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A Phase-Field Model for Articular Cartilage Regeneration in Degradable Scaffolds

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Abstract

Degradable scaffolds represent a promising solution for tissue engineering of damaged or degenerated articular cartilage which due to its avascular nature, is characterized by a low self-repair capacity. To estimate the articular cartilage regeneration process employing degradable scaffolds, we propose a mathematical model as the extension of Olson and Haider’s work (Int. J. Pure Appl. Math. 53:333–353, 2009). The simulated tissue engineering procedure consists in (i) the explant of a cylindrical sample, (ii) the removal of the inner core region, and (iii) the filling of the inner region with hydrogels, degradable scaffolds enriched with nutrients, such as oxygen and glucose. The phase-field model simulates the cartilage regeneration process at the scaffold-cartilage interface. It embeds reaction-diffusion equations, which are used to model the nutrient and regenerated extracellular matrix. The equations are solved using an unconditionally stable hybrid numerical scheme. Cartilage repair processes with full-thickness defects, which are controlled by properties of hydrogel materials and cartilage explant culture based on biological interest are observed. The implemented mathematical model shows the capability to simulate cartilage repairing processes, which can be virtually controlled evaluating hydrogel and cartilage material properties including nutrient supply and defected magnitude. In particular, the adopted methodology is able to explain the regeneration time of cartilage within hydrogel environments. With the numerical scheme, the numerical simulations are demonstrated for the potential improvement of hydrogel structures.

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Acknowledgements

The first author (A. Yun) was supported by National Junior research fellowship from the National Research Foundation of Korea grant funded by the Korea government (No. 2011-00012258). The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0027580). The authors also wish to thank the anonymous referee for the constructive and helpful comments on the revision of this article.

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Authors and Affiliations

  1. Department of Mathematics, Korea University, Seoul, 136-713, Republic of Korea

    Ana Yun & Junseok Kim

  2. Department of Orthopedic Surgery, Korea University College of Medicine, Seoul, 136-705, Republic of Korea

    Soon-Hyuck Lee

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  1. Ana Yun
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Correspondence to Junseok Kim.

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Yun, A., Lee, SH. & Kim, J. A Phase-Field Model for Articular Cartilage Regeneration in Degradable Scaffolds. Bull Math Biol 75, 2389–2409 (2013). https://doi.org/10.1007/s11538-013-9897-3

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  • DOI: https://doi.org/10.1007/s11538-013-9897-3

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