Stability analysis of a model for foreign body fibrotic reactions.

*(English)*Zbl 1339.92029Summary: 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 \((M\Phi)\) 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.

##### MSC:

92C50 | Medical applications (general) |

34D20 | Stability of solutions to ordinary differential equations |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

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\textit{A. Ibraguimov} et al., Comput. Math. Methods Med. 2012, Article ID 809864, 11 p. (2012; Zbl 1339.92029)

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