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17
Param eter Estimation in Biochemical
Engineering M odels
A number of examples from biochemical engineer ing are presented in this
chapter. T he mathematical models are either algebraic or differential and they
cover a wide area of topics. These models are often employed in biochemical en-
gineering for the development of bioreactor models for the production of bio-
pharmaceuticals or in the environmental engineering field. In this chapter w e have
also included an example dealing with th e determination of the average specific
production rate from batch and continuous runs.
17 . 1 A L G E B R A I C E Q U A T I O N M O D E L S
17 .1 . 1 Biological Oxygen Demand
Data on biological oxygen demand versus time are modeled by the follow-
ing equation
y = kl[l-exp(-k2t)] (17 .1)
where k i is the ultimate carbonaceous oxygen demand (mg/L) and k 2 is the BODreaction rate constant (</ ') . A set of BOD data were obtained by 3
rdyear Environ-
mental Engineering students at the Technical Universi ty of Crete and are given in
Table 4.2.
Although this is a dynamic experiment w here data are collected over time, it
is considered as a simple algebraic equation model with two unk no w n parameters.
322
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Parameter Estimation in Biochemical Engineering 323
Using an initial guess of k | =350 and k 2 = l th e Gauss-Newton method converged in
five iterations without the need for Marquardt 's modification. T he estimated pa-
rameters are k,= 334.27±2.10% and k 2=0.38075±5.78%. T he model-calculated
values are compared with th e experimental data in Table 17.1. A s seen th e agree-ment is very good in this case.
T he quadratic convergence of the Gauss-Newton method is shown in Table
17.2 where the reduction of the LS objective function is shown for an initial guess
o f k , = 1 0 0 a n d k 2 = 0 . 1 .
Table 17.1 BOD Data: Experimental Data and Model
Calculated Values
T im e
(d )
1
2
3
4
5
6
7
8
B O D - Exper i -
mental Data
(mg /L )110
180
230
260
280
290
310
330
BO D - Mo delCalculations
(mg /L )105.8
178.2
227.6
261.4
284 .5
300.2
311 .0
318.4
Table 17.2 BOD Da ta: Reduction of the LS Objective Function
Iteration
0
1
2
3
4
5
6
Objective function
390384
380140
19017.3
12160.3
475 .7
289.0
288.9
k i
100
70.78
249.5
271.9
331.8
334.3
334.3
k 2
0.1
0.1897
1.169
0 .4454
0.3686
0.3803
0.3807
17 . 1 . 2 Enzyme Kinetics
Let us consider the determinat ion of tw o parameters, th e maximum reaction
rate (rmax ) and the saturation constant (K m) in an enzyme-catalyzed reaction fol-lowing Michaelis-Menten kinet ics. The Michaelis-Menten kinetic rate equat ion
relates th e reaction rate (r) to the substrate concentrations (S) by
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32 4 Chapter 17
r =Km + S
(17.2)
T he parameters are usual ly obtained from a series of initial rate exper iments
performed at various substrate concentrations. Data for the hydrolysis of benzoyl-L-tyrosine ethyl ester (BTEE) by trypsin at 30°C and pH 7.5 are given in Table
17.3
Table 17.3 Enzyme Kinetics: E xperim ental Data for the hy -
drolysis of benzoyl-L-tyrosine ethyl ester (BTEE)
by trypsin at 30 ° C andpH 7.5
s
faM)r( p M / m i n )
20
330
1 5
300
1 0
260
5.0
220
2.5
1 1 0
Source: Blanch and Clark (1996).
A s w e have discussed in Chapter 8, th is is a typical t ransformably l inear
system. Using the wel l -known Lineweaver-Burk transformation, Equat ion 17.2
becomes
K,
r r I S' m a x ' m a xvJ
(17.3)
T he results from th e simple l inear regression of (1/r) versus (1/S) are shown
graphically in Figure 17.1.
0,010 ,008
0 ,006
0,004
0,002
0
0,000
Lineweauer-Burk Plot
y = 0 , 0 1 6 9 x + 0,002
R2 = 0,9632
0,100 0 , 2 0 0 0 , 3 0 0
1 /S
0,400 0,500
Figure 17.1 Enzyme Kinetics: Results from th e Lineweaver-Burk trans-
format ion.
