Model structure and control of bone remodeling: a theoretical study

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.

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.

Chemotherapy and adjuvant therapies' impact on the internal remodeling process of bone and its mechanical behavior for breast cancer patients

Breast cancer is a significant public health issue affecting women worldwide. While advancements in treatment options have led to improved survival rates, the impact of breast cancer and its treatments on bone health cannot be overlooked. Bone remodeling is a complex process regulated by the delicate balance between bone formation and resorption. Any disruption to this balance can lead to decreased bone density, increased fracture risk, and compromised physical function. To investigate the effects of breast cancer and its treatments on bone remodeling, a finite element model was developed in this study. This model incorporated bone remodeling equations to simulate the mechanical behavior of bone under different conditions. The ABAQUS/UMAT software was used to simulate the behavior of bone tissue under the influence of breast cancer and treatments. Our findings suggest that bone loss is more pronounced after secondary breast cancer and treatment, leading to bone loss (6%-19% decrease in BV/TV), reduced bone stimulation, and decreased effectiveness of physical activity on recovery. These results highlight the importance of early intervention and management of bone health in breast cancer patients to mitigate the negative impact of cancer and treatment on bone remodeling.

Aging effects on osteoclast progenitor dynamics affect variability in bone turnover via feedback regulation

Bone turnover markers (BTMs) are commonly used in osteoporosis treatment as indicators of cell activities of bone-resorbing osteoclasts and bone-forming osteoblasts. The wide variability in their values due to multiple factors, such as aging and diseases, makes it difficult for physicians to utilize them for clinical decision-making. The progenitors of osteoclasts and osteoblasts are indispensable for a comprehensive interpretation of the variability in BTM values because these upstream progenitors strongly regulate the downstream cell activities of bone turnover. However, understanding the complex interactions among the multiple populations of bone cells is challenging. In this study, we aimed to gain a fundamental understanding of the mechanism by which the progenitor dynamics affect the variability in bone turnover through in silico experiments by exploring the cell dynamics with aging effects on osteoporosis. Negative feedback control driven by the consumptive loss of progenitors prevents rapid bone loss due to excessive bone turnover, and through feedback regulation, aging effects on osteoclast differentiation and osteoclast progenitor proliferation cause variability in the osteoclast and osteoblast activity balance and its temporal transition. By expressing the variability in the bone turnover status, our model describes the individualities of patients based on their clinical backgrounds. Therefore, our model could play a powerful role in assisting tailored treatment and has the potential to resolve the various health problems associated with osteoporosis worldwide.

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.

A Novel Computational Biomechanics Framework to Model Vascular Mechanopropagation in Deep Bone Marrow

The mechanical stimuli generated by body exercise can be transmitted from cortical bone into the deep bone marrow (mechanopropagation). Excitingly, a mechanosensitive perivascular stem cell niche is recently identified within the bone marrow for osteogenesis and lymphopoiesis. Although it is long known that they are maintained by exercise-induced mechanical stimulation, the mechanopropagation from compact bone to deep bone marrow vasculature remains elusive of this fundamental mechanobiology field. No experimental system is available yet to directly understand such exercise-induced mechanopropagation at the bone-vessel interface. To this end, taking advantage of the revolutionary in vivo 3D deep bone imaging, an integrated computational biomechanics framework to quantitatively evaluate the mechanopropagation capabilities for bone marrow arterioles, arteries, and sinusoids is devised. As a highlight, the 3D geometries of blood vessels are smoothly reconstructed in the presence of vessel wall thickness and intravascular pulse pressure. By implementing the 5-parameter Mooney-Rivlin model that simulates the hyperelastic vessel properties, finite element analysis to thoroughly investigate the mechanical effects of exercise-induced intravascular vibratory stretching on bone marrow vasculature is performed. In addition, the blood pressure and cortical bone bending effects on vascular mechanoproperties are examined. For the first time, movement-induced mechanopropagation from the hard cortical bone to the soft vasculature in the bone marrow is numerically simulated. It is concluded that arterioles and arteries are much more efficient in propagating mechanical force than sinusoids due to their stiffness. In the future, this in-silico approach can be combined with other clinical imaging modalities for subject/patient-specific vascular reconstruction and biomechanical analysis, providing large-scale phenotypic data for personalized mechanobiology discovery.

