Mathematical modelling of flow and nutrient transport in hollow fibre membrane bioreactors
Keywords: solid mechanics
bioreactor
boundary layer analysis
hollow fibre membrane bioreactor
fluid mechanics
tissue engineering
asymptotic analysis

Hollow fibre membrane bioreactors comprise permeable tubes situated within a bioreactor chamber. Nutrients and growth factors flow through the fibre lumen, transporting across the porous membrane to external cells, while external flow removes residual nutrients and cellular waste. Given cellular sensitivity to their local microenvironment, mathematical modelling can be used to predict and optimise bioreactor performance.
In this thesis, we develop a series of continuum models describing fluid flow and metabolite transport in poroelastic hollow fibre membrane bioreactors, inspired by human vasculature, for applications in tissue engineering. We model flow across the membrane wall using Darcy's law, which is coupled to flow through the bioreactor governed by Navier-Stokes equations. The membrane is able to deform under the action of fluid stress, modelled as a poroelastic material using Terzaghi's principle. Exploiting disparate parameter ratios such as the long thin bioreactor geometry, large membrane stiffness and small cell layer thickness, we utilise asymptotic techniques to systematically reduce the complexity of our system, revealing mechanistic insight into the local chemical and mechanical environment of the tissue culture.
We extend existing models by considering fluid-structure interaction in a poroelastic hollow fibre membrane under steady viscous flow. We later extend consideration to pulsatile flow through our bioreactor system, where we derive generalisations of the Womersley solution for a deformable porous tube wall. These models reveal how fluid pressure, membrane compressibility, permeability and thickness, effect distributions of stress and strain through the tissue scaffold, and thus modulate the local cell environment. Coupling to advection-diffusion-reaction equations for generalised nutrient uptake kinetics, we reveal how the membrane permeability can be tuned to achieve uniform nutrient uptake and facilitate uniform tissue growth. Finally, we consider the canonical problem of high laminar Reynolds flow through two long thin channels mediated by a porous wall. We do so to emulate larger, more physiologically-relevant fluid shear stresses that cells would likely experience in arteries and large-diameter vessels. We explore the intricate boundary layer structure that emerges across a range of laminar Reynolds number regimes, validated by full numerical simulations, which provides an in-depth analysis of the coupled transitions from plug to lubrication flow.
The analysis we present in this thesis provides novel methods to investigate flow and metabolite transport in a poroelastic hollow fibre membrane bioreactor setup. Furthermore, our modelling framework facilitates quick and efficient sensitivity analysis to understand the impact of varying key structural and operating parameters on the local chemical and mechanical cell environment, which can be used to inform and optimise experimental bioreactor design.