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Parameter Estimation in Biochemical Engineering 325
If instead we use the Eadie-Hofstee t ransformation, Equat ion 17.2 becomes
r = rn (17 .4)
T he results from th e s imple l inear regression of (r) versus (r/S) are shown
graphically in Figure 17.2.
Eadie-Hofstee Plot
60
50
40
-3020 y =-0,1363x + 63,363
10 R2
= 0,7895
0
0 50 100 150 200 250 300 350
r/S
Figure 17.2 Enzym e Kinetics: Results from the Eadie-Hofstee
transformation.
Similarly we can use the Hanes transformation whereby Equat ion 17.2 be-
comes
(17 .5)
T he results from th e s imple linear regression of (S/r) versus (S) are shown
graphically in Figure 17.3. T he original parameters rm ax and K m can be obtained
from th e slopes and intercepts determined by the simple linear least squares for all
three cases and are shown in Table 17.4. As seen there is significant variation in
the estimates and in general they are not very reliable. Nonetheless, these esti-
mates provide excellent initial guesses for the Gauss-Newton method.
Table 17.4 Enz yme Kinetics: Est imated Parameter Values
Estimation Method
Lineweaver-Burk plot
Eadie-Hofstee plotHanes plot
Nonlinear Regress ion
k ,
500
63.36434 .8
420 .2
k 2
8.45
0.1366.35
5.705
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32 6 Chapter 17
0 , 0 8
0 , 0 6
3 5 °>04
0 , 0 2
0 , 0 0
Hanes Plot
y = 0,0023x + 0 , 0 1 4 6
R2
= 0,9835
1 0 1 5
S
20 25
Figure 17.3 Enzym e Kinetics: Results from th e I lanes transforma-
tion.
Indeed, using the Gauss-Newton method with an initial estimate of
k<0 )
=(450, 7) convergence to the opt imum w as achieved in three iterations with no
need to employ Marquardt 's modification. The optimal parameter estimates are
k ,= 420.2+ 8.68% and k 2= 5.705±24.58%. I t should be noted however that this
type of a model can often lead to i l l-conditioned estimation problems if the data
have not been collected both at low and high values of the independent variable.
The convergence to the opt imum is shown in Table 17.5starting with th e initial
guess k(0 )
=( l , 1).
Table 17.5 Enzym e Kinetics: Reduction of the LS Objective Function
Iteration
0
1
2>
3
4
5
6
7
8
9
Object ive Function
3 2 4 8 1 6
3 1 2 3 8 8
308393295058
196487
3 9 1 3 1 .7
3026.12
981.260
974.582
974 .582
k ,
1
370.8
36.5223.50
7 1 . 9 2
386.7
3 5 9 . 1
4 1 7 . 8
420 .2
420.2
k 2
1
740 .3
43 .387.696
1 . 6 9 3
1 2 . 6 5
4 .356
5.685
5.706
5.705
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Parameter Estimation in Biochemical Engineering 32 7
17 . 1 . 3 Determination of Mass Transfer Coefficient (k La) in a Munic ipa l
Wastewater Treatment Plant (with PULSAR aerators)
T he P U L S A R u n i t s are high efficiency static aerators that have been devel-
oped for municipal wastewater treatment plants and have successfully been used
over extended periods of t ime without any operational problems such as unstableoperation or plugging up during intermittent operation of the air pumps (Chourda-
kis, 1999). Data have been collected from a pilot plant unit at the Wastewater
Treatment plant of the Industrial Park (Herak le ion , Crete). A series of experimentswere conducted for the determination of the mass transfer coefficient (k La) and are
shown in Figure 17.4.T he data are also available in tabular form as part of the
parameter estimation input files provided with th e enclosed C D .
In a typical experiment by the dynamic gassing-in/gassing-out method dur-
ing the normal operation of the plant, the air supply is shut off and the dissolved
oxygen (DO) concentration is monitored as the DO is depleted. The dissolvedoxygen dynamics dur ing the gassing-out part of the experiment are described by
dC O 2
dt(17.6)
where CO2 is the dissolved oxygen concentration, x v is the viable cel l concentra-
tion and q 02 is the specific oxygen uptake rate by the cells. For the very short pe-
riod of this experiment we can assume that xv is constant. In addition, when the
dissolved oxygen concentration is above a critical value (about 1.5 mg/L ) , the spe-
cific oxygen uptake rate is essentially constant and hence Equat ion 17.6 becomes,
50 100 150 200 250
Time (min)
Figure 17.4: PULSAR: Measurements of Dissolved Oxygen (DO) Concentra-
tion During a Dynamic Gass ing- in/Gass ing-out Experiment .
