# American Institute of Mathematical Sciences

April  2021, 14(2): 283-301. doi: 10.3934/krm.2021005

## Mathematical modelling of collagen fibres rearrangement during the tendon healing process

 1 Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom 2 Inria-Bordeaux, Team CARDAMOM, Office B426,200 av. de la vieille tour, 33405 Talence Cedex, Bordeaux, France 3 Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland 4 ICM, University of Warsaw, ul. Tyniecka 15/17, 02-630 Warsaw, Poland

* Corresponding author: z.szymanska@icm.edu.pl

Received  April 2020 Revised  September 2020 Published  April 2021 Early access  January 2021

Tendon injuries present a clinical challenge to modern medicine as they heal slowly and rarely is there full restoration to healthy tendon structure and mechanical strength. Moreover, the process of healing is not fully elucidated. To improve understanding of tendon function and the healing process, we propose a new model of collagen fibres rearrangement during tendon healing. The model consists of an integro-differential equation describing the dynamics of collagen fibres distribution. We further reduce the model in a suitable asym-ptotic regime leading to a nonlinear non-local Fokker-Planck type equation for the spatial and orientation distribution of collagen fibre bundles. Due to its simplicity, the reduced model allows for possible parameter estimation based on data. We showcase some of the qualitative properties of this model simulating its long time asymptotic behaviour and the total time for tendon fibres to align in terms of the model parameters. A possible biological interpretation of the numerical experiments performed leads us to the working hypothesis of the importance of tendon cell size in patient recovery.

