Because of transport limitations, nutrient and growth factor concentrations vary significantly inside scaffolds and nonideal bioreactors, leading to large differences in the state and behavior of cells. However, population heterogeneity also results from the complex regulatory mechanisms that occur at the single-cell level. In addition, the single-cell and scaffold/bioreactor scales are tightly coupled through complex cell-cell and cell-environment interactions, as well as interactions occurring among the numerous cellular components. Clearly, comprehensive mathematical models are necessary for integrating the biological processes occurring at the bioreactor, tissue, cell and sub-cellular scales.
This study will first investigate the hypothesis that time-dependent addition of nutrients and/or growth factor to the media can be used to modulate the structure of regenerating tissues. To test this hypothesis, we will use a hybrid multi-scale comprehensive model consisting of (a) partial differential equations quantifying the simultaneous diffusion, convection and consumption of nutrients and growth factors in the interior of the scaffolds; and (b) a discrete model that tracks the migration and proliferation of heterogeneous cell populations on a three-dimensional cubic lattice. An implicit-explicit finite difference method is employed to discretize the transient PDE's and the resulting sparse linear system is solved with a preconditioned GMRES method. The computed local concentrations of nutrients or growth factors are used to modulate the cell locomotory properties and their division times. We assume here that tissue growth is limited by the availability of a key nutrient. Experimentally determined algebraic relations (Monod) are used in this case to modulate the cell proliferation rates and migration speeds according to the local concentrations of the critical nutrient.
To account for different cell phenotypes, the model considers cell populations that exhibit bimodal distributions of migration speeds, persistence and division times. The model also allows for cell death when the nutrient concentrations drop below a critical level for a given period of time. These two parameters can be different for each cell phenotype.
Simulation of 3D tissue growth is a computationally challenging problem requiring large grids to capture the growth of tissues of practically significant size. To meet this challenge, our algorithm has been parallelized to run on a Cray XD1 computer cluster. The computational domain is partitioned into subdomains along one dimension (slab decomposition) and each subdomain is assigned to a different processor.
For a given set of locomotory and metabolic parameter values, we will show that the initial conditions (cell seeding and initial nutrient concentration profile) influence the spatial distribution of the two cell subpopulations, as the cells segregate in different regions according to their phenotype. The structure of regenerated tissues can also be modulated by temporal variations of the boundary conditions (which can be of the Dirichlet, Neumann or mixed type). Boundary conditions can be dynamically manipulated by varying either the nutrient concentration in the culture media surrounding the scaffold or other operating conditions (like stirring speed or external fluid flow) that directly affect the external mass transfer coefficient. These results provide us with guidelines for a rational design of experiments that will lead to the formation of tissues with desired architecture.
Finally, we explore the effect of vascularization on the growth rates and structure of regenerated tissues. “Artificial” vessels with cylindrical or other shapes are placed in the 3D scaffolds to enhance the transport of nutrients to the interior of developing tissues. We will present simulation results that show how tissue development is affected by (a) the geometry of the artificial vessels, and (b) temporal variations in the flow rates of culture media or the concentrations of limiting nutrients.