Applying Mechanistic Models in Bioprocess Development - Measurement, Monitoring, Modelling, and Control of Bioprocesses - page 146

can be estimated with high degree of confidence based on the available experi-
mental measurements.
The main purpose of such an identifiability analysis is in fact to increase the
reliability of parameter estimation efforts from a given set of data [ 17 ]. One
method available to perform such an analysis is the two-step procedure based on
sensitivity and collinearity index analysis proposed by Brun and colleagues [ 18 ].
Accordingly, the method calculates two identifiability measures: (1) the parameter
importance index (d) that reflects the sensitivity of the model outputs to single
parameters, and (2) the collinearity index (c) which reflects the degree of near-
linear dependence of the sensitivity functions of parameter subsets. A parameter
subset (a combination of model parameters) is said to be identifiable if (1) the data
are sufficiently sensitive to the parameter subset (above a cutoff value), and (2) the
collinearity index is sufficiently low (below a cutoff value).

2.2.1 Local Sensitivity Analysis: Parameter Importance Indices d

The local importance of an individual parameter to a model output for small
changes (Dh) in the parameter values (h) at a specific location (h 0 ) can be mea-
sured by the estimation of a dimension-free scaled sensitivity matrix S sc = {s ij },
where the index i refers to a specific model variable (output) and j denotes the
model parameter. For further details, the reader is referred to the original paper of
Brun and colleagues [ 18 ]. The mean squared norm of column s j , denoted by d j ,isa
measure of the importance of parameter h j (see Eqs 8 - 10 ). A large norm indicates
that the parameter is identifiable with the available data if all other parameters are
fixed. A parameter importance ranking can be obtained by ranking the parameters
according to their d indices. The lower the value of d, the lower the importance of
that parameter.
For this first analysis, the parameter values (Table 4 ) provided in the original
paper [ 1 ] are used as nominal values at which sensitivity functions are calculated.
The scaled sensitivity matrix S and the resulting rank of d importance indices were
calculated using Eqs. 8 - 10 , and are graphically compared in Fig. 2 . It is note-
worthy that the d indices are very sensitive to: (1) the choice of variation range
defined for each parameter (Dh), (2) scaling factors (sc) used to calculate the
sensitivity matrix, and (3) the original set of parameters (h 0 ), naturally as this is a
local analysis. In this example the sc were defined as the mean of the experimental
observations for each variable.

h h 0

v ij ¼ o g i ð h j Þ
oh j

ð 8 Þ

Dh j
SC j

S ij ¼ v ij ð 9 Þ