Citation: 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
##### References:
 [1] A. R. Akintunde and K. S. Miller, Evaluation of microstructurally motivated constitutive models to describe age-dependent tendon healing, Biomech Model Mechanobiol., 17 (2018), 793-814.  doi: 10.1007/s10237-017-0993-4. [2] A. R. Akintunde, K. S. Miller and D. E. Schiavazzi, Bayesian inference of constitutive model parameters from uncertain uniaxial experiments on murine tendons, J Mech Behav Biomed Mater., 96 (2019), 285-300.  doi: 10.1016/j.jmbbm.2019.04.037. [3] A. R. Akintunde, D. E. Schiavazzi and K. S. Miller, Mathematical Model of Age-Specific Tendon Healing, Computer Methods, Imaging and Visualization in Biomechanics and Biomedical Engineering, Springer International Publishing, 36 (2020), 288-296. doi: 10.1007/978-3-030-43195-2_23. [4] J. Banasiak and M. Lachowicz, Kinetic Model of Alignment, Methods of Small Parameter in Mathematical Biology, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, 2014. doi: 10.1007/978-3-319-05140-6. [5] 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. [6] R. B. Bird, Ch. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of polymeric liquids, Volume 1: Fluid mechanics, Wiley, (1987). [7] R. B. Bird, Ch. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of polymeric liquids, Volume 2: Kinetic Theory, Wiley, (1987). [8] J. A. Carrillo, S. Cordier and G. Toscani, Over-populated tails for conservative-in-the-mean inelastic Maxwell models, Discrete Contin. Dyn. Syst. A., 24 (2009), 59-81.  doi: 10.3934/dcds.2009.24.59. [9] J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model,, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290. [10] J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Model. Simul., 11 (2013), 336-361.  doi: 10.1137/110851687. [11] J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun Comput Phys., 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a. [12] J. A. Carrillo, R. Eftimie and F. Hoffmann, Non-local kinetic and macroscopic models for self-organised animal aggregations, Kinet. Relat. Models., 8 (2015), 413-441.  doi: 10.3934/krm.2015.8.413. [13] C. Chainais-Hillairet and F. Filbet, Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model, IMA J. Numer. Anal., 27 (2007), 689-716.  doi: 10.1093/imanum/drl045. [14] A. Chauviere, L. Preziosi and T. Hillen, Modeling the motion of a cell population in the extracellular matrix, Discrete Contin. Dyn. Syst. A., (2007), 250-259. [15] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J Stat Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0. [16] S. L. Curwin, Rehabilitation after tendon injuries, Tendon Injuries, Springer-Verlag, 24 (2005), 242-266. doi: 10.1007/1-84628-050-8_24. [17] L. E. Dahners, Growth and development of tendons, Tendon Injuries, Springer-Verlag, 3 (2005), 22-24. doi: 10.1007/1-84628-050-8_3. [18] P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci., 2 (1992), 167-182.  doi: 10.1142/S0218202592000119. [19] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005. [20] D. Docheva, S.A. Müller, M. Majewski and H. E. Evans, Biologics for tendon repair, Adv. Drug Deliv. Rev., 84 (2015), 222-239.  doi: 10.1016/j.addr.2014.11.015. [21] M. Doumic, B. Perthame and J. P. Zubelli, Numerical solution of an inverse problem in size-structured population dynamics, Inverse Probl., 25 (2009), 045008, 25 pp. doi: 10.1088/0266-5611/25/4/045008. [22] M. Doumic, P. Maia and J. P. Zubelli, On the calibration of a size-structured population model from experimental data, Acta Biotheor., 58 (2010), 405-413.  doi: 10.1007/s10441-010-9114-9. [23] M. Doumic, A. Marciniak-Czochra, B. Perthame and J. P. Zubelli, A structured population model of cell differentiation, SIAM J Appl Math., 71 (2011), 1918-1940.  doi: 10.1137/100816584. [24] M. Doumic, M. Hoffmann, N. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree, BERNOULLI, 21 (2015), 1760-1799.  doi: 10.3150/14-BEJ623. [25] G. Dudziuk, M. Lachowicz, H. Leszczyński and Z. Szymańska, A simple model of collagen remodeling, Discrete Contin. Dyn. Syst. Ser. B., 24 (2019), 2205-2217.  