Prediction of osteoporotic degradation of tibia human bone at trabecular scale

A theoretical numerical model is proposed to predict patient dependent osteoporotic bone degradation. The model parameters are identified through a particle swarm optimization algorithm and based on individual patient high resolution peripherical quantitative computer tomography (HRpQCT) scan data. The degradation model is based on cellular activity initiated by the elastic strain energy developed in the bone microstructure through patient's body weight. The macro (organ scale) and meso (trabecular scale) scale analyses are carried out and predicted bone volume fraction and microstructure evolution are compared with in-vivo experimental bone degradation for four elderly women over a period of 10 years. A significant correlation (r > 0.9) is observed between the model predictions and in-vivo experiments in all cases with an average deviation error of 1.46%. The model can easily be extended to other patients and provide good predictions for different population categories such as ethnicity, gender, age, etc.

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.

Pharmacometrics: The Already-Present Future of Precision Pharmacology

The use of mathematical modeling to represent, analyze, make predictions or providing information on data obtained in drug research and development has made pharmacometrics an area of great prominence and importance. The main purpose of pharmacometrics is to provide information relevant to the search for efficacy and safety improvements in pharmacotherapy. Regulatory agencies have adopted pharmacometrics analysis to justify their regulatory decisions, making those decisions more efficient. Demand for specialists trained in the field is therefore growing. In this review, we describe the meaning, history, and development of pharmacometrics, analyzing the challenges faced in the training of professionals. Examples of applications in current use, perspectives for the future, and the importance of pharmacometrics for the development and growth of precision pharmacology are also presented.

Application of finite element analysis to tissue differentiation and bone remodelling approaches and their use in design optimization of orthopaedic implants: A review

Post-operative bone growth and long-term bone adaptation around the orthopaedic implants are simulated using the mechanoregulation based tissue-differentiation and adaptive bone remodelling algorithms, respectively. The primary objective of these algorithms was to assess biomechanical feasibility and reliability of orthopaedic implants. This article aims to offer a comprehensive review of the developments in mathematical models of tissue-differentiation and bone adaptation and their applications in studies involving design optimization of orthopaedic implants over three decades. Despite the different mechanoregulatory models developed, existing literature confirm that none of the models can be highly regarded or completely disregarded over each other. Not much development in mathematical formulations has been observed from the current state of knowledge due to the lack of in vivo studies involving clinically relevant animal models, which further retarded the development of such models to use in translational research at a fast pace. Future investigations involving artificial intelligence (AI), soft-computing techniques and combined tissue-differentiation and bone-adaptation studies involving animal subjects for model verification are needed to formulate more sophisticated mathematical models to enhance the accuracy of pre-clinical testing of orthopaedic implants.

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.

Assessment of Strategies for Safe Drug Discontinuation and Transition of Denosumab Treatment in PMO—Insights From a Mechanistic PK/PD Model of Bone Turnover

Denosumab (Dmab) treatment against postmenopausal osteoporosis (PMO) has proven very efficient in increasing bone mineral density (BMD) and reducing the risk of bone fractures. However, concerns have been recently raised regarding safety when drug treatment is discontinued. Mechanistic pharmacokinetic-pharmacodynamic (PK-PD) models are the most sophisticated tools to develop patient specific drug treatments of PMO to restore bone mass. However, only a few PK-PD models have addressed the effect of Dmab drug holidays on changes in BMD. We showed that using a standard bone cell population model (BCPM) of bone remodelling it is not possible to account for the spike in osteoclast numbers observed after Dmab discontinuation. We show that inclusion of a variable osteoclast precursor pool in BCPMs is essential to predict the experimentally observed rapid rise in osteoclast numbers and the associated increases in bone resorption. This new model also showed that Dmab withdrawal leads to a rapid increase of damage in the bone matrix, which in turn decreases the local safety factor for fatigue failure. Our simulation results show that changes in BMD strongly depend on Dmab concentration in the central compartment. Consequently, bone weight (BW) might play an important factor in calculating effective Dmab doses. The currently clinically prescribed constant Dmab dose of 60 mg injected every 6 months is less effective in increasing BMD for patients with high BW (2.5% for 80 kg in contrast to 8% for 60 kg after 6 years of treatment). However, bone loss observed 24 months after Dmab withdrawal is less pronounced in patients with high BW (3.5% for 80kg and 8.5% for 60 kg). Finally, we studied how to safely discontinue Dmab treatment by exploring several transitional and combined drug treatment strategies. Our simulation results indicate that using transitional reduced Dmab doses are not effective in reducing rapid bone loss. However, we identify that use of a bisphosphonate (BP) is highly effective in avoiding rapid bone loss and increase in bone tissue damage compared to abrupt withdrawal of Dmab. Furthermore, the final values of BMD and damage were not sensitive to the time of administration of the BP.

