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Computational and Mathematical Methods in Medicine
Volume 2012, Article ID 809864, 11 pages
http://dx.doi.org/10.1155/2012/809864
Research Article

Stability Analysis of a Model for Foreign Body Fibrotic Reactions

1Department of Mathematics and Statistics, Texas Tech University, P.O. Box 41042, Lubbock, TX 79409-1042, USA
2Department of Mathematics, The University of Texas at Arlington, Arlington, TX 76019, USA
3Department of Bioengineering, The University of Texas at Arlington, Arlington, TX 76019, USA

Received 30 April 2012; Revised 1 August 2012; Accepted 6 August 2012

Academic Editor: Gary C. An

Copyright © 2012 A. Ibraguimov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Implanted medical devices often trigger immunological and inflammatory reactions from surrounding tissues. The foreign body-mediated tissue responses may result in varying degrees of fibrotic tissue formation. There is an intensive research interest in the area of wound healing modeling, and quantitative methods are proposed to systematically study the behavior of this complex system of multiple cells, proteins, and enzymes. This paper introduces a kinetics-based model for analyzing reactions of various cells/proteins and biochemical processes as well as their transient behavior during the implant healing in 2-dimensional space. In particular, we provide a detailed modeling study of different roles of macrophages () and their effects on fibrotic reactions. The main mathematical result indicates that the stability of the inflamed steady state depends primarily on the reaction dynamics of the system. However, if the said equilibrium is unstable by its reaction-only system, the spatial diffusion and chemotactic effects can help to stabilize when the model is dominated by classical and regulatory macrophages over the inflammatory macrophages. The mathematical proof and counter examples are given for these conclusions.