A phase-field model for articular cartilage regeneration in degradable scaffolds.

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.

Mathematical model and numerical simulation of the cell growth in scaffolds.

A scaffold is a three-dimensional matrix that provides a structural base to fill tissue lesion and provides cells with a suitable environment for proliferation and differentiation. Cell-seeded scaffolds can be implanted immediately or be cultured in vitro for a period of time before implantation. To obtain uniform cell growth throughout the entire volume of the scaffolds, an optimal strategy on cell seeding into scaffolds is important. We propose an efficient and accurate numerical scheme for a mathematical model to predict the growth and distribution of cells in scaffolds. The proposed numerical algorithm is a hybrid method which uses both finite difference approximations and analytic closed-form solutions. The effects of each parameter in the mathematical model are numerically investigated. Moreover, we propose an optimization algorithm which finds the best set of model parameters that minimize a discrete l(2) error between numerical and experimental data. Using the mathematical model and its efficient and accurate numerical simulations, we could interpret experimental results and identify dominating mechanisms.

An immersed boundary method for simulating a single axisymmetric cell growth and division.

For an animal cell, cytokinesis is the process by which a cell divides its cytoplasm to produce two daughter cells. We propose a new mathematical model for simulating cytokinesis. The proposed model is robust and realistic in deciding the position of the cleavage furrow and in defining the contractile force leading to cell division. We use an immersed boundary method to track the morphology of cell membrane during cytokinesis. For accurate calculation, we adaptively add and delete the immersed boundary points. We perform numerical simulations on the axisymmetric domain to have sufficient resolution and to incorporate three-dimensional effects such as anisotropic surface tension. Finally, we investigate the effects of each model parameter and compare a numerical result with the experimental data to demonstrate the efficiency and accuracy of our proposed method.