#### Setup

The fibers in a HFB fiber bundle are assumed to be Krogh cylinders, so that each fiber is identical and surrounded by an annulus of ECS containing a homogeneous distribution of cells (Krogh, 1918). The interstitial space between the Krogh cylinders is neglected as a modeling assumption. In this study, we consider transport in a single Krogh cylinder unit of a HFB bundle. This unit consists of a central lumen with a synthetic porous wall (referred to as the membrane), and surrounding ECS containing cells. Let *z* be the axial direction down the lumen centerline, starting at the lumen inlet (*z* = 0) with the lumen outlet denoted by *z* = *L*. We denote the radius of the lumen by *d*, the depth of the membrane by *s* and the depth of the ECS by *l*. Typical values are *L* = 10 cm, *d* = 100 µm, *s* = 20 µm, and *l* = 600 µm (Ye et al., 2007), although these should be varied as part of the bioreactor design process. A schematic of the setup is given in Figure 1.

Culture medium is pumped through the lumen at an imposed flowrate. There is no flow through the inlet to the membrane or ECS, so that fluid enters the system through the lumen only. Although this medium includes a mixture of solutes and proteins, we consider the transport of oxygen alone in this article. This is a widely adopted approach in the literature as oxygen is generally considered to be the rate-limiting nutrient, and reduces the complexity of the modeling process (Martin and Vermette, 2005; Piret and Cooney, 1991). Oxygen is transported by both advection (by the fluid) and diffusion in the lumen. Furthermore, oxygen diffuses through the membrane and ECS, where it is taken up by the cell population. In the analysis that follows we assume that the cell population is homogeneously distributed throughout the ECS, and neglect expansion of the cell population so that the parameters describing oxygen uptake are constant in time.

Fluid flow in the lumen is described by Poiseuille's law whereas flow in the membrane and ECS is neglected (this is a common modeling assumption for small aspect ratio HFB when there is not a significant pressure drop across the membrane or ECS (Brotherton and Chau, 1996; Piret and Cooney, 1991)). We denote this fluid velocity in the lumen by , where *U* is the mean velocity (ms^{−1}), *r* is the radial coordinate, and **e**_{z} is the unit vector in the *z*-direction. The oxygen concentration and flux are denoted by *c* (mol m^{−3}) and **J** (mol m^{−2} s^{−1}), respectively, with subscripts *l*, *m*, and *e* denoting the values in the lumen, membrane, and ECS, respectively. The oxygen fluxes are

- (1)

where *D*_{l}, *D*_{m}, and *D*_{e} are the diffusion coefficients for oxygen in the lumen, wall, and ECS, respectively (all assumed constant, with units m^{2} s^{−1}). The lumen oxygen flux is comprised of advection due to the fluid velocity, together with diffusion; the membrane and ECS fluxes are comprised of diffusion only. The conservation equations for the concentration of oxygen in each of the regions are:

- (2)

where the reaction term *R*(*c*_{e}) captures the uptake of oxygen by the cells. We will assume Michaelis–Menten kinetics for this reaction term, so that

- (3)

It is necessary to prescribe boundary conditions on the internal and external boundaries of the bioreactor. On the lumen/membrane and membrane/ECS boundaries we prescribe continuity of concentration and flux, so that

- (4)

- (5)

where **n** is the unit outward pointing normal to the relevant surface. Finally we prescribe the oxygen concentration as *c*_{in} (mol m^{−3}) at the lumen inlet (where *c*_{in} may be chosen to suit the application under consideration), and impose no flux of concentration out of the outer ECS boundary,

- (6)

The assumption of no flux out of the outer boundary is analogous to a symmetry condition representation of a bundle of fibers. It compares directly to the Krogh cylinder approach used frequently in the literature.

Next the solution of the model (2)–(6) is considered using numerical or analytical techniques. For both strategies a steady-state solution is sought and it is assumed that a 2D axisymmetric geometry is described by the radial coordinate and the axial coordinate *z*.

#### Analytical Approach

To pursue an analytical approach, the system of equations given by (2)–(6) can be simplified with various assumptions. First of all the small aspect ratio of a fiber is exploited, defined by . It should be noted that whilst the lumen radius, *d* and fiber length, *L* can both be varied as part of the design process so that neither *d* nor *L* are fixed, *ε* ≪ 1 will be maintained throughout.