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32 8 Chapter 17
c02(t) = (17.7)
Equat ion 17.7 suggests that th e oxygen up take rate (qo2*v)can be obtained
by simple l inear regression of Co2(t) versus t ime. This is shown in Figure 17.5w here the oxygen uptake rate has been estimated to be 0.0813 mg/L-min.
10
9
o>E.
co£ 6
c0)ucooOQ
y = -0.0813x+ 19.329
R2= 0.9993
100
Figure 17.5:
120 140 160
Time (min)
18 0 200
PULSAR: Determinat ion of Oxygen Uptake Rate by Simple
Linear Least Squares During the Gassing-out Period.
Subsequently , during the gassing-in part of the experiment, we can deter-mine k La. In this case, th e dissolved oxygen dy nam ics are described by
dC 02
dt(17.8)
where CO2 is the dissolved oxygen concentration in equi l ibr ium with th e oxygen
in the air supply at the operating pressure and temperature of the system. This
value can be obtained from th e partial pressure of oxygen in the air supply andH e n r y ' s constant. However , Henry ' s constant values for wastewater are not read-
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Parameter Estimation in Biochemical Engineering 329
ily available and hence , w e shall consider th e equi l ibr ium concentration as an ad-
ditional unk no w n parameter.
Analytical solution of Equation 17.8 yields
or equivalent ly ,
CO2 ~
r02"
kL
a
= e-kLa[t-t0]q02
xv
— ,kLa
k , a~
C02-
( 17 .9 )
k L a
, - k L a [ t - t 0 ] (17JO)
Equation 17.10 can now be used to obtain the two unk no w n parameters (k La*
and C o 2 ) by fitting the data from th e gassing-in period of the experiment. Indeed,
using the Gauss-Newton method with an initial guess of (10, 10) convergence is
achieved in 7 iterations as s h o w n in Table 17.6. There was no need to employ
Marquardt 's modification. T he FORTRAN program used for the above calcula-
tions is also provided in Appendix 2.
Table 17.6 PULSAR: Reduction of the LS Objective Function and
Convergence to the Optimum (Two Parameters )
Iteration
0
1
23
4
5
6
7
Objective Function
383 .422
306.050
72.65799.01289
1.50009
0.678306
0 .338945
0.0891261
Standard Deviation (% )
k L a
10
1.8771
0 .73460.3058
0.1929
0.1540
0.1225
0 .1246
2.06
Co2
10
9.8173
8.51738.0977
8.2668
8.5971
9.1203
9.2033
0.586
While using Equat ion 17.10 for the above computations, it has been as-
sumed that C c o C t o )=
4.25 w as k n o w n precisely from th e measurement of the dis-
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330 Chapter 17
solved oxygen concentration taken at t0 = 207 m i n . We can re lax this assumpt ion
by treating CO2(to) as a third parameter. Indeed, by doing so a smaller value for the
least squares objective funct ion could be achieved as seen in Table 17.7 by a smal l
adjustment in C02(t0) from 4 .25 to 4 . 284 .
Table 17.7 PUL SAR: Reduction of the LS Objective Function and
Convergence to the Optimum (Three Parameters )
Iteration
0
1
23
4
5
6
7
8
9
10
1 1
12
13
14
Objec t ive Funct ion
383 .422
194.529
9.178051.92088
0.944376
0.508239
0.246071
0.138029
0 .101272
0.090090
0.086898
0.086018
0.085780
0.085716
0.085699
Standard Deviation (% )
k L a
r1 0
0.6643
0.30870.1880
0.1529
0.1375
0 .1299
0 .1259
0 .1239
0 .1228
0.1223
0 .1228
0 .1 2 1 9
0 .1218
0 . 1 2 1 8
1 . 4 5
*
CO2
10
9.6291
7.68678.3528
8.8860
9.0923
9.1794
9 .2173
9.2342
9.2420
9.2458
9.2420
9.2486
9.2490
9.2494
0.637
C02(to)
4 .25
4 .250
4.7024.428
4 .303
4.285
4 .283
4 .283
4 .283
4.283
4.283
4.283
4.284
4 .284
4 .284
0.667
In both cases th e agreement between the experimental data and the model
calculated values is very good.