Integrated computational and in vivo models reveal Key Insights into macrophage behavior during bone healing

Myeloid-derived monocyte and macrophages are key cells in the bone that contribute to remodeling and injury repair. However, their temporal polarization status and control of bone-resorbing osteoclasts and bone-forming osteoblasts responses is largely unknown. In this study, we focused on two aspects of monocyte/macrophage dynamics and polarization states over time: 1) the injury-triggered pro- and anti-inflammatory monocytes/macrophages temporal profiles, 2) the contributions of pro- versus anti-inflammatory monocytes/macrophages in coordinating healing response. Bone healing is a complex multicellular dynamic process. While traditional in vitro and in vivo experimentation may capture the behavior of select populations with high resolution, they cannot simultaneously track the behavior of multiple populations. To address this, we have used an integrated coupled ordinary differential equations (ODEs)-based framework describing multiple cellular species to in vivo bone injury data in order to identify and test various hypotheses regarding bone cell populations dynamics. Our approach allowed us to infer several biological insights including, but not limited to,: 1) anti-inflammatory macrophages are key for early osteoclast inhibition and pro-inflammatory macrophage suppression, 2) pro-inflammatory macrophages are involved in osteoclast bone resorptive activity, whereas osteoblasts promote osteoclast differentiation, 3) Pro-inflammatory monocytes/macrophages rise during two expansion waves, which can be explained by the anti-inflammatory macrophages-mediated inhibition phase between the two waves. In addition, we further tested the robustness of the mathematical model by comparing simulation results to an independent experimental dataset. Taken together, this novel comprehensive mathematical framework allowed us to identify biological mechanisms that best recapitulate bone injury data and that explain the coupled cellular population dynamics involved in the process. Furthermore, our hypothesis testing methodology could be used in other contexts to decipher mechanisms in complex multicellular processes.

Myeloid-derived monocytes/macrophages are key cells for bone remodeling and injury repair. However, their temporal polarization status and control of bone-resorbing osteoclasts and bone-forming osteoblasts responses is largely unknown. In this study, we focused on two aspects of monocyte/macrophage population dynamics: 1) the injury-triggered pro- and anti-inflammatory monocytes/macrophages temporal profiles, 2) the contributions of pro- versus anti-inflammatory monocytes/macrophages in coordinating healing response. In order to test various hypotheses regarding bone cell populations dynamics, we have integrated a coupled ordinary differential equations-based framework describing multiple cellular species to in vivo bone injury data. Our approach allowed us to infer several biological insights including: 1) anti-inflammatory macrophages are key for early osteoclast inhibition and pro-inflammatory macrophage suppression, 2) pro-inflammatory macrophages are involved in osteoclast bone resorptive activity, whereas osteoblasts promote osteoclast differentiation, 3) Pro-inflammatory monocytes/macrophages rise during two expansion waves, which can be explained by the anti-inflammatory macrophages-mediated inhibition phase between the two waves. Taken together, this mathematical framework allowed us to identify biological mechanisms that recapitulate bone injury data and that explain the coupled cellular population dynamics involved in the process.

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

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

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

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

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.