doi: 10.3934/dcdsb.2019091. [26] R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, PNAS, 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104. [27] R. C. Fetecau, Collective behavior of biological aggregations in two dimensions: A nonlocal kinetic model, Math. Modelels Methods Appl. Sci., 21 (2011), 1539-1569.  doi: 10.1142/S0218202511005489. [28] R. C. Fetecau and R. Eftimie, An investigation of a nonlocal hyperbolic model for self-organization of biological groups, J. Math. Biol., 61 (2009), 545-579.  doi: 10.1007/s00285-009-0311-6. [29] G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Models Methods Appl. Sci., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048. [30] Y. Hyon, J. A. Carrillo, Q. Du and Ch. Liu, A maximum entropy principle based closure method for macro-micro models of polymeric materials, Kinet. Relat. Models., 1 (2008), 171-184.  doi: 10.3934/krm.2008.1.171. [31] G. Jull, A. Moore, D. Falla, J. Lewis, C. McCarthy and M. Sterling, Grieve's Modern Musculoskeletal Physiotherapy, 4$^{th}$ ed., Elsevier, 2015. [32] D. Kader, M. Mosconi, F. Benazzo and N. Maffulli, Achilles tendon rupture, Tendon Injuries, Springer-Verlag, 20 (2005), 187-200. doi: 10.1007/1-84628-050-8_20. [33] M. Kjær, Role of extracellular matrix in adaptation of tendon and skeletal muscle to mechanical loading, Physiol. Rev, 84 (2004), 649-698. [34] 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. [35] H. Y. Li and Y. H. Hua, Achilles tendinopathy: Current concepts about the basic science and clinical treatments, Biomed Res Int., 2016 (2016), 6492597, 9 pp. doi: 10.1155/2016/6492597. [36] 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. [37] N. Loy and L. Preziosi, Modelling physical limits of migration by a kinetic model with non-local sensing, J Math Biol., 80 (2020), 1759-1801.  doi: 10.1007/s00285-020-01479-w. [38] G. Nourissat, X. Houard, J. Sellam, D. Duprez and F. Berenbaum, Use of autologous growth factors in aging tendon and chronic tendinopathy, Front. Biosci., E5 (2013), 911-921.  doi: 10.2741/E670. [39] M. O'Brian, Anatomy of tendon, Tendon Injuries, Springer-Verlag, 1 (2005), 3-13. [40] 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. [41] H. G. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772. [42] M. Parisot and M. Lachowicz, A kinetic model for the formation of swarms with nonlinear interactions, Kinet. Relat. Models., 9 (2016), 131-164.  doi: 10.3934/krm.2016.9.131. [43] P. Sharma and N. Maffulli, Biology of tendon injury: Healing, modeling and remodeling, J Musculoskelet Neuronal Interact., 6 (2006), 181-190. [44] P. Sharma and N. Maffulli, Tendinopathy and tendon injury: The future, Disabil Rehabil., 30 (2008), 1733-1745.  doi: 10.1080/09638280701788274. [45] J. G. Snedeker and J. Foolen, Tendon injury and repair - A perspective on the basic mechanisms of tendon disease and future clinical therapy, Acta Biomater., 63 (2017), 18-36.  doi: 10.1016/j.actbio.2017.08.032. [46] B. Perthame and J. P. Zubelli, On the inverse problem for a size-structured population model, Inverse Probl., 23 (2007), 1037-1052.  doi: 10.1088/0266-5611/23/3/012. [47] N. Takahashi, P. Tangkawattana, Y. Ootomo, T. Hirose, J. Minaguchi, H. Ueda, M. Yamada and K. Takehana, Morphometric analysis of growing tenocytes in the superficial digital flexor tendon of piglets, J Vet Med Sci., 79 (2017), 1960-1967.  doi: 10.1292/jvms.17-0436. [48] C. T. Thorpe and H. R. C. Screen, Tendon structure and composition, Metabolic Influences on Risk for Tendon Disorders, Advances in Experimental Medicine and Biology, Springer, 920 (2016), 3-10. doi: 10.1007/978-3-319-33943-6_1. [49] G. Toscani, The grazing collisions asymptotics of the non-cut-off Kac equation, Esaim Math Model Numer Anal., 32 (1998), 763-772.  doi: 10.1051/m2an/1998320607631. [50] G. Toscani, One-dimensional kinetic models of granular flows, Esaim Math Model Numer Anal., 34 (2000), 1277-1291.  doi: 10.1051/m2an:2000127. [51] G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1. [52] F. Wu, M. Nerlich and D. Docheva, Tendon injuries: Basic science and new repair proposals, EFORT Open Rev., 2 (2017), 332-342.  doi: 10.1302/2058-5241.2.160075. [53] 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. [54] K. A. Young, J. A. Wise, P. DeSaix, D. H. Kruse, B. Poe, E. Johnson, J. E. Johnson, O. Korol, J. Gordon Betts and M. Womble, Anatomy & Physiology, OpenStax, 2013.

