Applying Mechanistic Models in Bioprocess Development - Measurement, Monitoring, Modelling, and Control of Bioprocesses - page 156

2.5 Sensitivity Analysis: Linear Regression
of Monte Carlo Simulations

Based on the Monte Carlo simulations, a global sensitivity analysis can be con-
ducted. The aim of the sensitivity analysis is to break down the output uncertainty
with respect to input (parameter) uncertainty. The linear regression method is a
rather simple yet powerful analysis that assumes a linear relation between the
parameter values and the model outputs. The sensitivity of the model outputs to the
individual parameters, for a given time point, is summarized by a ranking of
parameters according to the absolute value for the standardized regression coef-
ficient (SRC). In a dimensionless form, the linear regression is described by
Eq. 12 , where sy ik is the scalar value for the kth output, b jk is the SRC of the jth
input parameter, h j , for the kth model output, y k , and its magnitude relates to how
strongly the input parameter contributes to the output.

¼ X

M

sy ik l sy k
r sy k

b jk h ij l hj
r h j

þ e ik ð 12 Þ

j ¼ 1

In the case of nonlinear dependence of the model variable on a parameter, this
method can still be used, although with caution. As a rule of thumb, if the model
coefficient of determination (R 2 ) is lower than 0.7, this analysis is not conclusive.
The SRC for each parameter has, by definition, a value between -1 and 1, where a
negative sign indicates that the output value will decrease when there is an
increase in the value of the parameter. Oppositely, a positive SRC indicates direct
proportionality between the parameter value and the model output. Sin et al. [ 12 ]
describe further details on how to perform the analysis.
In the model example, different growth phases are described, and therefore the
importance of the parameters is expected to change with time. Therefore, the
analysis was performed for a selection of time points up to 62 h.
The suitability of applying the linear regression method was in this case also
assessed for each time point and each output. The R 2 values are presented in Fig. 7
as a function of time.
While the regression method seems to be suitable for all time points in the case
of biomass, the same is not observed for glucose, ethanol, and oxygen.. With
regard to glucose, the model uncertainty is very small (narrow spread of the model
predictions plotted in Fig. 6 ). The depletion of glucose is estimated to occur at
time of approximately 22 h for all cases. The sensitivity analysis when the glucose
concentrations are virtually zero is not expected to be significant, and it is thus not
surprising that the R 2 value decreases abruptly at approximately the same time
point that glucose is depleted. Simultaneously, the uncertainty in ethanol con-
centration predictions increases substantially. This may explain the temporary
drop in the R 2 value for ethanol at this time point. A similar drop in R 2 is observed
for oxygen around the time that ethanol is depleted, and a sudden rise in the
dissolved oxygen concentration is observed. Upon ethanol depletion, the R 2 value