Combined Effects of Exercise and Denosumab Treatment on Local Failure in Post-menopausal Osteoporosis-Insights from Bone Remodelling Simulations Accounting for Mineralisation and Damage

Denosumab has been shown to increase bone mineral density (BMD) and reduce the fracture risk in patients with post-menopausal osteoporosis (PMO). Increase in BMD is linked with an increase in bone matrix mineralisation due to suppression of bone remodelling. However, denosumab anti-resorptive action also leads to an increase in fatigue microdamage, which may ultimately lead to an increased fracture risk. A novel mechanobiological model of bone remodelling was developed to investigate how these counter-acting mechanisms are affected both by exercise and long-term denosumab treatment. This model incorporates Frost's mechanostat feedback, a bone mineralisation algorithm and an evolution law for microdamage accumulation. Mechanical disuse and microdamage were assumed to stimulate RANKL production, which modulates activation frequency of basic multicellular units in bone remodelling. This mechanical feedback mechanism controls removal of excess bone mass and microdamage. Furthermore, a novel measure of bone local failure due to instantaneous overloading was developed. Numerical simulations indicate that trabecular bone volume fraction and bone matrix damage are determined by the respective bone turnover and homeostatic loading conditions. PMO patients treated with the currently WHO-approved dose of denosumab (60 mg administrated every 6 months) exhibit increased BMD, increased bone ash fraction and damage. In untreated patients, BMD will significantly decrease, as will ash fraction; while damage will increase. The model predicted that, depending on the time elapsed between the onset of PMO and the beginning of treatment, BMD slowly converges to the same steady-state value, while damage is low in patients treated soon after the onset of the disease and high in patients having PMO for a longer period. The simulations show that late treatment PMO patients have a significantly higher risk of local failure compared to patients that are treated soon after the onset of the disease. Furthermore, overloading resulted in an increase of BMD, but also in a faster increase of damage, which may consequently promote the risk of fracture, specially in late treatment scenarios. In case of mechanical disuse, the model predicted reduced BMD gains due to denosumab, while no significant change in damage occurred, thus leading to an increased risk of local failure compared to habitual loading.

Finite element analysis of bone remodelling with piezoelectric effects using an open-source framework

Bone tissue exhibits piezoelectric properties and thus is capable of transforming mechanical stress into electrical potential. Piezoelectricity has been shown to play a vital role in bone adaptation and remodelling processes. Therefore, to better understand the interplay between mechanical and electrical stimulation during these processes, strain-adaptive bone remodelling models without and with considering the piezoelectric effect were simulated using the Python-based open-source software framework. To discretise numerical attributes, the finite element method (FEM) was used for the spatial variables and an explicit Euler scheme for the temporal derivatives. The predicted bone apparent density distributions were qualitatively and quantitatively evaluated against the radiographic scan of a human proximal femur and the bone apparent density calculated using a bone mineral density (BMD) calibration phantom, respectively. Additionally, the effect of the initial bone density on the resulting predicted density distribution was investigated globally and locally. The simulation results showed that the electrically stimulated bone surface enhanced bone deposition and these are in good agreement with previous findings from the literature. Moreover, mechanical stimuli due to daily physical activities could be supported by therapeutic electrical stimulation to reduce bone loss in case of physical impairment or osteoporosis. The bone remodelling algorithm implemented using an open-source software framework facilitates easy accessibility and reproducibility of finite element analysis made.

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.

Computational modeling reveals a key role for polarized myeloid cells in controlling osteoclast activity during bone injury repair

Bone-forming osteoblasts and -resorbing osteoclasts control bone injury repair, and myeloid-derived cells such as monocytes and macrophages are known to influence their behavior. However, precisely how these multiple cell types coordinate and regulate each other over time within the bone marrow to restore bone is difficult to dissect using biological approaches. Conversely, mathematical modeling lends itself well to this challenge. Therefore, we generated an ordinary differential equation (ODE) model powered by experimental data (osteoblast, osteoclast, bone volume, pro- and anti-inflammatory myeloid cells) obtained from intra-tibially injured mice. Initial ODE results using only osteoblast/osteoclast populations demonstrated that bone homeostasis could not be recovered after injury, but this issue was resolved upon integration of pro- and anti-inflammatory myeloid population dynamics. Surprisingly, the ODE revealed temporal disconnects between the peak of total bone mineralization/resorption, and osteoblast/osteoclast numbers. Specifically, the model indicated that osteoclast activity must vary greatly (> 17-fold) to return the bone volume to baseline after injury and suggest that osteoblast/osteoclast number alone is insufficient to predict bone the trajectory of bone repair. Importantly, the values of osteoclast activity fall within those published previously. These data underscore the value of mathematical modeling approaches to understand and reveal new insights into complex biological processes.

