Mathematical Model of Tissue Remodeling During Osseointegration

The purpose of this work is a design of mathematical model, which describes the morphological features in the bone-implant interface during osseointegration. The model is a system of ordinary linear differential equations and initial conditions similar to the ones for chemical kinetics or population dynamics. We verify of our model by microscopy images of bone tissue histological slices in the area of contact with the implant. These data were obtained at the 6th and 9th months after implant placement. We segment images into constituents which are a fibrous (dense connective) tissue, a bone tissue, cell nuclei and unanalyzed areas. The model presents the dynamics of the “concentration” (percentage) of constituents. The temporary alterations of constituents were considered as a tissue remodeling. We find the time dependencies for our “concentrations” of constituents and the unknown parameters involved in these dependencies. Using the found parameters we present the solution of differential equation system as the plots. The obtained solutions describe a successful osseointegration.