# 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
##### References:
 [1] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhäuser, Boston, 2014. doi: 10.1007/978-3-319-05140-6.  Google Scholar [2] N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math Models Methods Appl Sci., 22 (2012), 1140006, 29pp. doi: 10.1142/S0218202511400069.  Google Scholar [3] P. K. Beredjiklian, Biologic aspects of flexor tendon laceration and repair, J Bone Joint Surg Am., 85 (2003), 539-550.  doi: 10.2106/00004623-200303000-00025.  Google Scholar [4] J. C. Dallon and J. A. Sherratt, A mathematical model for fibroblast and collagen orientation, Bull Math Biol., 60 (1998), 101-129.  doi: 10.1006/bulm.1997.0027.  Google Scholar [5] J. C. Dallon, J. A. Sherratt and P. K. Maini, Mathematical modelling of extracellular matrix dynamics using discrete cells: fiber orientation and tissue regeneration, J Theor Biol., 199 (1999), 449-471.  doi: 10.1006/jtbi.1999.0971.  Google Scholar [6] D. Docheva, S. A. Müller, M. Majewski and Ch. H. Evans, Biologics for tendon repair, Adv Drug Deliv Rev., 84 (2015), 222-239.  doi: 10.1016/j.addr.2014.11.015.  Google Scholar [7] E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model, J Math Biol., 46 (2003), 537-563.  doi: 10.1007/s00285-002-0187-1.  Google Scholar [8] L. Geris, A. Gerisch and R. C. Schugart, Mathematical modeling in wound healing, bone regeneration and tissue engineering, Acta Biotheor., 58 (2010), 355-367.  doi: 10.1007/s10441-010-9112-y.  Google Scholar [9] K. Kang, B. Perthame, A. Stevens and J. J. L. Velázquez, An integro-differential equation model for alignment and orientational aggregation, J Diff Eqs., 246 (2009), 1387-1421.  doi: 10.1016/j.jde.2008.11.006.  Google Scholar [10] M. Lachowicz, H. Leszczyński and M. Parisot, A simple kinetic equation of swarm formation: Blow-up and global existence, Appl Math Letters, 57 (2016), 104-107.  doi: 10.1016/j.aml.2016.01.008.  Google Scholar [11] M. Lachowicz, H. Leszczyński and M. Parisot, Blow-up and global existence for a kinetic equation of swarm formation, Math Models Methods Appl Sci., 27 (2017), 1153-1175.  doi: 10.1142/S0218202517400115.  Google Scholar [12] T. W. Lin, L. Cardenas and L. J. Soslowsky, Biomechanics of tendon injury and repair, J Biomech., 37 (2004), 865-877.  doi: 10.1016/j.jbiomech.2003.11.005.  Google Scholar [13] S. McDougall, J. C. Dallon, J. A. Sherratt and P. K. Maini, Fibroblast migration and collagen deposition during dermal wound healing: mathematical modelling and clinical implications, Philos Trans A Math Phys Eng Sci., 364 (2006), 1385-1405.  doi: 10.1098/rsta.2006.1773.  Google Scholar [14] M. O'Brian, Anatomy of tendon, in Tendon Injuries (eds. N. Maffulli, P. Renström and W.B. Leadbetter), Springer-Verlag, (2005), 3–13. Google Scholar [15] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J Math Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar [16] M. Parisot and M. Lachowicz, A kinetic model for the formation of swarms with nonlinear interactions, Kinetic & Related Models, 9 (2016), 131-164.  doi: 10.3934/krm.2016.9.131.  Google Scholar [17] P. Sharma and N. Maffulli, Biology of tendon injury: Healing, modeling and remodeling, J Musculoskelet Neuronal Interact., 6 (2006), 181-190.   Google Scholar [18] J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc Biol Sci., 241 (1990), 29-36.   Google Scholar [19] J. A. Sherratt and J. C. Dallon, Theoretical models of wound healing: Past successes and future challenges, C R Biol., 325 (2002), 557-564.  doi: 10.1016/S1631-0691(02)01464-6.  Google Scholar [20] R. T. Tranquillo and J. D. Murray, Continuum model of fibroblast-driven wound contraction: Inflammation-mediation, J Theor Biol., 158 (1992), 135-172.  doi: 10.1016/S0022-5193(05)80715-5.  Google Scholar [21] G. Yang, B. B. Rothrauff and R. S. Tuan, Tendon and ligament regeneration and repair: Clinical relevance and developmental paradigm, Birth Defects Res C Embryo Today., 99 (2013), 203-222.  doi: 10.1002/bdrc.21041.  Google Scholar

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##### References:
 [1] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhäuser, Boston, 2014. doi: 10.1007/978-3-319-05140-6.  Google Scholar [2] N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math Models Methods Appl Sci., 22 (2012), 1140006, 29pp. doi: 10.1142/S0218202511400069.  Google Scholar [3] P. K. Beredjiklian, Biologic aspects of flexor tendon laceration and repair, J Bone Joint Surg Am., 85 (2003), 539-550.  doi: 10.2106/00004623-200303000-00025.  Google Scholar [4] J. C. Dallon and J. A. Sherratt, A mathematical model for fibroblast and collagen orientation, Bull Math Biol., 60 (1998), 101-129.  doi: 10.1006/bulm.1997.0027.  Google Scholar [5] J. C. Dallon, J. A. Sherratt and P. K. Maini, Mathematical modelling of extracellular matrix dynamics using discrete cells: fiber orientation and tissue regeneration, J Theor Biol., 199 (1999), 449-471.  doi: 10.1006/jtbi.1999.0971.  Google Scholar [6] D. Docheva, S. A. Müller, M. Majewski and Ch. H. Evans, Biologics for tendon repair, Adv Drug Deliv Rev., 84 (2015), 222-239.  doi: 10.1016/j.addr.2014.11.015.  Google Scholar [7] E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model, J Math Biol., 46 (2003), 537-563.  doi: 10.1007/s00285-002-0187-1.  Google Scholar [8] L. Geris, A. Gerisch and R. C. Schugart, Mathematical modeling in wound healing, bone regeneration and tissue engineering, Acta Biotheor., 58 (2010), 355-367.  doi: 10.1007/s10441-010-9112-y.  Google Scholar [9] K. Kang, B. Perthame, A. Stevens and J. J. L. Velázquez, An integro-differential equation model for alignment and orientational aggregation, J Diff Eqs., 246 (2009), 1387-1421.  doi: 10.1016/j.jde.2008.11.006.  Google Scholar [10] M. Lachowicz, H. Leszczyński and M. Parisot, A simple kinetic equation of swarm formation: Blow-up and global existence, Appl Math Letters, 57 (2016), 104-107.  doi: 10.1016/j.aml.2016.01.008.  Google Scholar [11] M. Lachowicz, H. Leszczyński and M. Parisot, Blow-up and global existence for a kinetic equation of swarm formation, Math Models Methods Appl Sci., 27 (2017), 1153-1175.  doi: 10.1142/S0218202517400115.  Google Scholar [12] T. W. Lin, L. Cardenas and L. J. Soslowsky, Biomechanics of tendon injury and repair, J Biomech., 37 (2004), 865-877.  doi: 10.1016/j.jbiomech.2003.11.005.  Google Scholar [13] S. McDougall, J. C. Dallon, J. A. Sherratt and P. K. Maini, Fibroblast migration and collagen deposition during dermal wound healing: mathematical modelling and clinical implications, Philos Trans A Math Phys Eng Sci., 364 (2006), 1385-1405.  doi: 10.1098/rsta.2006.1773.  Google Scholar [14] M. O'Brian, Anatomy of tendon, in Tendon Injuries (eds. N. Maffulli, P. Renström and W.B. Leadbetter), Springer-Verlag, (2005), 3–13. Google Scholar [15] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J Math Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar [16] M. Parisot and M. Lachowicz, A kinetic model for the formation of swarms with nonlinear interactions, Kinetic & Related Models, 9 (2016), 131-164.  doi: 10.3934/krm.2016.9.131.  Google Scholar [17] P. Sharma and N. Maffulli, Biology of tendon injury: Healing, modeling and remodeling, J Musculoskelet Neuronal Interact., 6 (2006), 181-190.   Google Scholar [18] J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc Biol Sci., 241 (1990), 29-36.   Google Scholar [19] J. A. Sherratt and J. C. Dallon, Theoretical models of wound healing: Past successes and future challenges, C R Biol., 325 (2002), 557-564.  doi: 10.1016/S1631-0691(02)01464-6.  Google Scholar [20] R. T. Tranquillo and J. D. Murray, Continuum model of fibroblast-driven wound contraction: Inflammation-mediation, J Theor Biol., 158 (1992), 135-172.  doi: 10.1016/S0022-5193(05)80715-5.  Google Scholar [21] G. Yang, B. B. Rothrauff and R. S. Tuan, Tendon and ligament regeneration and repair: Clinical relevance and developmental paradigm, Birth Defects Res C Embryo Today., 99 (2013), 203-222.  doi: 10.1002/bdrc.21041.  Google Scholar
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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