## Mathematical Representation of Bioreactor Operation

A popular form of operation of bioreactors employing living cells involves the use of a well-mixed reactor, the mixing being accomplished by mechanical agitation and/or fluid motion. The uniformity of composition and temperature in the reactor allows its representation as a lumped parameter system. Although well-mixed reactors are almost a norm in cell cultivation, tubular reactors are used in bioprocesses involving immobilized cells and immobilized enzymes, these bioreactors being distributed parameter systems. In view of this, the focus in this chapter is on lumped parameter systems. The dynamics of bioreactors that can be viewed as lumped parameter systems can be described succinctly as dx

with x denoting the set of variables which represent the status of the cell culture in the bioreactor, the so-called state variables, and u and d representing the set of external variables which indirectly influence the status of the cell culture, the so-called input variables. The input variables are further classified into inputs that are manipulated (u) and inputs that are not manipulated (d), the so-called disturbance variables. Let n, m and p denote the number of state variables, manipulated inputs and disturbance variables. The right hand side in Eq. 2.1 contains information on how the temporal variations in state variables are influenced by the state and input variables, these influences in general being nonlinear (f - a nonlinear function of x, u and d). In the subsequent sections, illustrations will be provided on how descriptions of different operating modes for bioreactors and bioprocesses with varying levels of complexity as concerns description of kinetics of cellular and extracellular processes can be concisely represented in the form of Eq. 2.1. As these illustrations are discussed, it will be evident that not all state variables can be measured or estimated. There can be a variety of reasons for not monitoring variations in a state variable, including lack of availability of an assay (procedure for analysis) / measuring device/sensor, monitoring difficulties due to rapid fluctuations in the state variable, and costs associated with frequent measurement of the same. The specifics of an assay (analytical procedure) or measuring device may impose limitations on frequency of measurement of certain state variables. In such situations, the measurements of the state variable at discrete times may have to be supplanted by estimations of the same (based on these measurements) at times when no measurements were made. Some of the state variables, which cannot be measured as frequently as desired, can be estimated from measurement of certain other parameters, which in turn can be measured as frequently as desired. An example of such a state variable is the biomass concentration in the culture (usually expressed as dry mass of cells per unit culture volume). Direct estimation of this variable requires time-intensive separation (of the abiotic and biotic phases via cen-trifugation or filtration) and gravimetric procedures. A reliable procedure for monitoring biomass concentration involves determination of turbidity of cell culture via measurement of optical density (OD) of the culture and estimating the biomass concentration using a predetermined correlation between the biomass concentration and OD. The advantage of this estimation is rendered by one's ability to measure OD as frequently as possible. It must therefore be realized that only some of the state variables may be monitored or estimated. The set of variables which can be measured will be referred to as bioreactor outputs, y, with the number of outputs being 1. The relations among the state variables, the input variables and the output (measured) variables can then be succinctly stated as y = g(x,u,d). (2.2)