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Identification and Estimation of the Influential Parameters in Bioreaction Systems
Mordechai ShachamBen Gurion University of the Negev
Beer-Sheva, Israel
Neima BraunerTel-Aviv University
Tel Aviv, Israel
Identifiability of Kinetic Model Parameters
*Kadam et al., Biotechnol. Prog. 20, 698-705 (2004)
Closed-system enzymatic hydrolysis of lignocellulosic biomass*
32
312
21
053.1111.1
056.1
rrdt
dG
rrdt
dG
rrdt
dS
233
3
2233
2222
2
2122
1121
2
111
1
1
1
GK
X
K
GK
GEkr
K
X
K
G
K
GSREEk
r
K
X
K
G
K
GSREk
r
IXIG
M
Fr
IXIGIG
SBBr
IXIGIG
SBr
12 Model parameters to be determined by regression of experimental data
S, G2, G –substrate, cellobiose and glucose concentrations
A Motivating Example
Identifiability of Kinetic Model Parameters
*Sin et al., Computers and Chemical Engineering, 34, 1385 – 1392 (2010)
Kadam et. al and Sin et al.* used the same experimental data set in order to identify the 12 parameters of the model.
Observe large differences in parameter values Confidence intervals greater (in absolute value) than the parameters
Kadam et al.
No. Symbol Definition Value Value Conf. Int.
1 k 1r Reaction rate constant 22.3 19.16 ±830.39
2 K 1IG 2 Inhibition constant for cellobiose 0.015 0.07 ±3.3
3 K 1IG Inhibition constant for glucose 0.1 0.03 ±1.214 K 1IX Inhibition constant for xylose 0.1 0.16 ±6.84
5 k 2r Reaction rate constant 7.18 5.74 ±140.31
6 K 2IG 2 Inhibition constant for cellobiose 132 91.14 ±11265.3
7 K 2IG Inhibition constant for glucose 0.04 0.07 ±1.63
8 K 2IX Inhibition constant for xylose 0.2 0.93 ±23.11
9 k 3r Reaction rate constant 285.5 98.38 ±1775.56
10 K 3M Cellobiose saturation constant 24.3 104.86 ±2335.93
11 K 3IG Inhibition constant for glucose 3.9 0.77 ±3.64
12 K 3IX Inhibition constant for xylose 201 326.62 ±14306
Sin et al.*
Identifiability of Kinetic Model Parameters
Confidence interval (CIi) greater (in absolute value) than the parameter (PVi) suggests that PVi=0 is statistically justified.
CIi ≥ PVi the corresponding parameters are unidentifiable.
For this problem Sin et al (2010) found (based on CIi / PVi ratios) that there are 45 Identifiable Parameter Subsets (IPS) containing only 6 (out of 12) parameters.
Relying solely on statistical considerations for determining the optimal IPS for nonlinear regression problems is insufficient and may result in an unacceptable physical model.
Demonstration of Some Model Structure Constrains in IPS selection
32
312
21
053.1111.1
056.1
rrdt
dG
rrdt
dG
rrdt
dS
233
3
2233
2222
2
2122
1121
2
111
1
1
1
GK
X
K
GK
GEkr
K
X
K
G
K
GSREEk
r
K
X
K
G
K
GSREk
r
IXIG
M
Fr
IXIGIG
SBBr
IXIGIG
SBr
H(ierarchy) = 1 params
H = 2 parameters
H =3 parameters
Parameters that cannot be set at 0
SREEkSREkdtdy SBBrSBr 212111 /
2 1 1/ 1.056 r B SdG dt k E R S
2 1 2/ 1.111 r B B SdG dt k E E R S
An initial feasible model with 2 parameters
Stepwise Regression-in Dynamic Parameter Estimation Problems
Using stepwise regression, a complex model is built sequentially by adding one parameter (with the associated terms of the explanatory variables) to the model at every step, until a stopping criterion is satisfied.
Initially, parameters constrained to nonzero values and parameters (and the associated terms) required to represent the available data at t = 0 are included in the model.
Consequently the parameter whose addition causes the greatest reduction of the objective function value is selected for inclusion in the model at every step.