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.

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.

Establishment of a Disease-Drug Trial Model for Postmenopausal Osteoporosis: A Zoledronic Acid Case Study

Costly and lengthy clinical trials hinder the development of safe and effective treatments for postmenopausal osteoporosis. To reduce the expense associated with these trials, we established a mechanistic disease-drug trial model for postmenopausal osteoporosis that can predict phase 3 trial outcome based on short-term bone turnover marker data. To this end, we applied a previously developed model for tibolone to bisphosphonates using zoledronic acid as paradigm compound by (1) linking the mechanistic bone cell interaction model to bone turnover markers as well as bone mineral density in lumbar spine and total hip, (2) employing a mechanistic disease progression function, and (3) accounting for zoledronic acid's mechanism of action. Once developed, we fitted the model to clinical trial data of 581 postmenopausal women receiving (1) 5-mg zoledronic acid in year 1 and saline in year 2, (2) 5-mg zoledronic acid in year 1 and year 2, or (3) placebo (saline), calcium (500 mg), and vitamin D (400 IU). All biomarker data was fitted reasonably well, with no apparent bias or model misspecification. Age, years since menopause, and body mass index at baseline were identified as significant covariates. In the future, the model can be modified to explore the link between short-term biomarkers and fracture risk.

Influence of cancellous bone material and dead zone on stress-strain, bone stimulus and bone remodelling around the tibia for total ankle replacement

Extreme bone resorption due to bone remodelling is one of the reasons for ankle component loosening. Finite element (FE) analysis has been effectively used nowadays for pre-clinical analysis of orthopaedic implants. For FE modelling, the selection of bone material and dead zone play a vital role to understand the bone remodelling. This study deals with the effects of different cancellous elastic modulus-density relationships and dead zone on bone remodelling around the tibia owing to total ankle replacement (TAR), using finite element analysis with physiological loading conditions. This study also investigated the bone stimulus distribution in the tibia to identify the initial indication of bone density changes due to bone remodelling. Additionally, the Hoffman failure criterion was used to investigate the chances of implant-bone interface failure due to different cancellous bone material modelling and bone remodelling. The present bone remodelling study consists of three different dead or lazy zones (±0.75, ±0.60 and ±0.35) to examine the influence of the dead zone on bone remodelling. Differences in stress/strain distribution were observed in the tibia bone due to different cancellous bone material modelling. Despite little variations, bone density changes due to bone remodelling were found to be almost similar for two FE models having different cancellous bone material. Similar to these results, the effect of different dead zone on bone density changes due to bone remodelling was found to be minimal. Bone stimulus distribution in the cancellous bone was found to be almost similar for FE models having different cancellous bone material modelling and different dead zones. To understand the stress/strain and interface related failure of the tibial component, cancellous bone material modelling plays a crucial role. However, cancellous bone material modelling and dead zone have minimal influence on bone remodelling around the tibia cancellous bone due to TAR.

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.

Are drug holidays a safe option in treatment of osteoporosis? - Insights from an in silico mechanistic PK-PD model of denosumab treatment of postmenopausal osteoporosis

Recent reviews by the clinical bone research community suggest caution with prescription of drug holidays for patients with postmenopausal osteoporosis (PMO) treated with denosumab for an extended period of time. Main reasons for this suggestion are based on the fact that discontinuation of denosumab treatment leads to a relapse of osteoclastic bone resorption and a loss of bone mineral density (BMD) to pre-treatment levels at only 12-28 months. The question remains what is the best treatment option for cases where it is required to discontinue and/or reduce the drug dose and what are the consequences on BMD and bone turnover markers (BTMs). The latter questions are difficult to be addressed using clinical trials alone given the large number of parameter combinations involved to answer this problem. In this paper, we apply a recently developed in silico mechanistic pharmacokinetic-pharmacodynamic (PK-PD) model of the effect of denosumab on bone remodelling in PMO. To address the above clinical relevant questions, we design a wide range of current and virtual treatment regimens to study the effect of drug holiday duration and therapy resumption on the evolution of BTMs, BMD and mineral content. Our numerical simulation results indicate the symptomatic effect of denosumab, which is lost once treatment is stopped. This effect is most clearly seen on rapid loss of BMD to pre-treatment levels 12 months after the last injection (8% and 3.6% per year in the lumbar spine and femoral neck, respectively). Also, we identify that independently of the duration of drug holiday (i.e. 12, 16 or 18 months) resuming treatment can restore BMD quite effectively. However, the latter result does not consider the possibility of potential fractures that can occur during the drug holiday. Finally, we identify a treatment case most promising for achieving maintenance of BMD and mineral content, while moderately increasing BTMs. The latter case uses no drug holiday, but reduces the most commonly prescribed denosumab dose (60 mg every 6 months) by half at same interval.

