Mathematical Modelling of Collagen Fibers Rearrangement During Tendon Healing Process

José A. Carrillo (1) , Martin Parisot (2) , Zuzanna Szymańska (3)

1 MI - Mathematical Institute [Oxford]
2 CARDAMOM - Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts
3 PAN - Institut of Mathematics - Polish Academy of Sciences

Tendon injuries present a significant clinical challenge to modern medicine as they heal slowly and rarely recover the structure and mechanical strength of a healthy tendon. Moreover, tendon represents a highly under-researched tissue with the process of healing not fully elucidated. To improve the understanding of tendon function and healing process we propose a new model of collagen fibers rearrangement during tendon 1 healing. The model consists of integro-differential equation describing the dynamics of collagen fibers distribution. We further reduce the model in a suitable asymptotic regime leading to a nonlinear non-local Fokker-Planck type equation for the spatial and orientation distribution of col-lagen fiber bundles. The reduced model allows for possible parameter estimation based on data due to its simplicity. We showcase some of the qualitative properties of this model simulating its long time asymptotic behavior and the total time for tendon fibers 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 the tendon cell size in patients' recovery.

Mots clés: Collagen remodelling; Mathematical model; Tendon healing; Integro-differential equations; Alignment

Domaines: Equations aux dérivées partielles [math.AP]; Biomécanique [physics.med-ph]

José A. Carrillo, Martin Parisot, Zuzanna Szymańska. Mathematical Modelling of Collagen Fibers Rearrangement During Tendon Healing Process. Kinetic and Related Models , 2021, ⟨10.3934/krm.2021005⟩. ⟨hal-02537528⟩