Sensitivity analysis of a novel mathematical model identifies factors determining bone resorption rates

Advances in computational modeling of cytokine and growth factor dynamics in bone healing: a scoping review

Bone healing is a complex process regulated by intricate biological and mechanical factors and spatially varied regions over time. This scoping review synthesizes current computational models that incorporate cytokines and growth factors, examining their role in bone healing. Through a systematic analysis of 71 studies, this review identifies and categorizes the modeling methodologies used, including mathematical, finite element, agent-based, mechanobiological, pharmacobiological, and hybrid approaches. The findings highlight the predominant use of mathematical models while noting a recent shift toward more sophisticated techniques like finite element and agent-based models. Key cytokines and growth factors, such as TGF-β, RANK-RANKL-OPG, and PTH, are repeatedly used, underscoring their essential roles in regulating cellular processes. This review also analyzes parameter selection and validation strategies, identifying gaps in current practices and emphasizing the need for high-quality experimental validation to improve model reliability. Some bibliometric analyses provide insights into citation networks and keyword co-occurrence, illustrating influential studies in the field and central themes. The findings offer a foundation for future research to enhance model accuracy, aiming toward more predictive and clinically relevant models accounting for biology and mechanics in bone healing.

A review of mathematical modeling of bone remodeling from a systems biology perspective

Bone remodeling is an essential, delicately balanced physiological process of coordinated activity of bone cells that remove and deposit new bone tissue in the adult skeleton. Due to the complex nature of this process, many mathematical models of bone remodeling have been developed. Each of these models has unique features, but they have underlying patterns. In this review, the authors highlight the important aspects frequently found in mathematical models for bone remodeling and discuss how and why these aspects are included when considering the physiology of the bone basic multicellular unit, which is the term used for the collection of cells responsible for bone remodeling. The review also emphasizes the view of bone remodeling from a systems biology perspective. Understanding the systemic mechanisms involved in remodeling will help provide information on bone pathology associated with aging, endocrine disorders, cancers, and inflammatory conditions and enhance systems pharmacology. Furthermore, some features of the bone remodeling cycle and interactions with other organ systems that have not yet been modeled mathematically are discussed as promising future directions in the field.

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.

Age-related mechanical strength evolution of trabecular bone under fatigue damage for both genders: Fracture risk evaluation

Bone tissue is a living composite material, providing mechanical and homeostatic functions, and able to constantly adapt its microstructure to changes in long term loading. This adaptation is conducted by a physiological process, known as "bone remodeling". This latter is manifested by interactions between osteoclasts and osteoblasts, and can be influenced by many local factors, via effects on bone cell differentiation and proliferation. In the current work, age and gender effects on damage rate evolution, throughout life, have been investigated using a mechanobiological finite element modeling. To achieve the aim, a mathematical model has been developed, coupling both cell activities and mechanical behavior of trabecular bone, under cyclic loadings. A series of computational simulations (ABAQUS/UMAT) has been performed on a 3D human proximal femur, allowing to investigate the effects of mechanical and biological parameters on mechanical strength of trabecular bone, in order to evaluate the fracture risk resulting from fatigue damage. The obtained results revealed that mechanical stimulus amplitude affects bone resorption and formation rates, and indicated that age and gender are major factors in bone response to the applied loadings.

Age and gender effects on bone mass density variation: finite elements simulation

Bone remodeling is a physiological process by which bone constantly adapts its structure to changes in long-term loading manifested by interactions between osteoclasts and osteoblasts. This process can be influenced by many local factors, via effects on bone cells differentiation and proliferation, which are produced by bone cells and act in a paracrine or autocrine way. The aim of the current work is to provide mechanobiological finite elements modeling coupling both cellular activities and mechanical behavior in order to investigate age and gender effects on bone remodeling evolution. A series of computational simulations have been performed on a 2D and 3D human proximal femur. An age- and gender-related impacts on bulk density alteration of trabecular bone have been noticed, and the major actors responsible of this phenomenon have been then discussed.

