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

Kinetics of substrate utilization, metabolite production, and biomass formation in cell cultures
Structured models of cell growth kinetics

Unstructured models describe only the quantity of the biological phase. Such models neither account for nor reflect the COMPOSITION OF THE biophase, i.e., what can be defined as its qualitative characteristics. If the composition of The Cell population changes significantly during growth and this change affects cell growth kinetics, the analysis of such systems is only feasible using structured models. Since it is practically impossible to account for the material balances of all cell components in any single model, when developing a structured model (necessarily an approximate one), we must evidently carefully select the key variables and processes that play a vital role in the intended application of the model. As we will demonstrate through a series of Examples in the subsequent sections, structured models can be formulated based on various concepts.

In structured models, the mass (x_j) or molar (c_j) concentrations per unit volume of the biophase are typically used as biophase variables [15]. Under complete mixing conditions, the mass balance equation for component j can be written as follows:

Class="center">

where ρ_c is the cell density, equal to the mass of Cells per unit volume of cells; r_fj is the molar rate of formation of component j [(number of moles of j) ∙ (time)^-1 ∙ (unit volume of cells)^-1]; Φ_x is the mass of cells introduced into the Reactor per unit time; V_R is the culture volume; x is the cell mass concentration [(cell mass) ∙ (unit culture volume)^-1]; c_j is [(number of moles of j) ∙ (unit volume of cells)^-1]. The last term of equation (7.65) assumes that the cells introduced into the reactor (e.g., via recycle) or withdrawn from it are in the same state as the cell population inside the reactor. If this condition is not met (for example, In the second or one of the subsequent reactors in a CSTR cascade), the balance equation must be modified accordingly.

Assuming that the cell density ρ_c and culture volume remain constant over time, differentiation of equation (7.65) yields

For a batch reactor, Φ_x is equal to zero, and the expression in parentheses on the right-hand side of equation (7.66) is none other than the specific growth rate μ; consequently, for this case, equation (7.66) simplifies to:

The physical meaning of r_fj is straightforward; the expression -μs_j reflects the concentration decrease caused by dilution during cell population growth. Interestingly, evaluating the right-hand side of equation (7.66) for a CSTR under nutrient feed sterility conditions once again leads to equation (7.67).