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.

Modeling osteoinduction in titanium bone scaffold with a representative channel structure

Optimizing scaffold architecture for perfect osteointegration depends on good understanding of bone ingrowth in the porous space of implants. This study developed an immunoregulatory agent-based model to discover the osteoinduction mechanism in porous scaffolds. Immunoreaction, macrophage polarization, and the corresponding growth factors were combined in the model, and all played critical roles in recruiting osteogenic cells that migrated into the scaffolds. Angiogenesis was also considered in this model. The bone ingrowth predicted by the model coincides with results from published in vivo experiments. Simulation results suggested that the pore architecture affected the diffusion process of chemotactic factors in the scaffolds, subsequently influencing the complex reactions of diverse cells and the osteoinduction location. In flexural pore spaces, bone formation spread from the periphery into the center of scaffolds due to larger M2 phenotype macrophage populations colonizing boundary regions and the distribution of corresponding growth factors concentration. In straight channels, osteogenic cells migrated further inward and osteoinduction initiated in deeper position as a result of the deeper distribution of osteogenic cytokines concentration field.

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.

Numerical Analysis of Electromechanically Driven Bone Remodeling Using the Open-source Software Framework

Natural bone remodeling is the mechanism that regulates the relationship between bone morphology and external mechanical loads applied to it. This phenomenon has been studied extensively, including multiple numerical models that have been formulated to predict the density distribution and its evolution in several bone types. However, despite these models, bone remodeling mechanism under different stimuli is still not well understood. We implemented a recently proposed electromechanically driven bone remodeling model that encompasses both mechanical and therapeutic electrical stimuli using an open-source software framework, and studied a two-dimensional (2D) plate model and a femur bone model, respectively. For discretization, we employed the finite element method (FEM) for the spatial quantities and Euler scheme for the time derivatives. The simulation results demonstrate that the density distribution is changed under electrical stimulation, generally resulting in a greater mass deposition. This study supports the possibility of enhancing and accelerating the bone remodeling process via simultaneous application of electrical and mechanical stimulus.

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.

BONE ORTHOTROPIC REMODELING AS A THERMODYNAMICALLY-DRIVEN EVOLUTION

In this paper, we present and discuss a model of bone remodeling set up in the framework of the theory of generalized continuum mechanics which was first introduced by DiCarlo et al. [Sur le remodelage des tissus osseux anisotropes, Comptes Rendus Mécanique 334(11):651–661, 2006]. Bone is described as an orthotropic body experiencing remodeling as a rotation of its microstructure. Thus, the complete kinematic description of a material point is provided by its position in space and a rotation tensor describing the orientation of its microstructure. Material motion is driven by energetic considerations, namely by the application of the Clausius–Duhem inequality to the microstructured material. Within this framework of orthotropic remodeling, some key features of the remodeling equilibrium configurations are deduced in the case of homogeneous strain or stress loading conditions. First, it is shown that remodeling equilibrium configurations correspond to energy extrema. Second, stability of the remodeling equilibrium configurations is assessed in terms of the local convexity of the strain and complementary energy functionals hence recovering some classical energy theorems. Eventually, it is shown that the remodeling equilibrium configurations are not only highly dependent on the loading conditions, but also on the material properties.

