Process models can be classified into two groups: first principles (fundamental) models and data-based (empirical, black box) models. First principles models are based on fundamental theories or laws, such as the conservation of mass, energy and momentum. One of the most important reasons for using fundamental models is the analytical expressions they provide relating key features of the physical system to its dynamic behavior. Data-based models provide relations between measured inputs and outputs that describe how the process responds to changes in various inputs. They can be developed much faster than first principles models, but their accuracy, robustness, and usability are limited. They provide an inexpensive alternative to fundamental models in most monitoring, diagnosis and control tasks.

The central theme of mathematical modeling of bioprocesses is the abstraction of physical phenomena into a suitable simplified mathematical formalism [615]. Even the simplest living cell is a system of such complexity that any mathematical description of it is an extremely modest approximation [32]. For this reason, a fundamental understanding of the phenomena taking place in the cell is needed to develop an acceptable first principles model. In the context of biological systems, this requires the presumption of metabolic intermediates and pathways that are crucial to system behavior and the specific regulatory role they play. This approach may require a number of iterations since the pathways may consist of large number of biochemical reactions [615].

The first step in developing a bioprocess model is to specify model complexity. Model complexity depends primarily on the purpose the model such as description of specific intracellular events or biochemical reactions, effects of environmental variables and effects of bioreactor operating conditions on growth and product formation. Model specifications include the number of biochemical reactions in the model, specification of the stoi-chiometry for these reactions, and related assumptions and simplifications. In setting up bioprocess models, lumping of biochemical reactions is done while paying attention to the detail level appropriate for the intended use of the model developed. After the model complexity is specified, rates of biochemical reactions are described with appropriate mathematical expressions using linear or nonlinear formulations. The rates are defined as functions of bioprocess variables, namely the concentrations of substrate(s) and metabolic products. These functions are referred to as kinetic expressions. Biochemists traditionally use elemental balances as their basic models, formulated as reaction equations. These balances define the biochemical state identifying the components which change considerably during the process, and contain information on the yields of various species with respect to some reference species. Besides stoichiometric relationships, empirical relations discussed in Chapter 2 can also be used as black box expressions.

The second step in modeling is to develop mass, energy, and/or momentum balances based on bioreactor operation mode (batch, fed-batch, or continuous) and combine them with kinetic expressions of the bioprocess. In general, homogeneity is assumed within the bioreactor for the sake of simplicity. Detailed bioreactor models that include spatial non-uniformity are also available. The combination of kinetic and bioreactor models form the complete mathematical description of the bioprocess.

The final step in first principles model development is assigning values to operating and kinetic parameters. The former depend on operating conditions such as volumetric liquid/gas flow rates of inputs and outputs, rotational speed of the impeller, and environmental conditions. The latter are associated with the biological system under consideration. Parameter estimation algorithms are used to assign values to these parameters. The basic steps in developing first principles models of bioprocesses are summarized in Figure 1.1. Detailed discussion of first principles models of bioprocesses and case studies are presented in Chapter 2.

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