Network modelling of bioactive porous media

In this thesis we consider several approaches to modelling interactions between fluid flow and cell proliferation in a bioactive porous medium. This is motivated by models of cell growth within tissue engineering scaffolds placed in perfusion bioreactors. These scaffolds are porous materials used to facilitate nutrient transport to cells placed within them. Recent modelling efforts have sought to understand the influence that cells have on the effective permeability of these tissue scaffolds, and hence on the capacity of the porous medium to facilitate fluid and nutrient transport, which enhances the overall growth of tissue within a scaffold. Contemporary experimental and theoretical studies have emphasized the importance of mechanical forcing on cells due to their environment. We therefore consider simple models of cell growth that assume cells are affected by the fluid itself, either via shear stress, hydrostatic pressure, or local flow rate, and that cell growth influences the local permeability of the scaffold at higher cell densities via pore-blocking. Hence, we are interested in the feedback between cell growth and fluid flow within a bioactive porous medium. It is within this simplified context of fluid-growth interaction that we explore different approaches to modelling the spatial structure of the pore network, and the influence that this has on the cell density distribution throughout the scaffold.
Models in the literature are often spatially homogeneous (ODE) or spatially continuous (PDE), only implicitly accounting for the discrete pore network of the medium. The pore and scaffold length scales in typical experiments with perfusion bioreactors can lead to scaffolds with relatively few pores, and it is unclear that macroscopic spatially averaged (homogenized) continuous models will capture features present in small pore networks. We propose a suite of spatially continuous models, modified from existing PDE approaches in the literature, and compare these to discrete lattice (ODE) systems in order to elucidate differences between these modelling paradigms. We also use the structure of the lattice model to explore stochastic analogues that are computationally and theoretically amenable to analysis. We explore behaviours of these models via numerical simulations, bifurcation analyses, and asymptotic reductions to simpler systems.
Our lattice modelling approach provides a 'mesoscopic' perspective, where the effective equations governing the cell growth and fluid flow are prescribed as ODEs at each pore. These capture the microscale dynamics at the pore scale, as opposed to homogenization approaches where the microscale is explicitly related to macroscale equations. This also allows us to use tools from dynamical systems theory which are substantially more tractable in the finite dimensional setting of ODEs (with algebraic conditions for the fluid flow), as opposed to the infinite dimensional setting of PDEs (consisting of coupled elliptic and parabolic equations). Additionally, we do not have to worry about issues of numerical convergence or existence of solutions that are important in the spatially continuous setting. Our emphasis on mesoscopic governing equations makes this network-based approach somewhat unique in the tissue engineering community.
We demonstrate qualitative and quantitative differences between these continuum and network paradigms, and in particular show behaviours captured by explicitly accounting for the discrete pore network that are not captured in spatially continuous models. For each kind of fluid-cell interaction, we classify behaviours depending on nondimensional model parameters in order to elucidate in what regimes discrete models may provide useful insights into bioactive porous materials. We also discuss computational considerations for analyzing these kinds of models, and in particular suggest that stochastic and discrete models may be easier to simulate in some parameter regimes compared to typical spatially continuous models. Our results suggest several novel approaches to pursue in accounting for the finite discrete nature of bioactive porous media, and we highlight several useful further directions.

Advisor: Belyaev, D
Waters, S
van Gorder, R