17 . 1 .4 Determination of Monoclonal Antibody Productivity in a DialyzedChemostat
Linardos et al. (1992) investigated the growth of an SP2/0 derived mouse-
mouse hybridoma cell l i n e and the production of an anti-Lewish
IgM immuno-
globulin in a dialyzed continuous suspension culture using an 1.5 L Celligen bio-reactor. Growth medium supplemented with 1.5% serum was fed directly into the
bioreactor at a dilution rate of 0.45 d''. Dialysis tubing with a molecular weight
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Parameter Estimation in Biochemical Engineering 331
cut-off of 1000 w as coiled inside th e bioreactor . Fresh medium conta ining no se-
rum or serum substitutes was passed through th e dialysis tubing at flow rates of 2to 5 L/d . The objective was to r emove lo w molecu la r weight inhibitors such as
lactic acid and ammonia while retaining high molecular weight components such
as growth factors and antibody molecules. At the same t ime essential nutrients
such as glucose, glutamine and other aminoacids are replenished by the same
mechanism.
In the dialyzed batch start-up phase and the subsequent continuous operation
a substantial increase in viable cell density and monoclonal antibody ( M A b ) liter
w as observed compared to a conventional suspension culture. The raw data, pro-
files of the viable cell densi ty, viability and monoclona l antibody titer during the
batch start-up and the continuous operation with a dialysis flow rate of 5 L/d areshown in Figures 17.6 and 17.7. The raw data are also available in tabular form in
th e corresponding input file for the F O R T R A N program on data smooth ing for
short cut methods provided with th e enclosed CD .
O Raw Data + Smoothed ( 10% ) x Smoothed (5%)
200
O)
.§150
.0
I 100c<D
O
OO
J2
<
9 O
©
BatchStart-up <-—— continu
100 200 300
Time (h)
D U S operation
400 50 6 0 0
Figure 17.6: Dialyzed Chemosta t: Monoclonal antibody concentration (raw and
smoothed measurement s ) during initial batch start-up and subsequent
dialyzed continuous operation with a dialysis f low rate of 5 L/d. [re-
printed from th e Journal of Biotechnology & Bioengineering with per-
miss ion from J. Wileyj.
The objective of this exercise is to use the techniques developed in Section7.3 of this book to determine th e specific monoclonal antibody production rate
(q M ) during the batch start-up and the subsequent cont inuous operation.
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332 Chapter 17
Derivative Approach:
T he operat ion of the bioreactor was in the batch mode up to t ime t=212 h .
T he dialysis flow rate was kept at 2 L /d up to t ime t=91.5 h when a sharp drop in
th e viability w as observed. In order to increase further th e viable cell density, the
dialysis flow rate was increased to 4 L /d and at 180 A it was further increased to 5
L /d and kep t at this value for the rest of the experiment .
o Raw Data + Smo othed (10%) x Smoothed (5%)
§+ _ • _ ClGQ*
ooc 2o
BatchStart-up <4— — — *> continuous operation
100 200 300 400
T i m e ( h )
500 600
Figure 17.7: Dialyzed Chemostat : Viable cell density (raw and smoo thed m e as -
urements) during initial batch start-up and subsequent dialyzed
continuous operation with a dialysis /low rate o f 5 L /d [reprinted
from th e Journal of Biotechnology & Bioengineering with permis -
sion from J. Wiley].
A s described in Section 7.3.1 when the derivative method is used, th e spe-
cific M A b product ion rate at any t ime t dur ing the batch start-up period is deter-
mined by
dM(t)
X v ( t ) d t( 1 7 . 1 1 )
where Xv(t) and M(t)are the smoothed (filtered) values of viable cell density and
monoclonal antibody titer at tim e t.
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Parameter Estimation in Biochemical Engineering 333
A t t ime t= 2 1 2 h th e cont inuous feeding w as initiated at 5 L /d corresponding
to a dilut ion rate of 0.45 d1 Soon after cont inuous feeding started, a sharp in -
crease in the viabi l i ty was observed as a result of phys ical ly removing dead cells
that had accumulated in the bioreactor. T he viable cell density also increased as a
result of the initiation of direct feeding. A t t ime t«550 h a steady state ap peared to
have been reached as judged by the stability of the viable cell density and viability
for a period of at least 4 days. Linardos et al. (1992) used the steady state meas-
urements to analyze th e dialyzed chemostat . Our objective here is to use the tech-
niques developed in Chapter 7 to determine th e specific monoclonal antibody pro-
duction rate in the period 212 to 570 h where an osci l latory behavior of the MAb
liter is observed and examine whether it differs from the value computed dur ing
th e start-up phase.
