Applying Mechanistic Models in Bioprocess Development - Measurement, Monitoring, Modelling, and Control of Bioprocesses - page 145

Table 3 Nomenclature list
of matrices and vectors used
in the model formulation and
model analysis

Unit Description

Z Stoichiometric matrix

q Process rate vector

h Vector of model parameters

S sc

Scaled sensitivity matrix

s j Column vector of the sensitivity matrix:

corresponding to sensitivity
of the various model outputs to the parameter j

s ij Scaled sensitivity of the output i to the parameter j

d j Importance index of parameter j

sc Scaling factors

2.2 Parameter Identifiability Analysis

The model described in the previous sections has four variables—glucose (G),
ethanol (E), oxygen (O), and biomass (X)—and 11 parameters. In addition, the
oxygen saturation concentration in water (at growth temperature) is necessary for
solving the model. A list of the parameters and their descriptions is provided in
Table 4 .
The maximum specific growth rate on ethanol ð l E ; max Þ is defined as the product
of the yield of biomass on ethanol ð Y XE Þ and the maximum specific ethanol uptake
rate ð r Oxid

E ; max Þ . For consistency between parameters, the ethanol specific uptake rate
is used as a parameter in this example.
The number of parameters is considerably larger than the number of model
variables (or outputs), which is typical for this type of model. It is therefore
questionable whether all parameters can be estimated based on experimental data,
even if the four model variables were to be measured simultaneously. This is the
subject of identifiability analysis, which seeks to identify which of the parameters

Table 4 Model parameters, corresponding units, and numerical values [ 12 ]
Parameter Value Units

g G g -1 Xh -1

r G ; max Maximal specific glucose uptake rate 3.5

8 9 10 -3 mol O g -1 Xh -1

r O ; max Maximal specific oxygen uptake rate

g X g -1

Y Oxid
XG

Yield of biomass on glucose (oxidation) 0.49 G

g X g -1

Y red
XG

Yield of biomass on glucose (reduction) 0.05 G

g X g -1

Y XE Yield of biomass on ethanol 0.72 E

h -1 l E ; max Maximal specific growth rate on ethanol 0.17

g l -1 K G Saturation parameter for glucose uptake 0.5

1 9 10 -4 gl -1 K O Saturation parameter for oxygen uptake

g l -1

K E Saturation parameter for ethanol uptake 0.1

g l -1 K i Inhibition parameter: free glucose inhibits ethanol uptake 0.1

h -1

k L a Mass transfer coefficient 1,000