RESEARCH

The culture of mammalian cells by perfusion in a stirred tank fitted with a spinfilter as a retention device is a methodology aimed at retaining the highest number of cells inside the fermentor, which therefore allows the obtaining of high cell densities and, consequently, high product concentrations in a relatively short period of time, with high volumetric flows in small-scale facilities (1). The filtration time, in the case of the bioreactors of the Center for Molecular Immunology (Havana, Cuba) fitted with spinfilters for the perfusion culture of mammalian cells, is of 18 days. Filtration time, however, can be extended to 90 days according to literature (1-4). Given that no mathematical models for this system have been published, it was decided to model its behavior as a tool to guide further efforts in extending culture time (5) or filtration or perfusion time by varying the spin rate of the spinfilter and its filtration area (2, 3).

MATERIALS AND METHODS

Bioreactor

The fermentor used for the experimental runs (CMF 400, manufactured by Chemap AG) has a total volume of 41 L and an effective volume of 30 L, with a diameter of 0.27 m, a height of 0.7164 m and an effective height of 0.52 m. It has a propeller-type impellent with a diameter of 0.088 m (1, 3).

Spinfilters

Cell line

The study used the NSO/H7 host cell line (1, 3).

Culture medium

The study used the PFHM II protein-free culture medium (1).

RESULTS AND DISCUSSION

Derivation of hydrodynamic equations

At the Center for Molecular Immunology, perfusion cultures are usually performed in a stirred tank, using a spinfilter as a separation device. A spinfilter is a rotatory cylinder spinning on its axis that allows the continuous separation of cultured cells from the culture media and, therefore, the obtaining of a clarified culture supernatant.

Modeling a perfusion culture in a stirred tank with a spinfilter presents a number of challenges, derived from the simultaneous operation of different phenomena such as centrifugal effects, axial and sweeping forces, among others (3, 6, 7). However, the system can still be analyzed by decomposition into individual parts, modeling: 1) Its behavior as a rotatory filter; 2) Its behavior as a filtrating centrifuge and 3) Elements of mechanic energy balance in the interface of the outer and inner surface of the mesh (Figure 1).

By simultaneously using the equations corresponding to each individual part according to figure 1 and applying the filtration mechanism n = 3/2 (8, 9), the equation of Cozzeny-Karman (10) and the equation of Bernoulli (3, 8, 11, 12), the following is obtained:

Fp: perfusion flow (L/h) q₀*: constant (mL/s) Ks: separation power of the spinfilter tp: time of perfusion (s) ρ_SS: Density of the suspension (kg/m³) ϕ: viscosity of the fluid (Pa-s) The asterisk indicates the exit flow.

Therefore:

Fp= f(X(t), n, S, V, t...)

Where:

X(t): cell density in relationship to elapsed time n: spin rate of the spinfilter S: filtration area V: volume of the bioreactor (L) t: time (h)

Biomass balance equations

Previous work on this subject (1) dismissed the influence of the exchange flow through the filter (F) in order to simplify the calculations; however, this variable has a large influence on the process of clarification and the operation time (13). Therefore, it was decided to include this variable in the biomass balance corresponding to this system (Figures 2 and 3).

Modeling the exponential growth phase (E1) is simple, since it involves the same equations used for discontinuous cultures (1, 3, 14, 15):

Where:

X₀: starting biomass concentration in the bioreactor (cells/mL) μMAX: maximum specific growth rate (h^-1)

For edge conditions: X₀= 0.5 x 10⁶ cell/mL; X(t₁)= 10⁶ cell/mL; t₀= 24h and t= t₁

Modeling the E2 exponential growth phase under continuous flow

Exponential growth can be modeled in this case with the following differential equation:

Where:

Taking into account that feeding and extraction are described by the same function as that of perfusion flow capacity through the membrane, and assuming that the system reaches 90% retention instantaneously upon starting the perfusion, the differential equation can be arranged to depend on biomass concentration in the bioreactor (3):

Where:

C1= m_MAX= 0.003 h_-1
C*2= 8.561 x 10_-4 (g/cell)
C3= 0.02314 (mL/10₆ cells)
C4= r; r: density of the pure liquid (kg/m₃)
C5= 2 ρq* ₀(1-H); H: humidity of the cake

Where: η: filling coefficient, D: spinfilter diameter (m), α: specific resistance of the apparent cake (m/kg).