A semi-mechanistic model of bone mineral density and bone turnover based on a circular model of bone remodeling

Development of novel therapies for bone diseases can benefit from mathematical models that predict drug effect on bone remodeling biomarkers. Therefore, a bone cycle model (BCM) was developed that takes into consideration the concept of the basic multicellular unit and the dynamic equilibrium of bone remodeling. The model is a closed form cyclical model with four compartments representing resorption, formation, primary mineralization, and secondary mineralization. Equations describing the time course of bone turnover biomarkers were developed using the flow rate of bone cycle units (BCU) between the compartments or the amount of BCU in each compartment. A disease progression model representing bone loss in osteoporosis, a vitamin D and calcium supplementation (placebo) model, and a drug model for antiresorptive treatments were added to the model. Initial model parameter values were derived from published bone turnover data. The BCM accurately described biomarker-time profiles in postmenopausal women receiving either placebo or bisphosphonate treatment. The slow continual increase in bone mineral density (BMD) observed after 1 year of treatment was accurately described when changes in bone turnover were combined with increases in mineralization. For this purpose, the secondary mineralization compartment was replaced by three catenary chain compartments representing increasing mineral content. The refined BCM satisfactorily predicted biomarker profiles after long-term (10-year) bisphosphonate treatment. Furthermore, the model successfully described individual bone turnover markers and BMD results following treatment with denosumab in postmenopausal women. Analyses with this model could be used to optimize dosing regimens and to predict effects of novel osteoporotic treatments.

The role of osteocytes in targeted bone remodeling: a mathematical model

Until recently many studies of bone remodeling at the cellular level have focused on the behavior of mature osteoblasts and osteoclasts, and their respective precursor cells, with the role of osteocytes and bone lining cells left largely unexplored. This is particularly true with respect to the mathematical modeling of bone remodeling. However, there is increasing evidence that osteocytes play important roles in the cycle of targeted bone remodeling, in serving as a significant source of RANKL to support osteoclastogenesis, and in secreting the bone formation inhibitor sclerostin. Moreover, there is also increasing interest in sclerostin, an osteocyte-secreted bone formation inhibitor, and its role in regulating local response to changes in the bone microenvironment. Here we develop a cell population model of bone remodeling that includes the role of osteocytes, sclerostin, and allows for the possibility of RANKL expression by osteocyte cell populations. We have aimed to give a simple, yet still tractable, model that remains faithful to the underlying system based on the known literature. This model extends and complements many of the existing mathematical models for bone remodeling, but can be used to explore aspects of the process of bone remodeling that were previously beyond the scope of prior modeling work. Through numerical simulations we demonstrate that our model can be used to explore theoretically many of the qualitative features of the role of osteocytes in bone biology as presented in recent literature.

Implant stability and bone remodeling after 3 and 13 days of implantation with an initial static strain

OBJECTIVE

Bone is constantly exposed to dynamic and static loads, which induce both dynamic and static bone strains. Although numerous studies exist on the effect of dynamic strain on implant stability and bone remodeling, the effect of static strain needs further investigation. Therefore, the effect of two different static bone strain levels on implant stability and bone remodeling at two different implantation times was investigated in a rabbit model.

METHODS

Two different test implants with a diametrical expansion of 0.15 mm (group A) and 0.05 mm (group B) creating initial static bone strains of 0.045 and 0.015, respectively. The implants were inserted in the proximal tibial metaphysis of 24 rabbits to observe the biological response at implant removal. Both groups were compared to control implants (group C), with no diametrical increase. The insertion torque (ITQ) was measured to represent the initial stability and the removal torque (RTQ) was measured to analyze the effect that static strain had on implant stability and bone remodeling after 3 and 13 days of implantation time.

RESULTS

The ITQ and the RTQ values for test implants were significantly higher for both implantation times compared to control implants. A selection of histology samples was prepared to measure bone to implant contact (BIC). There was a tendency that the BIC values for test implants were higher compared to control implants.

CONCLUSION

These findings suggest that increased static bone strain creates higher implant stability at the time of insertion, and this increased stability is maintained throughout the observed period.

Towards a new spatial representation of bone remodeling

Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed. There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes. We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur in vitro and in vivo. By employing the level set approach it is possible to develop computational and mathematical representations of the spatial behavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as in vitro and in vivo. This makes it possible to perform in silica experiments more closely resembling experimental observations.

