8
Population Balance Equations
In Chapter 7, cell population balances are written in terms of a distribution of mass fractions of the
total biomass. This allows a direct combination of intracellularly structured models and population
models. However, the population balances based on mass fractions do not permit the incorporation
into the model of specific events in the cell cycle, and the single-cell models of Section 7.5.2 can
therefore not be used in connection with these population balances. Since there are numerous
examples that show a direct influence of certain specific events in the cell cycle on the overall
culture performance, e.g., the distribution of plasmids to daughter cells on cell division in
recombinant cultures, we need to derive a population balance based on cell number to obtain a
correct description of these processes.
In a population balance based on cell number, the basis is the individual cells. Thus, the cellular
content of the intracellular components has the unit grams per cell, and we can therefore not use the
composition vector X (unit: grams per gram dry weight). Instead the properties of the cell are
described by the vector y, which may also contain information about the cell's age, size, etc. The
distribution of cells in the population is given by
w hereby,/)
dy
represents the number of
cells per unit volume within the property space y to y +
dy
at time
t.
Thus the total number of cells
per unit volume in the population is given by
n( t) = \
f( y ,t) d y
(8.1)
vy
where
Vy
is the total property space. In general,
n
is determined from a mass balance for the limiting
substrate, as illustrated in Note 8.1.
Note 8.1 Determination of the total number of cells from a substrate balance
For a distribution of cells with different substrate uptake kinetics, the volumetric rate of substrate
consumption for a single limiting substrate is given by,
?r(0 = - f
rt (y ,s )f(y ,t)d y
(1)
4 yy
where
rs
(y ,j) is the rate of substrate consumption per cell per time. The mass balance for the limiting
substrate is therefore
315
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