Applying Mechanistic Models in Bioprocess Development - Measurement, Monitoring, Modelling, and Control of Bioprocesses - page 152

Table 7 Confidence intervals for the identifiable subset of parameters for 95 % confidence level
Parameter Estimated value Confidence interval Units

±9.8 9 10 -2 g G g -1 Xh -1 r G ; max 2.9 (3.4 %)

5.5 9 10 -3 ±6.3 9 10 -4 mol O g -1 Xh -1

r O ; max (11.6 %)

g E g -1 Xh -1 r E ; max 0.32 ±0.24 (75.7 %)

±3.1 9 10 -2 g X g -1 Y XE 0.47 (6.6 %) E

±8.4 9 10 -2 g G l -1

K G 0.17 (50.2 %)

g E l -1 K E 0.56 ±0.44 (78.9 %)

g G l -1

K i 0.31 ±0.30 (97.5 %)

h -1 k L a 930 ±49 (5.2 %)

calculated using Eq. 11 , where COV is the covariance matrix of the parameter
estimators, t(N - M, a/2) is the t-distribution value corresponding to the a/2
percentile, N is the total number of experimental observations (45 samples for the
two cultivations), and M is the total number of parameters. The confidence
intervals for the estimated parameters are presented in Table 7 .
h 1 a ¼ h

:

p

tN M ; a
2

diag ð COV ð h ÞÞ ð 11 Þ

None of the confidence intervals include zero, giving a first indication that all
parameters are significant to a certain degree and the model does not seem to be
overparameterized. In the case of the inhibition constant K i , the confidence interval
is rather large. This is most likely a consequence of the low sensitivity of model
outputs to this variable (Fig. 2 ). Furthermore, the confidence intervals of the
Monod half-saturation constants K G and K E are quite large as well, which might be
related to the fact that their estimated values are rather low. The latter means that
the collected data do not contain that many data points which can be used during
the parameter estimation for extracting information on the exact values of K G and
K E Indeed, only the data corresponding to relatively low glucose and ethanol
concentrations can be used, since the specific rates will be relatively constant and
close to maximum for higher substrate concentrations.
It is furthermore also a good idea to analyze the values of the parameter
confidence intervals simultaneously with the correlation matrix (Table 8 ); For
example, the correlation matrix shows that r E,max is correlated with K E and that
r O,max is correlated with K O . Both correlations are inherent to the model structure;
i.e., correlation between the parameters related to the maximum specific growth
rate and the substrate affinity constant in Monod-like kinetics expressions are quite
common, and point towards a structural identifiability issue.
Note also that the significant correlations found between some of the model
parameters (Table 8 ) seem to conflict with the results of the collinearity index
analysis which was reported earlier (Fig. 3 ; Table 5 ). That is one of the reasons
also for the identifiability analysis to be an iterative process.