During th e cont inuous operation of the bioreactor th e specific M A b produc-
tion rate at time t is determined by the derivative method as
In Figures 17.6 and 17.7 besides the raw data, the filtered values of MAb
titer and viable cell density are also given. The smoothed values have been ob-
tained us ing th e I M S L routine C S S M H assuming either a 10% or a 5% standard
error in the measurements and a value of 0.465 for the smoothing parameter s/N.
This value corresponds to a value of -2 of the input parameter S L E V E L in theF O R T R A N program provided in Appendix 2 for data smoothing. T he computed
derivatives from th e smoothed M Ab data are given in Figure 17.8. In Figure 17.9
the corresponding estimates of the specific M A b production rate (q M ) versus time
are given.Despite the differences between the estimated derivatives values , the com-
puted profi les of the specific M A b product ion rate are qui te similar . U p o n inspec-
tion of the data, it is seen that during the batch period (up to t=212 h), qM is de-
creasing almost monotonically. It has a mean value of about 0.5 jug/(106
cells-h).
Throughout the dialyzed cont inuous operation of the bioreactor, th e average q M isabout 0.6 j U g / ( I 0
6cells-h) and it stays constant during the steady state around t ime
t*550 h.In general, the computation of an instantaneous specific production rate is
not particularly useful . Quite often th e average production rate over specific peri-
ods of t ime is a more useful quanti ty to the experimentalist/analyst. Average ratesare better determined by the integral approach w h i c h is illustrated next .
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Parameter Estimation in Biochemical Engineering 335
In tegra l Approach:
A s described in Section 7 . 3 . 2 when the integral method is employed, th e averagespecific M Ab produc tion rate during any t ime interval of the batch start-up period
is estimated as the slope in a plot of M(t;) versus [ ' X v ( t )d t .J
t nIn order to implement the integral method, w e must compute numerical ly
th e integrals I X v ( t ) d t , t,= l , . . . , N where Xv is the smoothed value of the viable* o
cell density. A n efficient and robust way to perform the integration is through th e
use of the I M S L rout ine Q D A G S . T he necessary function calls to provide X v at
selected points in t ime are done by call ing th e I M S L routine C S V A L . O f course
the latter two are used once th e cubic splines coefficients and break points havebeen computed by C S S M H to smoo th the raw data. T he program that performs all
th e above calculations is also included in the enclosed CD. Two different valuesfo r th e weight ing factors (10% and 5%) have been used in the data smoothing.
I n Figures 1 7 . 1 0 and 17.11 the plots of M ( tj ) versus
fo r th e batch start-up period .
I'toX v ( t ) d t are shown
y = 0.4471x + 37.538
R2
= 0.9951
y = 0.7165x +4.4858
R2 = 0.9911
Figure 17.10: Dialyzed Chem osta t: Est imated values of specific M Ab production
rate versus t ime during the initial batch start-up period. A 10%
standard error in raw data was assumed fo r data smoo th ing .
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336 Chapter 17
The computation of an average specific production rate is part icular ly useful
to the analyst if the instantaneous rate is approximately constant during the time
segment of interest. By s imple visual inspection of the plot of M Ab ti ter versus the
integral of X v , one can readily identify segments of the data where the data points
are essentially on a straight l ine and hence, an average rate describes th e data satis-
factorily. For example upon inspection of Figure 17.10 or 17.11, three periods can
be identified. T he first five data points (corresponding to the t ime period 0 to 91 h)
yield an average q M of 0.7165 y.g/(l(f cells h) when a 10% weighting in used in
data smoothing or 0.7215 / u g / f l O6
cells'h) when a 5% weighting is used. The next
period where a lower rate can be identified corresponds to the t ime period 91 to
140 h (i.e., 5th, 6
th,..., 9
thdata point). In this segment of the data th e average spe-
cific M A b production rate is 0.4471 or 0.504 ng/ (106
cells h) w h e n a 10% or a 5%
weight ing is used respect ive ly . T he near unity values of the computed correlation
coefficient (R >0.99) suggests that th e selection of the data segment where th e
slope is determined by linear least squares estimation is most l ikely appropriate.The third period corresponds to the last five data points where it is obvious
that the assumption of a nearly constant q M is not valid as the slope changes es-
sentially from point to point. Such a segment can still be used and the integral
method wil l provide an average q M , however, it w ould not be representative of the
behavior of the culture during this t ime interval. I t w ould simply be a mathemati-
cal average of a t ime vary ing quanti ty .
