# American Institute of Mathematical Sciences

May  2019, 24(5): 2205-2217. doi: 10.3934/dcdsb.2019091

## A simple model of collagen remodeling

 1 ICM, University of Warsaw, ul. Tyniecka 15/17, 02-630 Warsaw, Poland 2 Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland 3 Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland 4 Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland

* Corresponding author: Zuzanna Szymańska

Received  January 2018 Revised  January 2019 Published  March 2019

Fund Project: G. D. and Z. S. were supported by the National Centre for Research and Development Grant STRATEGMED1/233224/10/NCBR/2014. M. L. was supported by the National Science Centre Poland Grant 2017/25/B/ST1/00051. Z. S. acknowledge the support from the National Science Centre Poland Grant 2017/26/M/ST1/00783

In the present paper we propose and study a simple model of collagen remodeling occurring in latter stage of tendon healing process. The model is an integro-differential equation describing the possibility of an alignment of collagen fibers in a finite time. We show that the solutions may either exist globally in time or blow-up in a finite time depending on initial data. The latter behavior can be related to the healing of injury without the scar formation in a finite time: a full alignment of collagen fibers. We believe that the present model is an essential ingredient of the full description of collagen remodeling.

Citation: Grzegorz Dudziuk, Mirosław Lachowicz, Henryk Leszczyński, Zuzanna Szymańska. A simple model of collagen remodeling. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2205-2217. doi: 10.3934/dcdsb.2019091
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##### References:
Model simulation for an initial condition with no plateau. Parameters β ≡ 1 and γ = 2 were assumed. Lower panels show a section of the solution at x = 0:0. In our opinion, due to high mass concentration, the last relevant time step of the simulation is t = 15:3
Model simulation for an initial condition with plateau present for each xD ("truncated tops"). Parameters β ≡ 1 and γ = 2 were assumed. Lower panels show a section of the solution at x = 0:0. Prior to the time t = 30:0, the solution attains a state which undergoes no further visible changes, and as such probably approximates an equilibrium of the model
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