the semimechanistic model. We aim at implementing the mechanistic model into clinical practice to assess its utility in real life clinical decision-making.

A mathematical biomechanical model for bone remodeling integrated with a radial point interpolating meshless method

Bone remodeling is a highly complex process, in which bone cells interact and regulate bone's apparent density as a response to several external and internal stimuli. In this work, this process is numerically described using a novel 2D biomechanical model. Some of the new features in this model are (i) the mathematical parameters used to determine bone's apparent density and cellular density; (ii) an automatic boundary recognition step to spatially control bone remodeling and (iii) an approach to mimic the mechanical transduction to osteoclasts and osteoblasts. Moreover, this model is combined with a meshless approach - the Radial Point Interpolation Method (RPIM). The use of RPIM is an asset for this application, especially in the construction of the boundary maps. This work studies bone's adaptation to a certain loading regime through bone resorption. The signaling pathways of bone cells are dependent on the level of strain energy density (SED) in bone. So, when SED changes, bone cells' functioning is affected, causing also changes on bone's apparent density. With this model, bone is able to achieve an equilibrium state, optimizing its structure to withstand the applied loads. Results suggest that this model has the potential to provide high quality solutions while being a simpler alternative to more complex bone remodeling models in the literature.

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.

Analysis of the uniqueness and stability of solutions to problems regarding the bone-remodeling process

Simulation of the bone remodeling process is extremely important because it makes possible the structure forecast of one or several bones when anomalous situations, such as prosthesis installation, occur. Thus, it is necessary that the mathematical model to simulate the bone remodeling process be reliable; that is, the numerical solution must be stable regardless of initial density field for a phenomenological approach to model the process. For several models found in the literature, this characteristic of stability is not observed, largely due to the discontinuities present in the property values of the models (e.g., Young's modulus and Poisson's ratio). In addition, checkerboard formation and the lazy zone prevent the uniqueness of the solution. To correct these difficulties, this study proposes a set of modifications to guarantee the uniqueness and stability of the solutions, when a phenomenological approach is used. The proposed modifications are: (a) change the rate of remodeling curve in the lazy zone region and (b) create transition functions to guarantee the continuity of the expressions used to describe Young's modulus and Poisson's ratio. Moreover, the stress smoothing process controls the checkerboard formation. Numerical analysis is used to simulate the solution behavior from each proposed modification. The results show that, when all proposed modifications are applied to the three-dimensional models simulated here, it is possible to observe the tendency toward a unique solution.

Simulation of bone remodeling around a femoral prosthesis using a model that accounts for biological and mechanical interactions

The present study focuses on a model for three-dimensional bone remodeling of the human femur that considers cellular dynamics to determine the volume fraction of new bone. The model considers the interaction among responsive osteoblasts, active osteoblasts, and osteoclasts, as well as signaling molecules and parathyroid hormone (PTH). The stimulus of the model has a systemic origin due to the PTH effect, and a local origin due to the action of cytokines, growth factors, and mechanical stimuli near the site of the bone cells. The present work considers that the mechanical stimulus that activates cellular activity is obtained from stresses acting on the bone tissue and the number of daily loading cycles. In addition to simulating the bone modeling process in an intact femur, the numerical model is used to simulate bone adaptation in relation to the stress shielding phenomenon after the implantation of a femoral prosthesis. The results showed that the simulations provide a distribution of bone density that is similar to a radiograph and, in addition, allows the visualization of osteoblast and osteoclast dynamics in bone adaptation response after prosthesis implantation.

A mechano-chemo-biological model for bone remodeling with a new mechano-chemo-transduction approach

Bone remodeling is a fundamental biological process that develops in bone tissue along its whole lifetime. It refers to a continuous bone transformation with new bone formation and old bone resorption that changes the internal microstructure and composition of the tissue. The main objectives of bone remodeling are: repair of the internal microcracks; adaptation of the macroscopic stiffness and strength to the actual changing mechanical demands; and control of the calcium homeostasis. Understanding this process and predicting its evolution is critical to reduce the effects of long-term disuse as happens during periods of reduced mobility. It is also important in the design of bone implants to avoid long-term stress shielding. Many mathematical models have been proposed from the earliest purely phenomenological to the latest that include biological knowledge. However, there still exists a lack of connection between the mechanical driving force and the biochemical and cell processes it triggers. Here, and following previous works that model independently the mechanobiological and biochemical processes in bone remodeling, we present a more complete model, useful for both cortical and trabecular bone, that uses a new mechanotransduction approach based on the effect of strains onto the bonding-unbonding rate of RANK/RANKL/OPG receptor-ligand reactions. We compare the results of this model with previous ones, showing a good agreement in similar conditions. We also apply it to realistic situations such as a femoral bone after implantation of a hip prosthesis, getting similar results to the clinical ones in the final bone density distribution. Finally, we extend this approach to the anisotropic case, getting not only the mean density, but also the directional homogenization of the microstructure. This biochemical approach permits, not only to predict the bone evolution under changes in the mechanical loads, but also, to consider anabolic and catabolic drugs to control bone density, such as those used in osteoporosis.

