Solute transport with Michaelis-Menten kinetics for in vitro cell culture

A traditional method of in vitro cell culture involves a monolayer of cells at the base of a petri dish filled with culture medium. While the primary role of the culture medium is to supply nutrients to the cells, drug or other solutes may be added, depending on the purpose of the experiment. Metabolism by cells of oxygen, nutrients and drug is typically governed by Michaelis-Menten (M-M) kinetics. In this paper, a mathematical model of solute transport with M-M kinetics is developed. Upon non-dimensionalization, the reaction/diffusion system is re-characterized in terms of Volterra integral equations, where a parameter $\beta $, the ratio of the initial solute concentration to the M-M constant, proves important: $\beta \ll 1$ is relevant to drug metabolism for the liver, whereas $\beta \gg 1$ is more appropriate in the case of oxygen metabolism. Regular perturbation expansions for both cases are obtained. A small-time expansion and steady-state solution are also presented. All results are compared against the numerical solution of the Volterra integral equations, and excellent agreement is found. The utility of the model and analytical solutions are discussed in the context of assisting experimental researchers to better understand the environment within in vitro cell culture experiments.