To assure convergence to a feasible physical model, the search should be constrained to a physically valid parameter values space
Stopping criterion- When for all remaining non-basic CIi ≥ PVi
Initial Model Analysis and Modification
for Nonlinear, Constrained Parameter Estimation Problems
Definition of new parameters to enable addition of one parameter at a time to the model and setting the rest of the parameters at zero
Classification of the parameters’ hierarchy within the model’s various expressions
Keeping parameters constrained to nonzero values in explicit forms
Selection of initial feasible set of parameter values from amongst the 1st hierarchy and nonzero constrained parameters, which is sufficient for representing the initial trend of the state variables and ensures that none of the state variables remains constant during the integration.
Description of the example problem: Kinetic Model of ethanol fermentation1
x - concentration of cell mass
s - concentration of glucose
p1 - concentration of ethanol
p2 - concentration of glycerol
0.1 ≤ Yp1/s ≤ 0.51 - ethanol yield factor
0.1 ≤ Yp2/s ≤ 0.2 - glycerol yield factor.
1 Liu and Wang, Comput. Chem. Eng. 33, 1851-1860, 2009.
xqdt
dp
xqdt
dp
Y
q
Y
q
dt
ds
xdt
dx
p
p
sp
p
sp
p
22
11
/2
2
/1
1
1 2
2 2 21 1 1 1 2 2 2 2
'1 1
1 ' 2 ' ' 2 '1 1 1 1
'2 2
2 '' 2 '' ' 2 '2 1 1 2
/ / /
/ /
/ /
p pm
s SI p p I p p I
p pp
s SI p p I
p pp
s SI p p I
K Ks
K s s K K p p K K p p K
s Kq
K s s K K p p K
s Kq
K s s K K p p K
19 parameters
Part of the Data Used for Parameter Identification1,2
1Wang, et al., I&EC Research, 40, 2876-2885, 2001. 2 Gennemark and Wedelin, Bioinformatics, 25, 780-786, 2009.
Point No.: i t (h) xi(g/L) si(g/L) p1i(g/L) p2i(g/L)
1 0 0.25794 197.938 0 0
2 1 0.35163 196.3455 0 0
3 2 0.50495 195.443 0 0
4 3 0.91191 192.0625 3.0597 0.2169
5 4 1.512 187.3675 4.8155 0.83731
6 5 2.8031 178.5835 7.5538 1.1603
7 6 4.4029 166.403 13.0554 1.6974
8 7 6.2635 144.1565 19.5166 2.4533
9 8 8.6109 124.5265 25.3135 3.0111
10 9 9.6147 102.8158 35.0076 3.8146
11 10 11.2518 85.5552 43.1744 4.5128
12 11 12.1692 68.7319 50.5293 5.1625
13 12 13.1241 51.4266 59.1247 5.8394
14 13 14.2433 36.2531 66.1442 6.2358
15 14 15.0539 24.549 71.8272 6.574
16 15 16.1507 8.4547 78.4983 6.8424
17 16 16.5711 2.1814 81.2294 6.9718
18 17 16.9961 0.52 85.8299 7.142
19 18 17.2255 0 85.9216 7.2436
x - cell mass concentration
s - glucose concentration
p1 - ethanol concentration
p2 - glycerol concentration
Example Problem - Initial Model Analysis and Modification
xqdt
dp
xqdt
dp
xqxqdt
ds
xdt
dx
p
p
pp
322
211
8271
2218217
2166
2
2119115
2145
1
2213212
2111110
294
1
1
1
1
1
1
1
1
1
1
1
1
ppss
sq
ppss
sq
ppppss
s
p
p
Definition of new parameters to enable addition of one parameter at a time to the model and setting the rest of the parameters at zero.
The 19 original parameters were replaced by new parameters π1, π2,… π19.
H(ierarchy) = 1 params
0.1 ≤ Yp1/s ≤ 0.51
0.1 ≤ Yp2/s ≤ 0.2π7 = π2/Yp1/s
π8 = π3/Yp2/s
The initial minimal parameter set includes Yp1/s, Yp2/s, π1 , π2 , π3.
