**Basic Assumptions and Formulation**

The theoretical model of the dilution-filtration system is developed (as shown in Fig.17) under the following assumptions: (1) Both intra- and extra-cellular solutions in cell suspension consist of water, a permeable CPA (e.g. glycerol) and an impermeable salt (e.g. NaCl); (2) Blood bag, hollow fibers and their connecting tubing are filled with cell suspension, and cells are uniformly distributed in the suspension; (3) Extracellular solution is diluted/filtrated immediately and evenly at the diluting/filtrating point when cell suspension circulates in the system; (4) Suspension flow is one dimensional, and the convection factors can be neglected.

Fig. 17. Theoretical modeling of the system. A: the overall system, and B: a control volume.

Based on the assumptions, a governing equation about the mass transfer process can be derived by focusing on the extracellular solution:

$$\frac{\partial \phi^{\epsilon}}{\partial t} = \frac{1}{A} \frac{\partial}{\partial \mathbf{x}} (DA \cdot \frac{\partial \phi^{\epsilon}}{\partial \mathbf{x}}) + S \tag{19}$$

where, A refers to effective mass transfer area, D refers to diffusion coefficient, *ϕe* refers to extracellular solute concentration (in osmolality), and S is the mass source/sink term, respectively.
