A simple model of collagen remodeling

Grzegorz Dudziuk; Mirosław Lachowicz; Henryk Leszczyński; Zuzanna Szymańska

doi:10.3934/dcdsb.2019091

Article Contents

2019, Volume 24, Issue 5: 2205-2217. Doi: 10.3934/dcdsb.2019091

This issue Previous Article Proof of the maximum principle for a problem with state constraints by the v-change of time variable Next Article On numerical methods for singular optimal control problems: An application to an AUV problem

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
^* Corresponding author: Zuzanna Szymańska

Received: January 2018

Revised: January 2019

Early access: March 2019

Published: March 2019

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35A01, 45K05, 92C50; Secondary: 92C30, 45J05.

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Figure 1. 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

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Figure 2. Model simulation for an initial condition with plateau present for each x ∈ D ("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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