# 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  May 2019 Early access  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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] M. O'Brian, Anatomy of tendon, in Tendon Injuries (eds. N. Maffulli, P. Renström and W.B. Leadbetter), Springer-Verlag, (2005), 3–13. [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. [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. [17] P. Sharma and N. Maffulli, Biology of tendon injury: Healing, modeling and remodeling, J Musculoskelet Neuronal Interact., 6 (2006), 181-190. [18] J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc Biol Sci., 241 (1990), 29-36. [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. [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. [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.

show all references

##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] M. O'Brian, Anatomy of tendon, in Tendon Injuries (eds. N. Maffulli, P. Renström and W.B. Leadbetter), Springer-Verlag, (2005), 3–13. [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. [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. [17] P. Sharma and N. Maffulli, Biology of tendon injury: Healing, modeling and remodeling, J Musculoskelet Neuronal Interact., 6 (2006), 181-190. [18] J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc Biol Sci., 241 (1990), 29-36. [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. [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. [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.
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
 [1] José Antonio Carrillo, Martin Parisot, Zuzanna Szymańska. Mathematical modelling of collagen fibres rearrangement during the tendon healing process. Kinetic and Related Models, 2021, 14 (2) : 283-301. doi: 10.3934/krm.2021005 [2] Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057 [3] Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021051 [4] Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems on degenerate integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022025 [5] Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907 [6] Jean-Michel Roquejoffre, Juan-Luis Vázquez. Ignition and propagation in an integro-differential model for spherical flames. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 379-387. doi: 10.3934/dcdsb.2002.2.379 [7] Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417 [8] Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 [9] Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053 [10] Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160 [11] Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053 [12] Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191 [13] Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541 [14] Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977 [15] Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065 [16] Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems and Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693 [17] Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015 [18] Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial and Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119 [19] Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control and Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044 [20] Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 639-657. doi: 10.3934/dcdsb.2021059

2021 Impact Factor: 1.497