Abstract
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.
Highlights
Tendon injuries, not directly threatening the lives of affected persons can significantly lower their quality of life [39]
The authors propose an integro-differential equation describing the dynamics of the probability density of collagen fibers in space and orientation
The obtained results suggest that the range of spatial interactions influences the dynamics a lot whereas, the size of the orientation kernel does not have a major impact on the result, see Figure 5, results corresponding to small range interactions both in space and orientation; small range interaction in space and long-range interactions in orientation; and long-range interaction in space and small range interactions in orientation, are shown in columns A, B, and C, respectively
Summary
Not directly threatening the lives of affected persons can significantly lower their quality of life [39]. With the development of regenerative medicine, new treatment perspectives have emerged Their effective implementation requires a thorough understanding of the healing process, which due to specific tendon morphology, low cellularity, and poor blood supply, is difficult and where mathematical modeling may provide additional insight [38]. The authors propose an integro-differential equation describing the dynamics of the probability density of collagen fibers in space and orientation They show that depending on initial data solutions may either exist globally in time or blow-up in L∞, i.e. concentration in orientation, in a finite time.
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