Mass Exchange and Advection Term in Bone Remodeling Process: Theory of Porous Media

Abstract

Bone remodeling and bone resorption are two of the most important processes during bone healing. There have been numerous experiments to understand the effects of mechanical loading on bone tissue. However, the progress is not much due to the complexity of the process. Although it is well accepted that bone is consisting of two phases, such as a solid and a fluid part, all experiments consider only the solid part for simplicity. Recent studies demonstrated that despite the induced strain field inside the solid part due to mechanical force, the fluid part plays a crucial role in the bone remodeling process as well. The interstitial fluid is pressed through the osteocyte canaliculi and produces a shear stress field that excites osteocytes to produce signaling molecules. These signals initiate the bone remodeling process within the bone. In addition, the strain field of the solid part stimulates osteoclast and osteoblast cells to commence bone resorption and apposition, respectively. A combination of these two processes could be the exact bone regeneration process. The purpose of this investigation is to examine the influence of the fluid stream inside the bone. Using theory of porous media, we considered the bone as a bi-phasic mixture consisting of a fluid and a solid part. Each constituent at a given spatial point has its own motion. Also, we assumed that this bi-phasic system is closed with respect to mass transfer but open with respect to the momentum. Furthermore, the characteristic time of chemical reactions is assumed several orders of magnitude greater than the characteristic time associated with the prefusion of the fluid flow, so the system is considered isothermal. We derived the balance of linear momentum for each constituent concerning these assumptions, resulting in coupled PDEs. Furthermore, the advection term is considered for the fluid part movement. The Finite Element Method (FEM) and the Finite Volume Method (FVM) are used to solve the balance of linear momentum for the solid and fluid parts, respectively. Finally, the results are compared with the theory of poroelasticity.