Mathematical modelling of the role of GADD45β in the pathogenesis of multiple myeloma

Multiple myeloma (MM) is an incurable disease with relatively high morbidity and mortality rates. Great efforts were made to develop nuclear factor-kappa B (NF-κB)-targeted therapies against MM disease. However, these treatments influence MM cells as well as normal cells, inevitably causing serious side effects. Further research showed that NF-κB signalling promotes the survival of MM cells by interacting with JNK signalling through growth arrest and DNA damage-inducible beta (GADD45β), the downstream module of NF-κB signalling. The GADD45β-targeted intervention was suggested to be an effective and MM cell-specific treatment. However, the underlying mechanism through which GADD45β promotes the survival of MM cells is usually ignored in the previous models. A mathematical model of MM is built in this paper to investigate how NF-κB signalling acts along with JNK signalling through GADD45β and MKK7 to promote the survival of MM cells. The model cannot only mimic the variations in bone cells, the bone volume and MM cells with time, but it can also examine how the NF-κB pathway acts with the JNK pathway to promote the development of MM cells. In addition, the model also investigates the efficacies of GADD45β- and NF-κB-targeted treatments, suggesting that GADD45β-targeted therapy is more effective but has no apparent side effects. The simulation results match the experimental observations. It is anticipated that this model could be employed as a useful tool to initially investigate and even explore potential therapies involving the NF-κB and JNK pathways in the future.

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.

In silico experiments of bone remodeling explore metabolic diseases and their drug treatment

Bone structure and function are maintained by well-regulated bone metabolism and remodeling. Although the underlying molecular and cellular mechanisms are now being understood, physiological and pathological states of bone are still difficult to predict due to the complexity of intercellular signaling. We have now developed a novel in silico experimental platform, V-Bone, to integratively explore bone remodeling by linking complex microscopic molecular/cellular interactions to macroscopic tissue/organ adaptations. Mechano-biochemical couplings modeled in V-Bone relate bone adaptation to mechanical loading and reproduce metabolic bone diseases such as osteoporosis and osteopetrosis. V-Bone also enables in silico perturbation on a specific signaling molecule to observe bone metabolic dynamics over time. We also demonstrate that this platform provides a powerful way to predict in silico therapeutic effects of drugs against metabolic bone diseases. We anticipate that these in silico experiments will substantially accelerate research into bone metabolism and remodeling.

A computational framework and sensitivity analysis for the hormonal treatment of bone

BACKGROUND

Osteoporosis is a bone disease identified by disordering of bone formation and resorption cells. It increases the risk of bone fragility and fracture. Autocrine and paracrine signalling of osteoclasts and osteoblasts plays an important role in the regulation of bone remodelling. Calcitonin is an approved pharmacologic agent for the treatment of osteoporosis.

METHODS

A novel mathematical model comprising the interaction among osteoclasts and osteoblasts cells with intermittent administration of calcitonin has been presented to study the dynamics of osteoporotic bone. The stability of model and sensitivity of parameters are also discussed.

FINDINGS

The population of osteoclastic and osteoblastic cells has been predicted via numerical simulation. The results of Monte Carlo sensitivity analysis are shown via tornado diagram.

INTERPRETATION

It is concluded that intermittent administration of calcitonin is an effective therapy for the treatment of osteoporosis.

Use of Computational Modeling to Study Joint Degeneration: A Review

Osteoarthritis (OA), a degenerative joint disease, is the most common chronic condition of the joints, which cannot be prevented effectively. Computational modeling of joint degradation allows to estimate the patient-specific progression of OA, which can aid clinicians to estimate the most suitable time window for surgical intervention in osteoarthritic patients. This paper gives an overview of the different approaches used to model different aspects of joint degeneration, thereby focusing mostly on the knee joint. The paper starts by discussing how OA affects the different components of the joint and how these are accounted for in the models. Subsequently, it discusses the different modeling approaches that can be used to answer questions related to OA etiology, progression and treatment. These models are ordered based on their underlying assumptions and technologies: musculoskeletal models, Finite Element models, (gene) regulatory models, multiscale models and data-driven models (artificial intelligence/machine learning). Finally, it is concluded that in the future, efforts should be made to integrate the different modeling techniques into a more robust computational framework that should not only be efficient to predict OA progression but also easily allow a patient’s individualized risk assessment as screening tool for use in clinical practice.

