A multiphase model for chemically- and mechanically- induced cell differentiation in a hollow fibre membrane bioreactor: minimising growth factor consumption

Abstract

We present a simplified two-dimensional model of fluid flow, solute transport, and cell distribution in a hollow fibre membrane bioreactor. We consider two cell populations, one undifferentiated and one differentiated, with differentiation stimulated either by growth factor alone, or by both growth factor and fluid shear stress. Two experimental configurations are considered, a 3-layer model in which the cells are seeded in a scaffold throughout the extracapillary space (ECS), and a 4-layer model in which the cell–scaffold construct occupies a layer surrounding the outside of the hollow fibre, only partially filling the ECS. Above this is a region of free-flowing fluid, referred to as the upper fluid layer. Following previous models by the authors (Pearson et al. in Math Med Biol, 2013, Biomech Model Mechanbiol 1–16, 2014a, we employ porous mixture theory to model the dynamics of, and interactions between, the cells, scaffold, and fluid in the cell–scaffold construct. We use this model to determine operating conditions (experiment end time, growth factor inlet concentration, and inlet fluid fluxes) which result in a required percentage of differentiated cells, as well as maximising the differentiated cell yield and minimising the consumption of expensive growth factor.

Appendix: Dimensionless equations

Following the non-dimensionalisation in Sect. 3.2, we present the dimensionless equations in the remaining layers of the modelling domain. Beginning in the lumen, we have

$$\begin{aligned}&\frac{\partial u_{\mathrm {l}}}{\partial x} + \frac{\partial v_{\mathrm {l}}}{\partial y} = 0,\quad -\frac{\partial p_{\mathrm {l}}}{\partial x} + \varepsilon ^2\frac{\partial ^2 u_{\mathrm {l}}}{\partial x^2} + \frac{\partial ^2 u_{\mathrm {l}}}{\partial y^2} = 0,\nonumber \\&-\frac{\partial p_{\mathrm {l}}}{\partial y} + \varepsilon ^4\frac{\partial ^2 v_{\mathrm {l}}}{\partial x^2} + \varepsilon ^2\frac{\partial ^2 v_{\mathrm {l}}}{\partial y^2} =0,\\&\varepsilon ^2\mathrm {Pe}\left( \lambda \varepsilon \frac{\partial c_{\mathrm {l}}}{\partial t} + \mathbf {\nabla }\cdot (c_{\mathrm {l}}\mathbf {u}_{\mathrm {l}})\right) = \varepsilon ^2\frac{\partial ^2 c_{\mathrm {l}}}{\partial x^2} + \frac{\partial ^2 c_{\mathrm {l}}}{\partial y^2}.\nonumber \end{aligned}$$

(6.1)

In the porous membrane, the dimensionless equations are

$$\begin{aligned}&u_{\mathrm {m}} = -\varepsilon ^2 \kappa _{\mathrm {m}}\frac{\partial p_{\mathrm {m}}}{\partial x},\quad v_{\mathrm {m}} = -\kappa _{\mathrm {m}}\frac{\partial p_{\mathrm {m}}}{\partial y},\nonumber \\&\quad \varepsilon ^2 \frac{\partial ^2 p_{\mathrm {m}}}{\partial x^2} + \frac{\partial ^2 p_{\mathrm {m}}}{\partial y^2} = 0,\\&\lambda \varepsilon ^3\mathrm {Pe}\left( \frac{\partial c_{\mathrm {m}}}{\partial t} + \mathbf {\nabla }\cdot (c_{\mathrm {m}}\mathbf {u}_{\mathrm {m}})\right) = \varepsilon ^2\frac{\partial ^2 c_{\mathrm {m}}}{\partial x^2} + \frac{\partial ^2 c_{\mathrm {m}}}{\partial y^2},\nonumber \end{aligned}$$

(6.2)

where $\kappa = k/(\lambda \varepsilon ^5 L^2)$ is the $\text {O}(1)$ dimensionless permeability (see Table 3). Finally, in the 4-layer model the upper fluid layer equations are

$$\begin{aligned}&\frac{\partial u_{\mathrm {f}}}{\partial x} + \frac{\partial v_{\mathrm {f}}}{\partial y} = 0,\quad -\frac{\partial p_{\mathrm {f}}}{\partial x} + \varepsilon ^2\frac{\partial ^2 u_{\mathrm {f}}}{\partial x^2} + \frac{\partial ^2 u_{\mathrm {f}}}{\partial y^2} = 0,\nonumber \\&\quad -\frac{\partial p_{\mathrm {f}}}{\partial y} + \varepsilon ^4\frac{\partial ^2 v_{\mathrm {f}}}{\partial x^2} + \varepsilon ^2\frac{\partial ^2 v_{\mathrm {f}}}{\partial y^2} =0,\\&\varepsilon ^2\mathrm {Pe}\left( \lambda \varepsilon \frac{\partial c_{\mathrm {f}}}{\partial t} + \mathbf {\nabla }\cdot (c_{\mathrm {f}}\mathbf {u}_{\mathrm {f}})\right) = \varepsilon ^2\frac{\partial ^2 c_{\mathrm {f}}}{\partial x^2} + \frac{\partial ^2 c_{\mathrm {f}}}{\partial y^2}.\nonumber \end{aligned}$$

(6.3)