Mathematical modelling of collagen fibres rearrangement during the tendon healing process

José Antonio Carrillo; Martin Parisot; Zuzanna Szymańska

doi:10.3934/krm.2021005

Article Contents

2021, Volume 14, Issue 2: 283-301. Doi: 10.3934/krm.2021005 open access

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

Received: April 2020

Revised: September 2020

Early access: January 2021

Published: April 2021

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35R09, 92-08, 92-10; Secondary: 92B05, 92C50.

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Figure 1. 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]

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Figure 2. 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

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Figure 3. 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

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Figure 4. 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 $

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Figure 5. 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

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Figure 6. 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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