Modeling of Growth Kinetics
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7.1 Model Structure and Complexity
Setting up a mathematical model involves several steps as illustrated in Fig. 7.1. A very
important first step is to specify the model complexity. This depends on the aim of the study,
i.e.
what the model is going to be used for. Specification of the model complexity involves defining
the number of reactions to be considered in the model, and specification of the stoichiometry for
these reactions. When the model complexity has been specified, rates of the cellular reactions
considered in the model are described in terms of a specified set of output variables that may be
concentrations in the reactor of substrates, metabolic products and certain biomass components.
Every reactant - substrate or product - which is suspected to influence the rate is included, but in
simple models mass balances for many of the output variables can be expressed in terms of other
output variables using assumptions such as constant yield coefficients. The expressions that
relate the rates to the output variables are normally referred to as
kinetic expressions
, since they
specify the kinetics of the reactions considered in the model in terms of a selected set of output
variables. This is an important step in the overall
modeling cycle
and in many cases different
kinetic expressions have to be examined before a satisfactory model is obtained. The next step in
the modeling process is to combine the kinetics of the cellular reactions with a model for the
reactor in which the cellular process occurs. Such a model specifies how the concentrations of
substrates, biomass, and metabolic products change with time, and what flows in and out of the
bioreactor. When no spatial variation occur in the reactor we speak of models for
ideal
bioreactors
, and these models, to be treated in Chapter 9, are normally represented in terms of
simple mass balances over the whole reactor. Mrife detailed reactor models may also be applied,
if inhomogeneity of the medium is likely to play a role (see Chapter 11). The combination of
mass balances, including kinetic rate equations, and the reactor model makes up a complete
mathematical description of the fermentation process, and this model can be used to simulate
how output variables depend on the set of input variables. The steady state model consists of a
set of algebraic equations that relate the output variables to the input variables. Dynamic - or
transient models consist of a set of differential equations. For an ideal bioreactor the output
variables are found as functions of time for any given set of input variables and the initial values
of the output variables. However, before this can be done it is necessary to assign values to the
parameters of the model. In order to do so one most compare model simulations with
experimental data, and hereby estimate a parameter set that gives the best fit of the model to the
experimental data. This is referred to as
parameter estimation.
The evaluation of the fit of the
model to the experimental data can be done by simple visual inspection of the fit, but generally it
is preferable to use a more rational procedure,
e.g.
by minimizing the sum of squared errors
between the model and the experimental data. If the model simulations are considered to
represent the experimental data sufficiently well the model is accepted, whereas if the fit is poor
even for the set of parameters that gives the best fit it is necessary to revise the kinetic model and
go through the modeling cycle again.