Analysis of a bone remodeling model with myeloma disease arising in cellular dynamics

In this work we study a bone remodeling model for the evolution of the myeloma disease. The biological problem is written as a coupled nonlinear system consisting of parabolic partial differential equations. They are written in terms of the concentrations of osteoblasts and osteoclasts, the density of the relative bone and the concentration of the tumor cells. Then, we deal with the numerical analysis of this variational problem, introducing a numerical approximation by using the finite element method and a hybrid combination of both implicit and explicit Euler schemes. We perform some a priori error estimates and show a few numerical simulations to demonstrate the accuracy of the approximation. Finally, we present the comparison with previous works and the behavior of the solution in two-dimensional examples.

Assessment of romosozumab efficacy in the treatment of postmenopausal osteoporosis: Results from a mechanistic PK-PD mechanostat model of bone remodeling

This paper introduces a theoretical framework for the study of the efficacy of romosozumab, a humanized monoclonal antibody targeting sclerostin for the treatment of osteoporosis. We developed a comprehensive mechanistic pharmacokinetic-pharmacodynamic (PK-PD) model of the effect of drug treatment on bone remodeling in postmenopausal osteoporosis (PMO). We utilized a one-compartment PK model to represent subcutaneous injections of romosozumab and subsequent absorption into serum. The PD model is based on a recently-developed bone cell population model describing the bone remodeling process at the tissue scale. The latter accounts for mechanical feedback by incorporating nitric oxide (NO) and sclerostin (Scl) as biochemical feedback molecules. Utilizing a competitive binding model, where Wnt and Scl compete for binding to LRP5/6, allows to regulate anabolic bone remodeling responses. Here, we extended this model with respect to romosozumab binding to sclerostin. For the currently approved monthly injections of 210 mg, the model predicted a 6.59%, 10.38% and 15.25% increase in BMD at the lumbar spine after 6, 12 and 24 months, respectively. These results are in good agreement with the data reported in the literature. Our model is also able to distinguish the bone-site specific drug effects. For instance, at the femoral neck, our model predicts a BMD increase of 3.85% after 12 months of 210 mg injections, which is consistent with literature observations. Finally, our simulations indicate rapid bone loss after treatment discontinuation, indicating that some additional interventions such as use of bisphosphonates are required to maintain bone mass.

Modeling Early Stages of Bone and Joint Infections Dynamics in Humans: A Multi-Agent, Multi-System Based Model

Diagnosis and management of bone and joint infections (BJI) is a challenging task. The high intra and inter patient's variability in terms of clinical presentation makes it impossible to rely on a systematic description or classical statistical analysis for its diagnosis. Advances can be achieved through a better understanding of the system behavior that results from the interactions between the components at a micro-scale level, which is difficult to mastered using traditional methods. Multiple studies from the literature report factors and interactions that affect the dynamics of the BJI system. The objectives of this study were (i) to perform a systematic review to identify relevant interactions between agents (cells, pathogens) and parameters values that characterize agents and interactions, and (ii) to develop a two dimensional computational model of the BJI system based on the results of the systematic review. The model would simulate the behavior resulting from the interactions on the cellular and molecular levels to explore the BJI dynamics, using an agent-based modeling approach. The BJI system's response to different microbial inoculum levels was simulated. The model succeeded in mimicking the dynamics of bacteria, the innate immune cells, and the bone mass during the first stage of infection and for different inoculum levels in a consistent manner. The simulation displayed the destruction in bone tissue as a result of the alteration in bone remodeling process during the infection. The model was used to generate different patterns of system behaviors that could be analyzed in further steps. Simulations results suggested evidence for the existence of latent infections. Finally, we presented a way to analyze and synthesize massive simulated data in a concise and comprehensive manner based on the semi-supervised identification of ordinary differential equations (ODE) systems. It allows to use the known framework for temporal and structural ODE analyses and therefore summarize the whole simulated system dynamical behavior. This first model is intended to be validated by in vivo or in vitro data and expected to generate hypotheses to be challenged by real data. Step by step, it can be modified and complexified based on the test/validation iteration cycles.