A macroscopic model for species transport during in vitro tissue growth obtained by the volume averaging method

In this work, we study the possibility of deriving a macroscopic model to describe reaction and transport by diffusion and convection of two species within a porous medium as encountered during in vitro tissue growth. The starting point is a boundary value problem of diffusion-advection and reaction in a three-phase system, the two species being identified as the nutrient for cell growth and the metabolic product within the framework of tissue culture. The method of volume averaging is applied to the set of microscopic equations. Under the local mass equilibrium assumption and a series of constraints on the parameters of the system that are identified, one obtains a one-equation macroscopic model corresponding to a dispersion-reaction equation. Associated closure problems allowing the computation of effective coefficients that appear in this macroscopic model are provided.