Study of the combined effects of PTH treatment and mechanical loading in postmenopausal osteoporosis using a new mechanistic PK-PD model

One of only a few approved and available anabolic treatments for severe osteoporosis is daily injections of PTH (1-34). This drug has a specific dual action which can act either anabolically or catabolically depending on the type of administration, i.e. intermittent or continuous, respectively. In this paper, we present a mechanistic pharmacokinetic-pharmacodynamic model of the action of PTH in postmenopausal osteoporosis. This model accounts for anabolic and catabolic activities in bone remodelling under intermittent and continuous administration of PTH. The model predicts evolution of common bone biomarkers and bone volume fraction (BV/TV) over time. We compared the relative changes in BV/TV resulting from a daily injection of 20 [Formula: see text]g of PTH with experimental data from the literature. Simulation results indicate a site-specific bone gain of 8.66[Formula: see text] (9.4 ± 1.13[Formula: see text]) at the lumbar spine and 3.14[Formula: see text] (2.82 ± 0.72[Formula: see text]) at the femoral neck. Bone gain depends nonlinearly on the administered dose, being, respectively, 0.68[Formula: see text], 3.4[Formula: see text] and 6.16[Formula: see text] for a 10, 20 and 40 [Formula: see text]g PTH dose at the FN over 2 years. Simulations were performed also taking into account a bone mechanical disuse to reproduce elderly frail subjects. The results show that mechanical disuse ablates the effects of PTH and leads to a 1.08% reduction of bone gain at the FN over a 2-year treatment period for the 20 [Formula: see text]g of PTH. The developed model can simulate a range of pathological conditions and treatments in bones including different PTH doses, different mechanical loading environments and combinations. Consequently, the model can be used for testing and generating hypotheses related to synergistic action between PTH treatment and physical activity.

Mathematical modelling of the role of Endo180 network in the development of metastatic bone disease in prostate cancer

Metastatic bone disease (MBD) is a common complication of advanced cancer and recent research suggests that Endo180 expression is dysregulated through the TGFβ-TGFβR-SMAD2/3 signalling pathway during the invasion of tumour cells in the development of MBD. We here provide a model for the dysregulation of the Endo180 network to demonstrate its vital contribution to bone destruction as well as tumour cell growth. The model consisted of a set of ordinary differential equations and reconstructed variations in the bone cells, resultant bone volume, and biochemical factors involved in the TGFβ-TGFβR-SMAD2/3 signalling pathway over time. The model also investigated the underlying mechanism in which the change of TGFβ affects the TGFβ-TGFβR-SMAD2/3 signalling pathway and the resultant Endo180 expression in osteoblastic and tumour cells. The model links the appearance of tumour cells with the inhibition of TGFβ binding to its receptors on osteoblastic cells, to affect TGFβ-TGFβR-SMAD2/3 signalling and Endo180 expression. Temporal variation in bone cells, bone volume, and the biochemical factors involved in the TGFβ-TGFβR-SMAD2/3 pathway as demonstrated in the model simulations agree with published experimental data. The model can be refined based on further discoveries but allows the influence of Endo180 network dysregulation on bone remodelling in MBD to be established. This model could aid in the development of Endo180 targeted therapies for MBD in the future.

An In Silico Model for the Prediction of Changes in Mineral Density in Cortical Bone Remodeling

An in silico model for the estimation of volumetric bone mineral density (vBMD) changes at a cortical bone site subjected to mechanobiological bone remodeling is proposed in this manuscript. Mechanisms of cell differentiation, receptor-ligand binding, mechanical signaling, and resorption or deposition of bone matrix were considered, therefore providing a comprehensive description of mechanobiological bone remodeling in the bone microenvironment and enabling the analysis of temporal evolution of disease or therapy scenarios. The proposed model is composed by five modules, namely, bone cells populations, mechanobiology, volume fractions and porosity, mineral density, and structural stiffness. The model is an extension of other models found in the literature because equations for the obtaining of cortical vBMD and the binding of parathyroid hormone (PTH) to parathyroid hormone 1 receptor are included. The proposed model showed a satisfactory agreement with the solutions of other in silico models found in the literature. Simulations of walking and running exercise routines were performed for the evaluation of model capability regarding the control of the numerical error and prediction of vBMD. The computational method used to solve the case study controlled the relative numerical error by less than 1 × 10-7 for approximately 1.7 × 106 time steps. The predicted values correlate with the concept of increasing BMD by vigorous physical activity; however, they contrast with the specific effect of physical activities on cortical vBMD.