A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease

BACKGROUND

Multiple myeloma is a hematologic malignancy associated with the development of a destructive osteolytic bone disease.

RESULTS

Mathematical models are developed for normal bone remodeling and for the dysregulated bone remodeling that occurs in myeloma bone disease. The models examine the critical signaling between osteoclasts (bone resorption) and osteoblasts (bone formation). The interactions of osteoclasts and osteoblasts are modeled as a system of differential equations for these cell populations, which exhibit stable oscillations in the normal case and unstable oscillations in the myeloma case. In the case of untreated myeloma, osteoclasts increase and osteoblasts decrease, with net bone loss as the tumor grows. The therapeutic effects of targeting both myeloma cells and cells of the bone marrow microenvironment on these dynamics are examined.

CONCLUSIONS

The current model accurately reflects myeloma bone disease and illustrates how treatment approaches may be investigated using such computational approaches.

REVIEWERS

This article was reviewed by Ariosto Silva and Mark P. Little.

Integrated cellular bone homeostasis model for denosumab pharmacodynamics in multiple myeloma patients

The purpose of this study is to couple a cellular bone homeostasis model with the pharmacokinetics (PK) and mechanism of action of denosumab, an inhibitor of receptor activator of nuclear factor-kappaB ligand, to characterize the time course of serum N-telopeptide (NTX), a bone resorption biomarker, following single escalating doses in multiple myeloma (MM) patients. Mean PK and median serum NTX temporal profiles were extracted from a previously conducted randomized, double-blind, double-dummy, active-controlled, multicenter study including 25 MM patients receiving escalating denosumab doses. Nonlinear denosumab PK profiles were well described by a target-mediated disposition model that includes rapid binding of drug to its pharmacological target. Fixed PK profiles were integrated into a previously reported theoretical cellular model of osteoblast-osteoclast interactions, and the NTX concentrations were linked to a resorbing active osteoclast (AOC) pool by a nonlinear transfer function. Reasonable fits were obtained for the NTX profiles from maximal likelihood estimation using the final model. Transfer function parameters, including the basal NTX level and the AOC concentration producing 50% of maximal NTX production, were estimated with good precision as 5.55 nM and 1.88 x 10(-5) pM. An indirect response model for inhibition of NTX production by denosumab was also used to characterize the data. Although this model adequately characterized the pharmacodynamic data, simulations conducted with the full model reveal that a cellular model coupled with clinical data has the distinct advantage of not only quantitatively describing data but also providing new testable hypotheses on the role of cellular system variables on drug response.

Cross-scale sensitivity analysis of a non-small cell lung cancer model: linking molecular signaling properties to cellular behavior

Sensitivity analysis is an effective tool for systematically identifying specific perturbations in parameters that have significant effects on the behavior of a given biosystem, at the scale investigated. In this work, using a two-dimensional, multiscale non-small cell lung cancer (NSCLC) model, we examine the effects of perturbations in system parameters which span both molecular and cellular levels, i.e. across scales of interest. This is achieved by first linking molecular and cellular activities and then assessing the influence of parameters at the molecular level on the tumor's spatio-temporal expansion rate, which serves as the output behavior at the cellular level. Overall, the algorithm operated reliably over relatively large variations of most parameters, hence confirming the robustness of the model. However, three pathway components (proteins PKC, MEK, and ERK) and eleven reaction steps were determined to be of critical importance by employing a sensitivity coefficient as an evaluation index. Each of these sensitive parameters exhibited a similar changing pattern in that a relatively larger increase or decrease in its value resulted in a lesser influence on the system's cellular performance. This study provides a novel cross-scaled approach to analyzing sensitivities of computational model parameters and proposes its application to interdisciplinary biomarker studies.

