Applying Mechanistic Models in Bioprocess Development - Measurement, Monitoring, Modelling, and Control of Bioprocesses - page 148

The prediction of biomass is also greatly affected by the yield of biomass on
ethanol, in addition to the yield on glucose (oxidative metabolism). The impact of
the saturation constants is rather limited for any of the model variables.

2.2.2 Identifiability of Parameter Subsets: Collinearity Index c K

In addition to understanding the importance of individual parameters to the model
output, it is necessary to take the joint influence of all parameters into account as
well ([h 1 , …, h j=J ]). If columns s j are nearly linearly dependent, the change of a
parameter h j can be compensated by a change in the other parameter values. This
means that the parameters [h 1 , …, h J ] are not uniquely identifiable.
The collinearity index c K assesses the degree of near-linear dependence
between a subset of K (2 B K B J) parameters, i.e., columns of the scaled sen-
sitivity matrix. 1
A high value of a collinearity index indicates that the parameter set is poorly
identifiable. In practice, c K is calculated for all subsets of K parameters out of the
11 parameters and is plotted in Fig. 3 . Also the subset size for each case is shown.
In this case, a subset was considered identifiable if the corresponding collinearity
index was smaller than 5. This threshold has to be defined a priori. Brun and
colleagues [ 18 ] suggested as a rule of thumb that this threshold should lie in the
range 5-20, where the lowest collinearity index corresponds to the strictest cri-
terion. In practice, this decision on the threshold value is dependent on prior
experience of the model user, and thus an iterative process.
All the model variables were considered in this analysis, implying as well that
all could be measured experimentally. As illustrated in Fig. 3 , a maximum of eight
parameters can be identified, and the collinearity index increases with the number
of parameters. The maximum collinearity index observed for combinations of
eight parameters was 22.34, while the best identifiable sets of eight parameters
correspond to a c K value of approximately 2.65. These parameter subsets are listed
in Table 5 .
It is indeed known that a change in the maximum uptake rate of glucose can be
compensated with a change of biomass yield coefficients. Also, based on the model
structure, it is clear that changes in yields for the oxidative and reductive con-
sumptions of glucose can compensate each other. It is therefore not surprising that
the parameter subsets that have higher collinearity index include these parameters.
When comparing the subset of six parameters with the lowest collinearity index
(last row in Table 5 ) with the ''best'' subset of eight parameters (shaded row in
Table 5 ), the two parameters that have been removed in the subset of six
parameters are the maximum uptake rates of ethanol and oxygen.

1

Further discussion and equations are provided in the paper by Brun et al. (2002).