A mathematical model of bone remodeling with delays

摘要

In all vertebrates, bone tissue is constantly regenerated. As part of a very complicated process, osteoclasts reabsorb bone tissue and then osteoblasts reconstruct it. The process is regulated by several chemical signals, and there are also other cells involved. Mathematical models try to reproduce the main characteristics of the process while keeping it simplified. One of the most important part of the remodeling is the periodicity of the process. Here we will consider a simplified model consisting of three ordinary differential equations and introduce delays. The delays appear because there is a lag in the change of the population of osteoblasts due to changes in the population of osteoclasts, and vice versa. We study the properties of the system, including stability and bifurcation and find that the delay differential equations have Hopf bifurcations that give periodic solutions. We calculate numerical solutions to illustrate the behavior of the solutions.