Bioreactor Design and Operation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 930

18.7.1. Batch Thermal Sterilization
Death is considered the failure of the cell, spore, or virus to reproduce or germinate when
placed in a favorable environment. The probability of extinction of the total population,
P 0 (t), is

P 0 ðtÞ¼½1pðtÞ N 0 (18.14)

where p(t) is the probability that an individual will still be viable at time, t,andN 0 is the
number of individuals initially present. The expected value of individuals present at time,
t, E[N(t)] is

E½NðtÞ ¼ N 0 pðtÞ (18.15)

While the variance of this expected value, V[N(t)] is

V½NðtÞ ¼ N 0 pðtÞ½1pðtÞ (18.16)

The specific death rate, k d is

1
E½NðtÞ

d
d

d
d

k d ¼

t E½NðtÞ ¼

ln pðtÞ (18.17)

t

So far in modeling cell death rate, we have been assuming a first order rate or a constant
specific death rate. When this first-order death model in which k d is constant is applied:

pðtÞ¼expðk d tÞ (18.18)

which is equivalent to

E½NðtÞ ¼ N 0 expðk d tÞ (18.19)

Decimal reduction time, t DR , is the time for number of viable cells to reduce by 10 fold.
That is

0:1 ¼ expðk d t DR Þ (18.20)

ln10
k d

t DR ¼ (18.21)

Fig. 18.8 is a plot of N(t)/N 0 , strictly speaking E[N(t)]/N 0 , versus time, which is called
a survival curve. The survival curve can be applied to indicate the relative sterility as
E[N(t)]/N 0 is the probability of surviving cells relative to the total cells initially. We often
drop E[ ] notation with an understanding that the number of cells need not be an integer in
computing the number of survived cells statistically. In practice, of course, there is no such
thing as 1.1 or 0.2 cells alive. To ensure the sterility of the culture, one must ensure a low
number (a small fraction of a cell) survival in absolute terms. Even one live spore can repopu-
late in the reactor. When the number of cells present initially is high, the survival ratio must
even be smaller for sterility.