Biochemical Engineering Fundamentals, Part 1 - Bailey J., Ollis D. 1989

Kinetics of substrate utilization, metabolite production, and biomass formation in cell cultures
Transient-state cell growth kinetics
Unstructured models of cell growth in batch processes

In the simplest approach to modeling Cell culture growth in a batch process, we assume that the growth rate of cell mass over time is a function solely of The Cell mass itself, i.e.,

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As we shall soon see, this equation by no means implies that we should ignore any changes occurring in the medium during cell growth.

One of the simplest models of type (7.49) is Malthus's law, which can be expressed mathematically as follows:

f(x) = μx      (7.50)

where μ is a constant. It is easy to see that this expression identically describes the familiar pattern of cell mass accumulation during the exponential growth phase. Malthus's prediction concerning the inevitable doom of humanity resulting from uncontrolled population growth has not come to pass (or perhaps has not yet come to pass?); similarly, in the case of microbial populations, the exponential growth phase is followed by a transition to a stationary population.

FIG. 7.19. Logistic curve (ß > 0; ß > 0).

This theory was further developed in the works of Verhulst (1844), as well as Pearl and Reed (1920), who incorporated an inhibitory factor into their population growth analysis. Assuming that inhibition is proportional to x², these researchers proposed the equation

This expression is a Riccati equation; its integration is straightforward and yields the equation of the logistic curve:

As shown in Fig. 7.19, the logistic curve is S-shaped and leads to a stationary population with a maximum biomass concentration x_s = 1/ß.

One possible explanation for the logistic curve is the assumption that The rate of toxin production is

proportional to the population growth rate:

And if

с_t( 0) = 0      (7.54)

then

c_t = а(х — х₀)      (7.55)

Typically, x0 is negligible compared to x; in such cases, substituting equation (7.55) into (7.42) yields an equation of type (7.51).

Another class of Unstructured models describing populations approaching a steady state is based on THE PRINCIPLE OF growth-limiting nutrient depletion. If, as in the derivation of the Monod equation, we assume that μ = μ(s) and that the overall yield coefficient Y_х/s is constant, the material balance equations for the nutrient and cellular matter can be combined into a single equation of type (7.49) (see Exercise 7.14).

A drawback of the logistic equation is that it neglects the decline phase in population mass (or size) after the stationary population has exhausted all its resources. This factor is accounted for in a model developed early this century by Volterra. In Volterra's model, equation (7.50) is supplemented by an integral term of the type

To a first approximation, the physical meaning of expression (7.56) can be interpreted as the Influence of the population's history K(t,r) on its growth rate. Expression (7.56) reflects the dependence of the growth rate at time t on all previous values of the population density. If K does not depend on t, expression (7.56) can be viewed as reflecting METABOLISM/18.html">The Influence of a culture component whose concentration change obeys the law

FIG. 7.20. Solution of the Volterra equation for various values of the parameter λ reflecting The history of the cell population,

Assuming that K is constant and equal to K₀, and supplementing equation (7.51) with expression (7.57), which accounts for the past History of the population, The change in population mass or density over time can be described by the following equation

In accordance with equation (7.58), K₀ has a negative value for an inhibitor and a positive value for a compound that promotes cell growth. This equation can be solved numerically; an example of such a solution is given in Fig. 7.20. For negative K₀, the population mass reaches a maximum and then decreases.

The unstructured models of cell growth described above have several drawbacks. Specifically, they fail to account for the lag phase, do not incorporate various variables affecting cell growth, and lack data on cellular metabolism and its regulation. In the subsequent sections, we will attempt to bridge the complex biochemistry of the cell with the phenomena observed during population growth in a batch process.