Mathematical Modeling in Bioprocessing

Models allow generalization to other situations within the boundaries of their validities. 

Models are mathematical formulas that can be manipulated to optimize processes. 

Models are used to identify unknown or previously disregarded process variables and parameters that may be significant variables. 

Models serve as controls in examining whether an effective separation of biological and physical phenomena has been achieved. 

As an indirect result, models may also help in clarifying reaction mechanisms. 

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There is an increasing interest in technologies that maximize the production of various essential enzymes and therapeutic proteins based on E. coli cultivation. The costs of developing mathematical models for bioprocesses improvements are often too high and the benefits are too low. The main reason for this is related to the intrinsic complexity and non-linearity of biological systems. The important part of model building is the choice of a certain optimization procedure for parameter estimation. The estimation of model parameters with high parameter accuracy is essential for successful model development. 

Simutis,R., Dors, M., Lubbert, A., Increasing the efficiency of hybrid models in bioprocess supervision and control, in preparation). The simultaneous utilization of mathematical models, heuristic rule systems, and artificial neural networks has been referred to as a hybrid model. 

Pseudokinetic phenomena become evident only when process kinetic analysis is carried out with mathematical models. Most bioprocesses are basically heterogeneous systems. Generally, pseudohomogeneous rates measured in L phase analyses are used, because they are thought to reflect directly the intrinsic reaction rate of metabolism in the solid phase (biomass). Even under steady-state conditions, however, this assumption is not necessarily valid. 

From kinetics studies of unicellular organisms, a set of mathematical expressions have been established to represent the most frequent phenomena in bioprocesses. These phenomena involve a limitation or inhibition of growth and product formation, caused by the presence of substrates, products, or byproducts in culture media. Many of these expressions do not derive from known kinetic mechanisms. In fact, they are simply mathematical expressions with fitted parameters that are able to reproduce experimentally observed kinetic profiles. These equations have been derived and used in many unstructured microbial or cell models. 

Figure 5.48 summarizes practical situations in bioprocessing (A. Moser, 1981). Classification of situations can be achieved by distinction between sequential and simultaneous utilization, with a transition case of overlapping utilization. While sequential or consecutive consumption of substrates (Monod, 1942, 1949) can often be analyzed in two separate growth phases, the simultaneous utilization encountered in biological waste water treatment is more difficult for mathematical modeling. 

The main purpose of this chapter is to understand and discover, through examples, the importance of mathematical modeling and to start constructing simple mathematical models. The second purpose is introduce you to process and bioprocess simulation. Again, in the case of simulation, you have already been exposed to and practiced simulation through some exercises in previous chapters of the... 

In this section, we will attempt to develop mathematical models for different situations that you will face as a process or bioprocess engineer. Although these examples are limited and tailored to your mathematical background, it will be interesting and rewarding for you to discover that you can be involved with fascinating and ever-challenging examples. 

Another efficient way to study chemical and bioprocesses in microreactors is by means of mathematical models. Such studies can be coupled with experimental observations to give a better understanding of the reaction phenomena. The complexity of mathematical models grew simultaneously with the complexity of the reactive systems. 

Sideman S, Pinczewski W (1975) Turbulent Heat and Mass Transfer at Interfaces Transport Models and Mechanisms. In Gutfinger C (ed) Topics in Transport Phenomena bioprocesses, mathematical treatment, mechanisms. Hemisphere, Washington... 

Simple bioprocess models are typically based on batch fermentation processes where the change in cell concentration with respect to time is proportional to the current cell concentration. This first-order rate equation is mathematically described by... 

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