Analysis of cell growth in a polymer scaffold using a moving boundary approach

Two mathematical models of chondrocyte generation and nutrient consumption are developed to analyze the behavior of cell growth in a biodegradable polymer matrix. Substrate reaction and diffusion are analyzed in two regions: one consisting of cells and nutrients and the other consisting of only nutrients. A pseudo‐steady state approximation for the transport of nutrients in these two regions is utilized. The rate of growth is determined by a moving boundary equation that equates the rate at which the interfacial region between the cells and the void space moves to a substrate dependent growth reaction. The change in the location of this interfacial region with time therefore depicts the rate at which the cells propagate. The two limiting cases discussed in this article represent extremes in how the cells will grow in the polymer matrix; one case assumes that cells grow inward from the external boundary, and the other case assumes that cells grow parallel to the external boundary. The results of both models are compared to experimental data found in the literature. It is found through these comparisons that the model parameters, including the unit cell spacing parameter L, the metabolic rate constant k, the growth rate constant kG, and external mass transfer coefficient, K, may vary as the thickness of the polymer matrix is changed, however, unrealistic and large changes in the diffusion coefficients were required to account for the full range of experimental data. Furthermore, these results suggest modification of the functional form of the growth kinetics to include substrate or product inhibition, or death terms. Based upon diffusion/reaction concepts, these models for cell growth in a biodegradable polymer give bounds for the upper and lower limits of the cellular growth rate and nutrient consumption in a polymer matrix and will aid in the development of more extensive models. © 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 422–432, 1997.