Analysis of cell growth kinetics and substrate diffusion in a polymer scaffold

The cultivation of cartilage cells (chondrocytes) in polymer scaffolds leads to implants that may potentially be used to repair damaged joint cartilage or for reconstructive surgery. For this technique to be medically applicable, the physical parameters that govern cell growth in a polymer scaffold must be understood. This understanding of cell behavior under in vitro conditions, where diffusion is the primary mode of transport of nutrients, may aid in the scale-up of the cartilage generation process. A mathematical model of chondrocyte generation and nutrient consumption is developed here to analyze the behavior of cell growth in a biodegradable polymer matrix for a series of different thickness polymers. Recent literature has implied that the diffusion of nutrients is a major factor that limits cell growth (Freed et al., 1994). In the present paper, a mathematical model is developed to directly relate the effects of increasing cell mass in the polymer matrix on the transport of nutrients. Reaction and diffusion of nutrients in the cell-polymer system are described using the fundamental species continuity equations and the volume averaging method. The volume averaging method is utilized to derive a single averaged nutrient continuity equation that includes the effective transport properties. This approach allows for the derivation of effective diffusion and rate coefficients as functions of the cell volume fraction. The cell volume fraction as a function of time is determined by solution of a material balance on cell mass. Growth functions including the Moser, a modified Contois, and an nth-order heterogeneous growth kinetic model are evaluated through a parameter analysis, and the results are compared to experimental data found in the literature. The results indicate that cellular functions in conjunction with mass transfer processes can account partially for the general trends in the cell growth behavior for various thickness polymers. The Contois growth function appeared to describe the data more accurately in terms of the lag period at early times and the long time limits. However, all kinetic growth functions required variations in the kinetic parameters to fully describe the effects of polymer thickness. This result implies that restricted diffusion of nutrients is not the sole factor limiting cell growth when the thickness of the polymer is changed. Therefore, further experimental data and model improvements are needed to accurately describe the cell growth process.