Mathematical modeling of elastodynamics and cell growth inside a deformable scaffold fitted to the periphery of a bioreactor

A mathematical model is developed for fluid flow, nutrient transport, and cell growth inside a bioreactor with a scaffold at the periphery and lumen at the centerline. The scaffold material is assumed to be deformable. In order to deal with the deformation of the solid phase and fluid phase inside the scaffold, biphasic mixture theory equations are adopted, which are derived from the theory of mixtures. The flow inside the lumen is governed by Stokes equation. Advection-diffusion-reaction equation is used for the mass balance of the nutrient within the scaffold region. Cell growth depends on the nutrient concentration and is expressed by the Contois equation that accounts for contact inhibition. We use lubrication approximation to reduce the system of hydrodynamic and nutrient transport equations. This leads to a coupled system of partial differential equations (PDEs) with time-dependent variables. Laplace transformation is used to deal with time-dependent terms, and Durbin's algorithm is used to retrieve the time dependency. We investigate the outreach of nutrients inside the scaffold region, which regulates the growth of cells at a particular time. Based on the available experimental data, we consider relevant reaction kinematics of the cells and observe the corresponding nutrient distribution inside the bioreactor. The factors that affect the nutrient concentration are lumen radius, porosity, and permeability of the scaffold, Thiele modulus, pressure gradient, and so forth. The total mass transfer rate is computed to understand the nutrient distribution across various regions of the bioreactor.

Automated summary

This paper develops a mathematical model to study fluid flow, nutrient transport, and cell growth inside a bioreactor. The model includes deformable scaffold material, biphasic mixture theory equations, Stokes equation for flow, advection-diffusion-reaction equation for nutrient balance, Contois equation for cell growth, and the use of Laplace transformation to handle time-dependent terms. The study investigates nutrient outreach in the scaffold region, factors affecting nutrient concentration, and computes total mass transfer rate for nutrient distribution across different regions of the bioreactor.