The role of mass transport is to ensure that cell metabolism is kept within a physiological range by provision of metabolic substrates and removal of toxic degradation products. Understanding how device geometry is related to convective or diffusive transport limitations is therefore a key element of bioreactor design.
The role of diffusion and convection
There is a range of substrates that need to be supplied to the biomass. The rate of diffusion is proportional to their concentration gradient, the constant of proportionality being the diffusion coefficient. The Stokes–Einstein equation relates the radius of the diffusing particle and temperature to the diffusion coefficient:
where k is Boltzmann's constant, T is temperature and R is the particle radius.24 The volume of a sphere is proportional to the cube of its radius, , so the diffusion coefficient is approximately inversely related to the cube root of molecular weight. Larger substrates including low-density lipoproteins, free fatty acids, transferrin and growth factors will have a lower diffusion coefficient. Their specific cellular uptake (mol s−1 per cell) is many orders of magnitude lower than oxygen, as is the molar concentration required for binding to cell surface receptors (nano- to picomolar range). The diffusion of growth factors is also limiting at low concentration, particularly if high density culture is not supplemented with additional growth factors.
The mass flux is also related to the gradient that can be generated at the cell/medium interface (Fick's law of diffusion). For a stagnant boundary around a cell the mass transfer is by diffusion alone, and is limited by the thickness of the stagnant layer and the concentration at the boundary of the stagnant layer. Mass transfer is increased by reducing the thickness of the stagnant boundary layer surrounding cells. The metabolite with the lowest solubility relative to specific uptake (Table 1) is oxygen: approximately 0.2 mmol L−1 in room air (partial pressure of oxygen is 0.2 atm) at 37 °C. Flask culture systems rely primarily on diffusion, and to a lesser extent natural convection, for transport of oxygen to cells. The depth of media in the flask limits the supply of oxygen from the gas phase.25
Within a polymer scaffold (e.g., polygycolic acid), such as those used to generate cartilage,26 the material properties of the system are spatially and temporally heterogeneous. Galban et al. considered a seeded polymer scaffold as consisting of two phases:27, 28 a void phase (β), which contains the nutrient fluid and some polymer matrix, and the cell phase (γ), which includes the cells, nutrient fluid, extracellular matrix and some polymer matrix. Diffusive transport within the two phase system is modelled by coupling boundary conditions and diffusion equations for both phases (β, γ). The volume-averaging method was utilized to derive a single averaged nutrient continuity equation that allows calculation of effective diffusion coefficients as a function of cell volume fraction and time. Leddy et al. measured the diffusivity of tissue-engineered cartilage as a function of scaffold material, culture conditions and time in culture.29 Diffusivity in these constructs was much greater than in native cartilage. A decrease in diffusivity over time was most likely related to new matrix synthesis and matrix contraction by cells in the fibrin and gelatin scaffolds.
The role of convection is to reduce diffusion limitations imposed by stagnant boundary layers surrounding cells. Hydrodynamic modelling can be used to predict the flow field for various geometries by solution of the Navier–Stokes equations. This is only trivial for simple geometries that generate well-defined flow fields such as parallel plates,30, 31 membranes32 or hollow-fibre bioreactors.33
Improving local perfusion of thick tissue constructs remains a significant challenge for scaffold-based devices. Static culture of cell-seeded 3D scaffolds typically produces thin tissue growth localized to the construct periphery.34 Scaffolds can be perfused by housing them within a flow-through column,35–37 or by suspending them within rotary culture devices or spinner flasks.38 While flow rate can be used to modulate media exchange, pore sizes, connectivity and anisotropy can impart vastly different rates of media exchange and shear stress on cells within the construct.
Calculating flow-mediated shear stress within a porous scaffold is not a trivial exercise. Porter et al. utilized computational fluid dynamics to model the flow of media through scaffolds.34 Micro-computed tomography was used to define scaffold geometry, with generation of a detailed simulation of the flow field within pores. The authors were able to estimate average shear stress within the scaffold and to estimate the fluid shear stresses that correlate with increased osteogenic proliferation. Zhao et al. extended this numerical approach to include terms for the diffusion and consumption of oxygen within the scaffold.22