20 0
Figure 17 . I I :
00 Oo
y = 0.504x + 27.052
R2= 0.9784
y = 0.7215x + 4.4018
R2= 0.9918
100 200 300 400 500
£ x v ( t ) d t
Dialyzed Chemostat : Es t imated values of specific M Ab produc-
tion rate versus tim e during the initial ba tch start-up period. A 5%
standard e rror in raw data was a s s umed fo r data smoo th ing .
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Parameter Estimation in Biochemical Engineering 337
Let us now turn our attention to the dialyzed cont inuous operation (212 to
570 h) . By the integral method, th e specific MAb product ion rate can be estimated
f fli 1 f
li
a s t h e slope in a p l o t o f ^ M ( t j ) + D M(t)dt \ versus X v ( t ) d t .[
J to J
Jto
I n this case besides the integral of Xv, th e integral of MAb liter must also becomputed. Obviously th e same F O R T R A N program used for the integration of Xv
can also be used for the c omp utations (see Appendix 2). The results by the integral
method are shown in Figures 17.12 and 17.13.
1400
y = 0.5233X + 320.3
R2
= 0.9953
400
400 600 800 1000 1200 1400 1600
Xv(t)dtJ to
Figure 17.12: Dialyzed Chemosta t: Est imated va lues of specific M Ab production
rate versus t ime during th e period o f continuous operation. A 10%
standard error in the raw data was assumed fo r data smoo th ing .
1400
1200
M ( t ; ) + 1 1000
" I" 'tO
800
600
400
400 600
y = 0.6171X + 291.52
R2
= 0.9976
y = 0.4906X + 345.9
R2
= 0.9975
800 1000 1200 1400 1600
x v ( t ) d tJt 0
Figure 17.13: Dialyzed Chem osta t: Est im ated va lues of specific M Ab production
rate versus t ime during th e period of continuous operation. A 5%
standard error in the raw data wa s assumed for data smoo th ing .
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338 Chapter 1 7
In both cases, one can readily identify tw o segments of the data. T he first
one is comprised by the first 8 data points that yie lded a q M equal to 0.5233 or
0.4906 jug/(106
cells h) when the weighting factors fo r data smoothing are 10% or
5% respectively. The second segment is comprised of the last 5 data points corre-
sponding to the steady state conditions used by Linardos et al. (1992) to estimate
th e specific MAb production rate. Using th e integral method on this data segment
q M was estimated at 0.5869 or 0.6171 jug/(106
cells h) as seen in Figure 17.13.
This is the integral approach for the estimation of specific rates in biological
systems. General ly speaking it is a simple and robust technique that allows a vis-
ual conformation of the compu tations by the analyst.
