Multiphase modelling of the effect of fluid shear stress on cell yield and distribution in a hollow fibre membrane bioreactor

Abstract

We present a simplified two-dimensional model of fluid flow, nutrient transport and cell distribution in a hollow fibre membrane bioreactor, with the aim of exploring how fluid flow can be used to control the distribution and yield of a cell population which is sensitive to both fluid shear stress and nutrient concentration. The cells are seeded in a scaffold in a layer on top of the hollow fibre, only partially occupying the extracapillary space. Above this layer is a region of free-flowing fluid which we refer to as the upper fluid layer. The flow in the lumen and upper fluid layer is described by the Stokes equations, whilst the flow in the porous fibre membrane is assumed to follow Darcy's law. Porous mixture theory is used to model the dynamics of and interactions between the cells, scaffold and fluid in the cell-scaffold construct. The concentration of a limiting nutrient (e.g. oxygen) is governed by an advection-reaction-diffusion equation in each region. Through exploitation of the small aspect ratio of each region and asymptotic analysis, we derive a coupled system of partial differential equations for the cell volume fraction and nutrient concentration. We use this model to investigate the effect of mechanotransduction on the distribution and yield of the cell population, by considering cases in which cell proliferation is either enhanced or limited by fluid shear stress and by varying experimentally controllable parameters such as flow rate and cell-scaffold construct thickness.

Fig. 1

Top Photograph of a single HFMB module (ruler scale in cm) as shown in Pearson et al. (2013). Bottom cross section of the boxed region (not to scale), where the lower half (not shown) has been excluded based on symmetry. This depicts the idealised two-dimensional modelling region with the $x$ axis running along the lumen centreline. The solid black arrows show the direction and location of the fluid fluxes into the system, and the star denotes the origin $(x,y)=(0,0)$

Fig. 2

Plot of $F(S)$, showing the dependence of cell proliferation on the fluid shear stress in the case where increasing shear stress first enhances cell proliferation (solid line, Sect. 4.1), and where increasing shear stress inhibits cell proliferation (dashed line, Sect. 4.2)

Fig. 3

Plots of ${\mathit{\theta }}_{\mathrm{n}}$ for a range of values of the upper fluid layer flux $Q_{\mathrm{f},\mathrm{in}}$, in the a shear-insensitive and b shear-sensitive (enhanced proliferation) cases. The lumen flux $Q_{\mathrm{l},\mathrm{in}}=0.1$, the cell layer width $h_3=1$ and other parameter values are as in Table 2

Fig. 4

Plots of (a) the mean $\mathit{\mu }$ and (b) the standard deviation $\mathit{\sigma }$ of ${\mathit{\theta }}_{\mathrm{n}}$, for a range of $Q_{\mathrm{f},\mathrm{in}}$ values in the shear-insensitive, shear-enhanced proliferation and shear-limited proliferation cases. The vertical dashed lines indicate the optimal flux range for the shear-enhanced case, within which the cell yield is higher than the shear-insensitive population. The vertical dash-dotted line indicates the critical flux value for the shear-limited case, below which the shear-limited and shear-insensitive cases are indistinguishable. The lumen flux $Q_{\mathrm{l},\mathrm{in}}=0.1$, the cell layer width $h_3=1$ and other parameter values are as in Table 2

Fig. 5

Plots of ${\mathit{\theta }}_{\mathrm{n}}$ in the shear-sensitive (limited proliferation) and shear-insensitive cases, for a range of values of the upper fluid layer flux $Q_{\mathrm{f},\mathrm{in}}$. The lumen flux $Q_{\mathrm{l},\mathrm{in}}=0.1$, the cell layer width $h_3=1$ and other parameter values are as in Table 2

Fig. 6

Plot of $\mathit{\partial }{p}_{\mathrm{w}}/\mathit{\partial }x$ versus the cell layer width $h_3$, for fixed $Q_{\mathrm{f},\mathrm{in}}=1$

Fig. 7

Plots of the mean $\mathit{\mu }$ and standard deviation $\mathit{\sigma }$ of ${\mathit{\theta }}_{\mathrm{n}}$, for a range of $Q_{\mathrm{f},\mathrm{in}}$ values in the shear-insensitive, shear-enhanced proliferation and shear-limited proliferation cases, for (a, b) $h_3=0.5$ and (c, d) $h_3=1.5$. The vertical dashed lines again indicate the optimal flux range for the shear-enhanced case, and the vertical dash-dotted line the critical flux value for the shear-limited case. The lumen flux $Q_{\mathrm{l},\mathrm{in}}=0.1$ and other parameter values are as in Table 2