IJPAM: Volume 53, No. 3 (2009)


Sarah D. Olson$^1$, Mansoor A. Haider$^2$
$^1$Department of Mathematics
Tulane University
6823, St. Charles Ave., New Orleans, LA 70118, USA
e-mail: [email protected]
$^2$Department of Mathematics
North Carolina State University
P.O. Box 8205, Raleigh, NC 27695-8205, USA
e-mail: [email protected]

Abstract.A mathematical model and numerical solutions are presented for an interface problem that models an in-vitro experiment for regeneration of articular cartilage in a localized defect region. In this experiment, a cylindrical cartilage explant has a core region removed and replaced with a nutrient-rich hydrogel. The gel-tissue aggregate is then immersed in media for a period of several weeks. An axisymmetric reaction-diffusion model of this experiment is developed to capture coupling between cell-mediated nutrient absorption and matrix biosynthesis, and diffusive transport of nutrients and matrix constituents. The reaction governing turnover of the hydrogel to newly synthesized tissue is modeled via a level set method that captures the moving gel-tissue interface, and local curvature effects are also considered. After nondimensionalization, finite difference numerical solutions are employed to simulate cartilage regeneration as a function of cell mediated reaction rates in the model. Both the cases of external media maintained at a homeostatic nutrient concentration, and at a higher concentration associated with the nutrient-rich hydrogel are considered. Via a detailed parametric analysis using the model, regeneration times required to completely degrade the hydrogel are determined. Potential effects of local curvature along the gel-tissue interface are briefly discussed.

Received: April 28, 2009

AMS Subject Classification: 92, 35, 65

Key Words and Phrases: reaction-diffusion model, cartilage regeneration, level set method, tissue engineering, hydrogel, interface problem

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 53
Issue: 3