## Fedbatch Culture

Yoshida et al. (1973) introduced the term fed-batch culture to describe batch cultures which are fed continuously, or sequentially, with medium, without the removal of culture fluid. A fed-batch culture is established initially in batch mode and is then fed according to one of the following feed strategies:

(i) The same medium used to establish the batch culture is added, resulting in an increase in volume.

(ii) A solution of the limiting substrate at the same concentration as that in the initial medium is added, resulting in an increase in volume.

(iii) A concentrated solution of the limiting substrate is added at a rate less than in (i) and (ii), resulting in an increase in volume.

(iv) A very concentrated solution of the limiting substrate is added at a rate less than in (i), (ii) and (iii), resulting in an insignificant increase in volume.

Fed-batch systems employing strategies (i) and (ii) are described as variable volume, whereas a system employing strategy (iv) is described as fixed volume. The use of strategy (iii) gives a culture intermediate between the two extremes of variable and fixed volume.

The kinetics of the two basic types of fed-batch culture, variable volume and fixed volume, will now be described.

### Variable volume fed-batch culture

The kinetics of variable volume fed-batch culture have been developed by Dunn and Mor (1975) and Pirt (1974, 1975, 1979). The following account is based on that of Pirt (1975). Consider a batch culture in which growth is limited by the concentration of one substrate; the biomass at any point in time will be described by the equation:

where xt is the biomass concentration after time, t hours, and x0 is the inoculum concentration. The final biomass concentration produced when 5 = 0 may be described as xmax and, provided that x0 is small compared with xraax:

Y-SR

If, at the time when x = xmax, a medium feed is started such that the dilution rate is less than /imax, virtually all the substrate will be consumed as fast as it enters the culture, thus:

where F is the flow rate of the medium feed, and X is the total biomass in the culture, described by X = xV, where V is the volume of the culture medium in the vessel at time t.

From equation (2.32) it may be concluded that input of substrate is equalled by consumption of substrate by the cells. Thus, (ds/dt) = 0. Although the total biomass in the culture (X) increases with time, cell concentration (x) remains virtually constant, that is {dx/dt) ~ 0 and therefore u — D. This situation is termed a quasi steady state. As time progresses the dilution rate will decrease as the volume increases and D will be given the expression:

where F0 is the original volume. Thus, according to Monod kinetics, residual substrate should decrease as D decreases resulting in an increase in the cell concentration. However, over most of the range of /x which will operate in fed-batch culture, SR will be much larger than Ks so that, for all practical purposes, the change in residual substrate concentration would be extremely small and may be considered as zero. Thus, provided that D is less than /xirrav and Ks is much smaller than SR, a quasi steady state may be achieved. The quasi steady state is illustrated in Fig. 2.13a. The major difference between the steady state of a chemo-stat and the quasi steady state of a fed-batch culture is that ¡jl is constant in the chemostat but decreases in the fed-batch.

Pirt (1979) has expressed the change in product concentration in variable volume fed-batch culture in the same way as for continuous culture (see equation 2.28):

Thus, product concentration changes according to the balance between production rate and dilution by the feed. However, in the genuine steady state of a chemostat, dilution rate and growth rate are constant whereas in a fed-batch quasi steady state they change over the time of the fermentation. Product concentration in the chemostat will reach a steady state, but in a fed-batch system the profile of the product concentration over the time of the fermentation will be dependent on the relationship between q and /z (hence D). If q is

Fig. 2.13. Time profiles of fed-batch cultures. fj. = specific growth rate x = biomass concentration S(GLS) — growth limiting substrate SN = any other substrate than S(GLS)

(a) Variable volume fed-batch culture.

(b) Fixed volume fed-batch culture. (Pirt, 1979).

strictly growth related then it will change as ¡x changes with D and, thus, the product concentration will remain constant. However, if qp is constant and independent of /x, then product concentration will decrease at the start of the cycle when Dp is greater than qpx but will rise with time as D decreases and q x becomes greater than Dp. These relationships are shown in Fig. 2.14a. If qp is related to /x in a complex manner, then the product concentration will vary according to that relationship. Thus, the feed strategy of a fed-batch system would be optimized according to the relationship between qp and ¡x.

### Fixed volume fed-batch culture

Pirt (1979) described the kinetics of fixed volume fed-batch culture as follows. Consider a batch culture in which the growth of the process organism has depleted the limiting substrate to a limiting level. If the limiting substrate is then added in a concentrated feed such that the broth volume remains almost constant, then:

- i 14 Product concentration (p) in fed-batch culture when qp ^growth related (-----> or non-growth related, i.e. qp

(""variable volume fed-batch culture, (b) Fixed volume fed-batch culture. (Modified'from Pirt, 1979.)

but substituting for x from equation (2.36) gives: dp./dt = qp(xt + GYt).

If qp is strictly growth-rate related then product concentration will rise linearly as for biomass. However, if qp is constant then the rate of increase in product concentration will rise as growth rate declines, i.e. as time progresses and x increases. These relationships are shown in Fig. 2.14b. If qp is related to ¡x in a complex manner then the product concentration will vary according to that relationship. As in the case of variable volume fed-batch the feed profile would be optimized according to the relationship between qp and ¡x.

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