show all references

##### References:
 [1] A. R. Akintunde and K. S. Miller, Evaluation of microstructurally motivated constitutive models to describe age-dependent tendon healing, Biomech Model Mechanobiol., 17 (2018), 793-814.  doi: 10.1007/s10237-017-0993-4. [2] A. R. Akintunde, K. S. Miller and D. E. Schiavazzi, Bayesian inference of constitutive model parameters from uncertain uniaxial experiments on murine tendons, J Mech Behav Biomed Mater., 96 (2019), 285-300.  doi: 10.1016/j.jmbbm.2019.04.037. [3] A. R. Akintunde, D. E. Schiavazzi and K. S. Miller, Mathematical Model of Age-Specific Tendon Healing, Computer Methods, Imaging and Visualization in Biomechanics and Biomedical Engineering, Springer International Publishing, 36 (2020), 288-296. doi: 10.1007/978-3-030-43195-2_23. [4] J. Banasiak and M. Lachowicz, Kinetic Model of Alignment, Methods of Small Parameter in Mathematical Biology, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, 2014. doi: 10.1007/978-3-319-05140-6. [5] 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. [6] R. B. Bird, Ch. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of polymeric liquids, Volume 1: Fluid mechanics, Wiley, (1987). [7] R. B. Bird, Ch. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of polymeric liquids, Volume 2: Kinetic Theory, Wiley, (1987). [8] J. A. Carrillo, S. Cordier and G. Toscani, Over-populated tails for conservative-in-the-mean inelastic Maxwell models, Discrete Contin. Dyn. Syst. A., 24 (2009), 59-81.  doi: 10.3934/dcds.2009.24.59. [9] J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model,, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290. [10] J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Model. Simul., 11 (2013), 336-361.  doi: 10.1137/110851687. [11] J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun Comput Phys., 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a. [12] J. A. Carrillo, R. Eftimie and F. Hoffmann, Non-local kinetic and macroscopic models for self-organised animal aggregations, Kinet. Relat. Models., 8 (2015), 413-441.  doi: 10.3934/krm.2015.8.413. [13] C. Chainais-Hillairet and F. Filbet, Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model, IMA J. Numer. Anal., 27 (2007), 689-716.  doi: 10.1093/imanum/drl045. [14] A. Chauviere, L. Preziosi and T. Hillen, Modeling the motion of a cell population in the extracellular matrix, Discrete Contin. Dyn. Syst. A., (2007), 250-259. [15] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J Stat Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0. [16] S. L. Curwin, Rehabilitation after tendon injuries, Tendon Injuries, Springer-Verlag, 24 (2005), 242-266. doi: 10.1007/1-84628-050-8_24. [17] L. E. Dahners, Growth and development of tendons, Tendon Injuries, Springer-Verlag, 3 (2005), 22-24. doi: 10.1007/1-84628-050-8_3. [18] P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci., 2 (1992), 167-182.  doi: 10.1142/S0218202592000119. [19] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005. [20] D. Docheva, S.A. Müller, M. Majewski and H. E. Evans, Biologics for tendon repair, Adv. Drug Deliv. Rev., 84 (2015), 222-239.  doi: 10.1016/j.addr.2014.11.015. [21] M. Doumic, B. Perthame and J. P. Zubelli, Numerical solution of an inverse problem in size-structured population dynamics, Inverse Probl., 25 (2009), 045008, 25 pp. doi: 10.1088/0266-5611/25/4/045008. [22] M. Doumic, P. Maia and J. P. Zubelli, On the calibration of a size-structured population model from experimental data, Acta Biotheor., 58 (2010), 405-413.  doi: 10.1007/s10441-010-9114-9. [23] M. Doumic, A. Marciniak-Czochra, B. Perthame and J. P. Zubelli, A structured population model of cell differentiation, SIAM J Appl Math., 71 (2011), 1918-1940.  doi: 10.1137/100816584. [24] M. Doumic, M. Hoffmann, N. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree, BERNOULLI, 21 (2015), 1760-1799.  doi: 10.3150/14-BEJ623. [25] G. Dudziuk, M. Lachowicz, H. Leszczyński and Z. Szymańska, A simple model of collagen remodeling, Discrete Contin. Dyn. Syst. Ser. B., 24 (2019), 2205-2217.  doi: 10.3934/dcdsb.2019091. [26] R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, PNAS, 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104. [27] R. C. Fetecau, Collective behavior of biological aggregations in two dimensions: A nonlocal kinetic model, Math. Modelels Methods Appl. Sci., 21 (2011), 1539-1569.  doi: 10.1142/S0218202511005489. [28] R. C. Fetecau and R. Eftimie, An investigation of a nonlocal hyperbolic model for self-organization of biological groups, J. Math. Biol., 61 (2009), 545-579.  doi: 10.1007/s00285-009-0311-6. [29] G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Models Methods Appl. Sci., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048. [30] Y. Hyon, J. A. Carrillo, Q. Du and Ch. Liu, A maximum entropy principle based closure method for macro-micro models of polymeric materials, Kinet. Relat. Models., 1 (2008), 171-184.  