Search for Optimal Parameter Values for a Particular Parameter Set
Subject to:
2
, ,1 1
*minn m
ci i i
i
x x W
maxmin
0)0(
,,
xx
xFx
tdt
d
1,{ }maxi iW x
0.06
0.07
0.08
0.09
0.1
0.11
0 5 10 15 20
Parameter No.
Ob
ject
ive
Fu
nct
ion
Initial Basic Parameter Set and Search for an Additional Parameter to Enter the Set
π12
Initial basis
Selected optimal parm.
No. Parameter
Estimate Optimal PV CI
1 Y p 1/s 0.51 0.50785 0.51 -
2 Y p 2/s 0.2 0.19606 0.2 -
3 π 1 0.002 0.0022795 2.580E-03 1.176E-04
4 π 2 0.01 0.010761 9.036E-03 6.049E-04
5 π 3 0.0008 0.0009852 8.386E-04 5.140E-05
6 π 12 1.125E-01 4.811E-02
0.72787 0.098133 0.067776 -
Initial basis
Objective function
1st stage
All PVi > CIi
Parameters Identifiable with Statistical JustificationNo. Parameter
PV CI PV CI
1 Y p 1/s 0.51 - 0.51 -
2 Y p 2/s 0.2 - 0.2 -
3 π 1 2.447E-02 1.257E-02 2.302E-02 1.214E-02
4 π 2 1.699E-02 3.047E-03 1.514E-01 1.742E-01
5 π 3 7.771E-04 3.346E-05 7.810E-04 3.114E-05
6 π 12 6.907E-01 1.452E-01 6.691E-01 1.381E-01
7 π 5 9.123E-03 3.452E-03 1.139E-01 1.322E-01
8 π 4 3.405E-02 1.842E-02 3.206E-02 1.808E-02
9 π 19 - - 3.107E-04 1.366E-04
0.029413 - 0.02465 -
3rd stage 4th stage
Objective function
xqdt
dp
xqdt
dp
xqxqdt
ds
xdt
dx
p
p
pp
322
211
8271
CIi > PVi
sq
s
sq
ps
s
p
p
2
51
2124
1
1
1
1
1
π7 = π2/Yp1/s
π8 = π3/Yp2/s
Plot of the Experimental Data and Calculated Values Using the Optimal 8 Parameters Model
Very good agreement between the calculated and experimental values
0
50
100
150
200
250
0 5 10 15 20
Time (min)
Co
nc
en
tra
tio
n (
g/L
)
s
p1
sc
p1c
s - concentration of glucose
p1 - concentration of ethanol
Plot of the Experimental Data and Calculated Values Using the Optimal 8 Parameters Model
Very good agreement between the calculated and experimental values
x - concentration of cell mass
p2 - concentration of glycerol
0
5
10
15
20
0 5 10 15 20
Time (min)
Co
nc
en
tra
tio
n (
g/L
)
x
p2
xc
p2c
ConclusionsIn many parameter estimation problems only a subset of the parameters can be identified (i.e., assigned significant value) using the available experimental data.
Considerations related to the model’s structure, parameter hierarchy, constrains on parameter values, initial trend of the data and reduction of the objective function value need to be employed in order to estimate the optimal values of the Identifiable Parameter Subset (IPS).
A new stepwise regression procedure for nonlinear models, based on these principles has been developed for detection of the IPS and estimating the optimal parameter values.
The proposed procedure has proven to be successful in identification of a kinetic model of ethanol fermentation
Plot of the Experimental Data and Calculated Values Using the Initial 5 Parameters Model
The initial model already follows well the trend of the data
s - concentration of glucose
p1 - concentration of ethanol
0
50
100
150
200
250
0 5 10 15 20
Time (min)
Co
nc
en
tra
tio
n (
g/L
)
s
p1
sc
p1c
Plot of the Experimental Data and Calculated Values Using the Initial 5 Parameters Model
x - concentration of cell mass
p2 - concentration of glycerol
0
5
10
15
20
0 5 10 15 20
Time (min)
Co
nc
en
tra
tio
n (
g/L
)
x
p2
xc
p2c
The initial model already follows well the trend of the data