It is not possible to make progress analytically using the nonlinear Michaelis–Menten reaction term given by (3). Therefore, we assume that *c*_{e} ≫ *K*_{m} so that the reaction term *R*(*c*_{e}) can be approximated by *V*_{max}. This is an important assumption and means that predictions of the analytical model are only valid when the ECS oxygen concentration is much larger than the half-maximal oxygen concentration. As such, for cell types where the demand for oxygen is similar to, or smaller than, *K*_{m} it will not be appropriate to use the analytical model (in this scenario a numerical approach should be used, as outlined later in the article).

Finally the relative importance of advection and diffusion in the lumen is evaluated by considering the Péclet number, *Pe* = *UL*/*D*_{l}. In fact it is the reduced Péclet number, , that is critical for this system, as it also takes account of the small aspect ratio of the lumen (it is analogous to the *reduced Reynolds number* that was used to characterize fluid transport for a similar study in Shipley et al. (2010)). A large reduced Péclet number indicates an advection-dominated regime, whereas a small reduced Péclet number indicate a diffusion-dominated regime. Typically for this system *U* ≈ 1 cm s^{−1}, *L* ≈ 10 cm, and *D* ≈ 10^{−9} m^{2} s^{−1}, giving *Pe*^{*} ≈ 1 so that advection and diffusion are both important in the lumen. It is assumed that *Pe*^{*} = ε^{2}*Pe* is of order 1 in the analysis that follows. For the mathematical detail of the reduction of (2)–(6) based on the assumptions above, together with the solution of the resulting model, please refer to the Supplementary Material A.

The outer radius of the lumen, membrane and ECS (each measured from the lumen centerline) are denoted by *R*_{l}, *R*_{w}, and *R*_{e} so that *R*_{l} = *d*, *R*_{m} = *d* + *s*, and *R*_{e} = *d* + *s* + *l*. The following dimensionless parameters are also defined:

- (7)

which capture the key physical features of the system. As described above, *Pe*^{*} is the reduced Péclet number and is assumed to be of order 1. The parameter *M* represents the balance of oxygen consumption versus diffusion in the ECS, and can take a range of values depending on the relative importance of these effects.

Although Equations (8)–(10) appear complex, the behavior that they describe is relatively straightforward to understand: the oxygen concentration in the lumen, membrane, and ECS depends on the radial distance from the lumen centerline. Each solution is also dependent on the distance down the lumen centerline, *z*, as a consequence of advection in the lumen. This is transmitted into the membrane and ECS regions through the function *B*(*z*), which is the lumen concentration value on the lumen wall (i.e., the solution in (8) when *r* = *R*_{l} = *d*). This function *B*(*z*) reveals that the concentration decays exponentially down the lumen from a maximum value at the inlet *z* = 0. The remaining terms in the solution for *c*_{m} and *c*_{e} in (9) and (10) describe the radial decay of the oxygen concentration from the outer surface of the membrane as a consequence of oxygen uptake by the cells in the ECS.

Through cell-specific design criteria, we must design the bioreactor to ensure that the oxygen concentration exceeds a prescribed minimum throughout the bioreactor. This minimum oxygen concentration will be achieved at the furthest distance from the inlet, that is, when *r* = *R*_{e} and *z* = *L*. Denoting this minimum value by *c*_{min}, the analytical method gives the following expression for *c*_{min}, in terms of experimentally controlled and cell-specific parameters:

- (13)

#### Numerical Approach

For the analytical approach, the full system given by (2)–(6) is solved using finite element method package “COMSOL Multiphysics 3.5a”1 to evaluate the dependence of the oxygen concentration on the underlying parameters. The numerical approach is valid for all concentration values; however, the full system of equations must be solved iteratively each time. This is a computationally intensive process and does not provide operating equations that describe the dependence of the minimum oxygen concentration on the underlying parameters. Therefore, the numerical approach will be used when the analytical approach is not valid, that is, when . The mesh used for the results in this article consists of approximately 7,000 finite elements (and refining the mesh to 29,312 elements did not change the results to three significant figures).