Modeling bone regeneration around endosseous implants

SummaryModeling bone regeneration around endosseous implantsPavel A. ProkharauThe present work is focused on mathematical modeling of bone regeneration.Various aspects of modeling are considered. In Chapter 2, a classicalsystem of partial differential equations (PDE’s) is analyzed, which is constructedto simulate bone healing around endosseous implants. The presentsystem is of the diffusion-advection-reaction type and is typical for mathematicalmodels for bone regeneration. The need of analyzing the PDE’sfollows from the appearance of the wave-like profiles, that are found in thenumerical solutions for the distribution of cells and of growth factors and,consequently, of newly formed bone matrix. Such predictions of the modelcontradict experimental observations. Hence it is critically important tounderstand why these wave-like patterns appear and how the model maybe modified in order to provide biologically relevant solutions. The linearstability analysis carried out around constant-state (i.e. homogeneous inthe physical space) solutions provides the answers to the stated questions.Stability of the constant-state solutions is determined by the values of themodel parameters. Explicit relations determining the stability region for theparameters are derived. It is concluded that if the model parameters havethe values outside of the stability region, then the constant-state solution isunstable and the exact solution of the current system will not converge toit. Hence formation of stable patterns is likely. The analytical results arevalidated by finite element simulations.Chapters 3 and 4 of the thesis are devoted to development of new approachesin modeling bone regeneration. In Chapter 3, the evolutionarydifferentiation model is introduced, that allows to incorporate the historyinto the differentiation of cells by defining an additional differentiation statevariable a ∈ [0, 1]. During bone regeneration, mesenchymal stem cells differv