1 7 . 2 P R O B L E M S W I T H A L G E B R A I C EQ U A T I O N M O D E L S
17 .2 .1 Effect of Glucose to Glutamine Ratio on MAb Productivity in a
Chemostat
At the Pharmaceutical Production Research Facility of the Universi ty of
Calgary experimental data have been collected (Linardos, 1991) to investigate the
effect of glucose to g lu t amine ratio on monoc lona l antibody (anti-Lewish
I g M )
productivity in a chemosta t and they are reproduced here in Tables 17.8,17.9 and
17.10. Data are provided for a 5:1 (standard for cell culture media), 5:2 and 5:3
glucose to glutamine ratio in the feed. The dilution rate w as kept constant at 0.45
d
1
. For each data set you are asked to use the integral approach to estimate the
fol lowing rates:
(a) Specific monoclonal antibody production rate
(b) Specific glucose uptak e rate
(c) Specific glu tamine uptake rate
(d) Specific lactate production rate, and
(e ) Specific am mon ia production rate
Does the analysis of data suggest that there is a significant effect of glucose
to glutamine ratio on MAb productivity?By computing the appropriate integrals with filtered data and generating the
corresponding plots, you must determine first which section of the data is best
suited for the estimation of the specific uptake and production rates.In th e next three tables, th e fol lowing notation has been used:
/ Elapsed Time (h)
X Y Viab le Cel l Densi ty ( /O6
cel l s /ml )
vh Viability (% )
M Ab M onoclonal Antibody Concentration (mg/L)
L ac Lactate (mmo l /L )
Gls Glucose (mmol /L)
Am m A mmo ni a (mmol /L)
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Parameter Estimation in Biochemical Engineering 339
Glm Glutamine (mmol /L)
Git Glutamate (mmol /L )
xd No nviable Cell Density ( I f f cells /mL)
Table 17.8: Glucose/Glutamine Ratio: Experimental Data from a Chemostat
Run with a Glucose to Glutam ine Ratio in the Fee d of 5:1
t
0.0
1 7 . 0
7 3 . 0
9 1 . 0118.0
1 6 0 . 0
1 9 1 . 3
2 1 0 . 4
305.9
3 1 1 . 0
330.0
339.0365.0
x v
1 . 9 1
1 . 9 9
1 . 4 2
2 . 1 31 . 1 6
1 . 5 7
1.6
1 . 6 6
2.03
2 . 2 1
2 . 0 3
2.52.4
V h
0.84
0.69
0 . 8 7
0 . 8 90 . 8 9
0.93
0 . 9 2
0.9
0.83
0.8
0 . 8 5
0.850 . 8 2
MAb
58.46
6 1 . 2 5
60.3
6063.86
5 2 . 8
5 4 . 3
54
66.6
5 8 . 7 1
6 1 . 5
7 3 . 2 167.6
L ac
24 .38
3 3 . 2 1
3 2 . 1 9
32.043 1 . 1 7
35.58
36.05
34.94
34.54
3 0 . 6 1
34.50
3 8 . 5 13 4 . 7 1
Gls
7.00
5 . 8 5
5 . 6 2
6 . 2 26 . 0 2
5 . 6 7
5 . 5 0
5 . 1 3
4 .75
3.61
3.96
5 . 5 05 . 9 2
A m m
1 . 2 5
1 531
1 . 4 5
1 . 6 4
1 5 5 1
1 521
1 . 6 3
1 2 8 1
1 121
1 071
0 . 9 8
1 . 1 41 051
Glm
0 . 0 8 1
0 . 0 4 1
0 . 1 3 1
0 . 1 80 . 2 1
0 . 1 6 1
0 . 1 6 1
0 . 1 4 1
0 . 0 6
0 . 0 4 1
0 . 0 3
0 . 0 7 10 . 0 5 1
Git
0 .141
0 . 1 3 1
0 . 1 4 1
0 . 1 4 10 . 1 4 1
0 . 1 6 1
0 . 1 5
0 . 1 3 1
0 . 1 3 1
0 .111
0 . 1 0 1
0 . 1 20 . 1 3 1
Xj
0.364
0.894
0 . 2 1 2
0.2630.143
0 . 1 1 8
0 . 1 3 9
0 . 1 8 4
0 . 4 1 6
0.553
0.358
0 . 4 4 10.527
Source: Linardos (1991) .
Table 17.9: Glucose/Glutam ine Ratio: Experimental Data from a ChemostatRun with a Glucose to Glutamine Ratio in the Fe ed of 5:2
t
1 8 9 . 8
203.0
2 1 2 . 0
231.8
2 5 1 . 8
298.0
309.3
322.8
334.0
346.8
• * V
1 . 4 9
2 . 0 5
1 . 9 2
1 . 7 0
1 . 9 0
1 . 7 4
2 . 0 9
1 . 9 9
1 . 6 0
2 . 1 5
v / ,
0 . 6 5
0.76
0 . 7 5
0 . 7 1
0.77
0.80
0 . 7 8
0 . 8 0
0 . 8 1
0.84
M Ab
55.50
53.65
55.90
53.65
53.65
3 8 . 7 1
1 5 . 8 1
3 5 . 7 1
23.00
3 1 . 0 0
L ac
3 7 . 1 2
37.53
4 1 . 0 1
37.57
36.58
36.22
35.97
40.35
40.07
42.93
Gls
4 . 9 2
5 . 3 4
5 . 4 9
4 . 7 7
5 . 1 7
5.99
4 . 0 0
6 . 6 7
5 . 2 9
6 . 4 5
A m m
1 . 6 4
2 . 5 0
2 . 3 2
2 . 5 3
1 . 6 5
1 .84
1 . 4 4
1 . 3 1
1 . 8 7
1 . 4 3
Glm
1 . 9 0
1 . 8 6
2 . 0 3
1 . 4 5
2 . 0 5
2 . 1 7
1 . 5 6
2 . 0 1
1 . 7 3
1 . 6 5
Git
0 . 0 9
0.09
0 . 0 9
0 . 1 1
0.08
0 . 0 9
0 . 0 6
0.09
0.08
0.09
Xj
0.802
0.647
0.640
0.694
0.568
0.435
0.589
0.498
0.375
0 . 4 1 0
Source: Linardos (1991) .