doi: 10.3934/krm.2008.1.171. [31] G. Jull, A. Moore, D. Falla, J. Lewis, C. McCarthy and M. Sterling, Grieve's Modern Musculoskeletal Physiotherapy, 4$^{th}$ ed., Elsevier, 2015. [32] D. Kader, M. Mosconi, F. Benazzo and N. Maffulli, Achilles tendon rupture, Tendon Injuries, Springer-Verlag, 20 (2005), 187-200. doi: 10.1007/1-84628-050-8_20. [33] M. Kjær, Role of extracellular matrix in adaptation of tendon and skeletal muscle to mechanical loading, Physiol. Rev, 84 (2004), 649-698. [34] 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. [35] H. Y. Li and Y. H. Hua, Achilles tendinopathy: Current concepts about the basic science and clinical treatments, Biomed Res Int., 2016 (2016), 6492597, 9 pp. doi: 10.1155/2016/6492597. [36] 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. [37] N. Loy and L. Preziosi, Modelling physical limits of migration by a kinetic model with non-local sensing, J Math Biol., 80 (2020), 1759-1801.  doi: 10.1007/s00285-020-01479-w. [38] G. Nourissat, X. Houard, J. Sellam, D. Duprez and F. Berenbaum, Use of autologous growth factors in aging tendon and chronic tendinopathy, Front. Biosci., E5 (2013), 911-921.  doi: 10.2741/E670. [39] M. O'Brian, Anatomy of tendon, Tendon Injuries, Springer-Verlag, 1 (2005), 3-13. [40] 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. [41] H. G. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772. [42] M. Parisot and M. Lachowicz, A kinetic model for the formation of swarms with nonlinear interactions, Kinet. Relat. Models., 9 (2016), 131-164.  doi: 10.3934/krm.2016.9.131. [43] P. Sharma and N. Maffulli, Biology of tendon injury: Healing, modeling and remodeling, J Musculoskelet Neuronal Interact., 6 (2006), 181-190. [44] P. Sharma and N. Maffulli, Tendinopathy and tendon injury: The future, Disabil Rehabil., 30 (2008), 1733-1745.  doi: 10.1080/09638280701788274. [45] J. G. Snedeker and J. Foolen, Tendon injury and repair - A perspective on the basic mechanisms of tendon disease and future clinical therapy, Acta Biomater., 63 (2017), 18-36.  doi: 10.1016/j.actbio.2017.08.032. [46] B. Perthame and J. P. Zubelli, On the inverse problem for a size-structured population model, Inverse Probl., 23 (2007), 1037-1052.  doi: 10.1088/0266-5611/23/3/012. [47] N. Takahashi, P. Tangkawattana, Y. Ootomo, T. Hirose, J. Minaguchi, H. Ueda, M. Yamada and K. Takehana, Morphometric analysis of growing tenocytes in the superficial digital flexor tendon of piglets, J Vet Med Sci., 79 (2017), 1960-1967.  doi: 10.1292/jvms.17-0436. [48] C. T. Thorpe and H. R. C. Screen, Tendon structure and composition, Metabolic Influences on Risk for Tendon Disorders, Advances in Experimental Medicine and Biology, Springer, 920 (2016), 3-10. doi: 10.1007/978-3-319-33943-6_1. [49] G. Toscani, The grazing collisions asymptotics of the non-cut-off Kac equation, Esaim Math Model Numer Anal., 32 (1998), 763-772.  doi: 10.1051/m2an/1998320607631. [50] G. Toscani, One-dimensional kinetic models of granular flows, Esaim Math Model Numer Anal., 34 (2000), 1277-1291.  doi: 10.1051/m2an:2000127. [51] G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1. [52] F. Wu, M. Nerlich and D. Docheva, Tendon injuries: Basic science and new repair proposals, EFORT Open Rev., 2 (2017), 332-342.  doi: 10.1302/2058-5241.2.160075. [53] 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. [54] K. A. Young, J. A. Wise, P. DeSaix, D. H. Kruse, B. Poe, E. Johnson, J. E. Johnson, O. Korol, J. Gordon Betts and M. Womble, Anatomy & Physiology, OpenStax, 2013.
Collagen within a tendon has a hierarchical structure of increasing complexity: fibrils, fibres (primary bundles), fascicles (secondary bundles), tertiary bundles and a tendon itself [31,43]
Sequence of sagittal sections of a ruptured Achilles tendon taken before the reconstruction and within the first year after the reconstructive surgery. Yellow arrows indicate the tendon
Sequence of cross-sections of a ruptured Achilles tendon taken before the reconstruction and within first year after the reconstructive surgery. For comparison, the last image shows the cross-section of a healthy tendon. Red arrows indicate the tendon
Illustration of the connection between mathematical and biological objects. The left side of the image shows a bundle of interacting collagen fibres, whereas the right side shows its magnification at point $x$. Turning rate $T(x, \phi', \phi)$ (blue arrow) models the probability that collagen fibre with orientation $\phi'$ (black dashed line) rearranges into a fibre with orientation $\phi$ (solid black line). This turning rate is influenced by all fibres in the neighbourhood whose orientation (example denoted by $\theta$ and red dashed line) is close enough to $\phi$. The reverse action, that is the rearrangement form orientation $\phi$ to $\phi'$ obviously exists and is expressed by a green arrow with the $T(x, \phi, \phi')$ label. The vertical dotted line corresponds to the reference direction $\phi = 0$
Time evolution of the solution to the model (7) performed for reorientation range ${\varepsilon} = 10^{-3}\pi$ and different tenocyte action ranges $R$ (value indicated at the top of each column). Colour scale represents the density value
Characteristic time of the dynamic $\tau$ as a function of the tenocyte action range $R$ and reorientation range ${\varepsilon}$ (log scale). The dashed red line indicates the minimum value for epsilon for the orientation resolution to be fine enough
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