Where:

X_K: cell density at perfusion start (around 10⁶ cell/mL)

And for the perfusion flow capacity:

For edge conditions: X(t₁)= 10⁶ cells/mL, X(t₂)= 9 x 10⁶ cells/mL

Modeling the phase of limited growth under continuous flow (stationary phase), E3

During the third phase the stationary state (regarding biomass) is finally reached; that is, biomass remains constant. Therefore:

And for the perfusion flow:

In this phase perfusion flow capacity depends only on time, and therefore:

Fp= Fp(t)

For edge conditions:

X(t³)= X(t²)= 9x10⁶ cells/mL t= t² , Fp= Fp (t²), t= t³ , Fp= Fp (t³)

Limitations of the model

These models have some limitations, since upon starting perfusion 1) the spinfilter is assumed to reach instantaneously a retention of 90%; 2) extraction flow is described by the same function as perfusion capacity through the mesh of the filter; 3) The study was performed only for 3 discrete spin rates of the spinfilter and the same number of different filtration areas; 4) The small-scale study was performed with only one spinfilter mesh; 5) The influence of the spin rate of the impellent over perfusion flow capacity is not taken into account; 6) The influence of the pressure inside the bioreactor on perfusion flow capacity is not taken into account; 7) The washing phase of the fermentor was not modeled and 8) The determination of limiting substrate was not performed, and neither μ = f(S, μ^MAX) nor O² balance nor product formation were adjusted.

Biomass simulation

Figure 4 shows the stages of exponential growth and exponential growth under constant flow, which for this model fall midrange between the actual run data (Figure 4C), revealing practically the same maximum specific growth rate. In other words, the curve of the model has the same slope as those of the actual runs, with the exception of run 3231TA-0207, which differs in this aspect due to a longer adaptation phase in comparison to the others. Something similar is observed for the stationary phase; although the actual run data follow a curve characterized by a fall that is not present in the model. While this difference is expected, given that the model did not take into account the washing phase that takes place upon complete clogging of the spinfilter, it still constitutes a weakness of the model.

Simulation of perfusion flow capacity

This simulation used as input variables the variation of cell density, the spin rate of the spinfilter, the area of the spinfilter and the operation time, setting the perfusion flow capacity of the system as output variable. Three discrete values for the filtration area were analyzed: that corresponds to the actual filter, and both larger or smaller values, varying on each case the spin rate of the spinfilter (100 rpm, 200 rpm and 300 rpm) (Figure 5) (18, 19).

A similar behavior was observed at the beginning (Figure 6A, B and C) if the fixed area is taken into account, since an increase of the spin rate of the rotatory filter results in an increase of both perfusion flow capacity and filtration time, with a non-linear dependence according to equation 4 (3). The perfusion flow from the pump (Fp_B), ranging from 7.5 to 30 L/day, would be added to each Fp value corresponding to the spin rate of the spinfilter. Although literature mentions this behavior (2, 13, 20-22), this phenomenon is not described analytically, and therefore the results reveal a direct influence of the spin rate on perfusion flow capacity. The effect probably derives from an enlarged zone of laminarity (Figure 7) due to the larger spin rate of the filter (8, 10, 13, 22), which allows a sweep of the cellular profile that approaches the membrane to occlude it, due to the drag of the exchange flow, that appears due to the centrifugal action of the filter itself on the cellular profile.

If the filtration area increases while the spin rate is kept constant, perfusion flow capacity increases almost exponentially (Figure 8). This effect is also described in literature (2, 13, 20-22) although it is not specified analytically, and constitutes another direct result of the influence of the spin rate of the spinfilter and the area on perfusion flow capacity. This must also be a result of the increase in the zone of laminarity (Figure 7) as the spin rate of the filter increases (8, 10, 13, 22) and of the filtration area, allowing a larger sweep of the cellular profile in spite of the drag due to the exchange flow, appearing due to the centrifugal action of the filter itself on the cellular profile.

CONCLUSIONS

This work obtained mathematical models that predict the behavior of perfusion cultures of mammalian cells in stirred tanks for each of their growth stages, coupling kinetic and hydrodynamic equations (1-3, 9). These novel results, which have no precedents in literature, allowed the analysis of the influence of the spin rate of the filter and the filtration area on the perfusion flow capacity of the system. In spite of the limitations of the models, it was possible to determine that an increase in spin rate at a constant filtration area results in an increase in the perfusion capacity of the system, which is even larger if the filtration area also increases, with a concomitant increase in filtration or perfusion times (2, 3, 13, 20).