Stability analysis of a Komarova type model for the interactions of osteoblast and osteoclast cells during bone remodeling

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.

Antigen receptor therapy in bone metastasis via optimal control for different human life stages

In this work we propose a bone metastasis model using power law growth functions in order to describe the biochemical interactions between bone cells and cancer cells. Experimental studies indicate that bone remodeling cycles are different for human life stages: childhood, young adulthood, and adulthood. In order to include such differences in our study, we estimate the model parameter values for each human life stage via bifurcation analysis. Results reveal an intrinsic relationship between the active period of remodeling cycles and the proliferation of cancer cells. Subsequently, using optimal control theory we analyze a possible antigen receptor therapy as a new treatment for bone metastasis. Theoretical results such as existence of optimal solutions are proved. Numerical simulations for late stages of bone metastasis are presented and a discussion of our results is carried out.

Strange attractors in discrete slow power-law models of bone remodeling

Recently, a family of nonlinear mathematical discrete systems to describe biological interactions was considered. Such interactions are modeled by power-law functions where the exponents involve regulation processes. Considering exponent values giving rise to hyperbolic equilibria, we show that the systems exhibit irregular behavior characterized by strange attractors. The systems are numerically analyzed for different parameter values. Depending on the initial conditions, the orbits of each system either diverge to infinity or approach a periodic orbit or a strange attractor. Such dynamical behavior is identified by their Lyapunov exponents and local dimension. Finally, an application to the biochemical process of bone remodeling is presented. The existence of deterministic chaos in this process reveals a possible explanation of reproducibility failure and variation of effects in clinical experiments.

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.

Fluctuating periodic solutions and moment boundedness of a stochastic model for the bone remodeling process

In this work, we model osteoclast-osteoblast population dynamics with random environmental fluctuations in order to understand the random variations of the bone remodeling process in real life. For this purpose, we construct a stochastic differential model for the interactions between the osteoclast and osteoblast cell populations using the parameter perturbation technique. We prove the existence of a globally attractive positive unique solution for the stochastically perturbed system. Also, the stochastic boundedness of the solution is demonstrated using its p-th order moments for p ≥ 1. Finally, we show that the introduction of noise in the deterministic model provides a fluctuating periodic solution. Numerical evidence supports our theoretical results and a discussion of the results is carried out.