Allosteric control model of bone remodelling containing periodical modes

To help to understand the modelling process that occurs when a scaffold is implanted it is vital to understand the rather complex bone remodelling process prevalent in native bone. We have formulated a mathematical model that predicts osteoactivity both in scaffolds, as well as in bone in vivo and could set a basis for the more detailed allosteric models. The model is extended towards a bio-cybernetic vision of basic multicellular unit (BMU) action, when some of the regulation loops have been modified to reflect the allosteric control mechanisms, developed by Michaels-Menten, Hill, Koshland-Nemethy-Filmer, Monod-Wyman-Changeux. By implementation of this approach a four-dimensional system was obtained that shows steady cyclic behaviour using a wide range of constants with clear biological meaning. We have observed that a local steady state appears as a limiting cycle in multi-dimensional phase space and this is discussed in this paper. Physiological interpretation of this limiting four-dimension cycle possibly related to a conservative-like value has been proposed. Analysis and simulation of the model has shown an analogy between this conservative value, as a kind of substrate-energy regenerative potential of the bone remodelling system with a molecular nature, and to the classical physical value--energy. This dynamic recovery potential is directed against both mechanical and biomechanical damage to the bone. Furthermore, the current model has credibility when compared to the normal bone remodelling process. In the framework of widely recognised Hill mechanisms of allosteric regulation the cyclic attractor, described formerly for a pure cellular model, prevails for different forms of feedback control. This result indicates the viability of the proposed existence of a conservative value (analogous to energy) that characterises the recovery potential of the bone remodelling cycle. Linear stability analysis has been performed in order to determine the robustness of the basic state, however, additional work is required to study a wider range of constants.

On allosteric control model of bone turnover cycle containing osteocyte regulation loop

One approach to developing a mathematical model that predicts osteoactivity both in bio-scaffolds, as well as the in bone tissue in vivo, is based on a bio-cybernetic vision of basic multicellular unit (BMU) action. In the case of the model presented in this paper, some of the loops of regulation have been modified to reflect the range of allosteric control mechanisms: Michaelis-Menten, Hill, Adair, Koshland-Nemethy-Filmer (KNF), Monod-Wyman-Changeux (MWC). This approach has resulted in a four-dimensional system that shows steady cyclic behaviour using a range of constants with clear biological meaning. The initial findings suggesting that a steady state appears as a cycle in multidimensional phase space and this is discussed in this paper. The existence of this cycle in the osteoclasts-osteoblasts-osteocytes-bone subspace indicates that there is a conservative value along steady trajectories for this dynamic system. Biophysical interpretation of this conservative value has been proposed as a kind of substrate-energy regenerative potential of the bone remodelling system with a similarity to the classical physical value-energy. Such a recovery "potential" is directed against both mechanical and biomechanical damage to the bone. The current model has credibility when compared to the normal bone remodelling process. In the framework of widely recognised Michaelis-Menten mechanisms of allosteric regulation the cyclic attractor, described formerly for a pure cellular model, prevails for different forms of feedback control. This finding demonstrates the viability of the suggestion of the subsistence of conservative value (analogous to energy) that characterises the recovery potential of the bone remodelling cycle. The results indicate that the robust behaviour of the model is maintained from the simple cellular level to the molecular biochemical level of regulation.

A novel mathematical model identifies potential factors regulating bone apposition

The development of pharmaceutical treatments for bone disease can be enhanced by mathematical models that predict their effects on matrix apposition during cancellous bone remodelling. Therefore, a mathematical model was constructed to simulate the rate of focal bone formation from the number of osteoid-forming osteoblasts at one microsite and their rate of activity. The number of mature osteoid-forming cells was simulated from a relationship describing the proliferation of preosteoblasts. Osteoblast activity was described by Michaelis-Menten enzyme kinetic equations adapted to describe cellular activity. The model incorporates the negative feedback effects on the rates of bone apposition due to the reduction in size of mature osteoblasts with continuing differentiation and the reduction in number of osteoid-forming cells with apoptosis and osteocyte formation. In addition, the rate of mineralisation is limited according to osteoid substrate availability. Results of sensitivity analysis revealed the amount of bone formed at one microsite to be more sensitive to changes in factors that controlled cell growth during proliferation and the number of mature osteoid-forming osteoblasts than to those that determined cellular activity. Matrix and osteocyte signalling were shown to have potentially important roles in controlling rates of osteoid apposition in normal, healthy bone. This simple model supports the critical role of controlled mitotic growth in normal bone apposition. It can also help to explain how the homeostatic processes of bone resorption and apposition during remodelling can be disrupted by growth factors that affect the mitotic fraction and division time of proliferative preosteoblast cells.