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340 Chapter 17
Table 17.10: Glucose/Glutamine Ratio: Experimental Data from a Chemos ta t
Run with a Glucose to Glutamine Ratio in the Feed of 5:3.
t
23.933.5
39.4
49.7
58.2
72.9
97.1
119.8
153.91 8 1 . 6
1 9 2 . 3
*,.
1.952.05
1 . 8 5
2.05
1.99
2.21
2.44
2.07
2.091.94
2.25
v *
0.830 . 8 2
0.80
0.80
0.81
0.78
0.81
0.78
0 . 7 10.70
0.69
MAb
3 2 . 7 1
1 9 . 6 1
29.90
30.00
1 6 . 2 0
32.00
25.50
28.31
24 .8137.05
23.81
L ac
42.6141 .46
42.25
43.29
38.51
26.18
24.06
24.99
22.9923.09
24.27
Gls
5.624.05
5.09
4 . 5 1
3.25
3.78
2.90
2.90
2.532.09
1 . 5 6
A m m
3 . 1 1
2.90
2 . 6 1
2.80
2.80
3.90
3.90
3.59
4.636.88
6.66
Glm
3.593.40
4.60
4.67
3.01
6.26
4.96
6.00
6.387.39
6.26
Git
0.030.09
0.03
0.55
0.52
0.57
0.56
0.59
0.600.68
0.66
Xj
0.3990.450
0.463
0 . 5 1 3
0.467
0.623
0.572
0.584
0.8540.831
1 O i l
Source: Linardos (1991).
17.2.2 Enzyme Inhibition Kinetics
Blanch and Clark (1996) reported th e fo l lowing data on an enzyme cata-
lyzed reaction in the presence of inhibitors A or B. The data are shown in Table
17.11.
Table 17.11 Inhibition Kinetics: Initial Reaction Rates in the
Presence of Inhibitors A or B at Different Sub-
strate Concentrations
SubstrateConcentration
(mM)0.2
0.33
0.5
1.0
2.5
4 .0
5.0
Reaction Rate (nM/min)
N o Inhibi tor
8.34
12.48
16.67
25.0
36.2
40.0
42.6
A at 5 nM
3.15
5.06
7 .12
13.3
26.2
28.9
31.8
B at 25 y .M
5.32
6.26
7.07
8.56
9.45
9.60
9.75
Source: Blanch and Clark (1996).
In this problem you are asked to do the following:
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Parameter Estimation in Biochemical Engineering 341
(a) U se the Michae l i s - Men ten kine t ic m o d e l and the data from th e inhibi tor
f ree-exper iments to estimate the tw o u n k n o w n parameters (rm ax and K m )
r =
K
(17 .13 )
where r is the measured reaction rate and S is the substrate concentration.
(b) For each data set with inhibi tor A or B , est imate th e inhibi t ion parameter K,
fo r each alternative inhibition model given below:
Competit ive Inhibition r = (17.14)
Uncompet it ive Inhibition r =
K, s i
( 1 7 . 1 5 )
Nonncompetil ive Inhibition ''max'-' (17 .16)
where I is the concentrat ion of inhibitor (A or B) and K J is the inhibi t ion k i-
netic constant.
(c) U se the appropriate model adequ acy tests to determine which one of theinhibition models is the best.
17.2.3 Determination of kLa in Bubble-free Bioreactors
Kalogerakis and B e h i e (1997) have reported exper imental data from dy -
namic gassing-in exper iments in three ident ical 1000 L bioreactors designed for
the cultivation of anchorage dependent animal cells. T he bioreactors have an in-
ternal conical aeration area w h i l e the remaining bubble-free region contains th e
microcarriers. A compartmental mathematical model w as developed by Kalo-gerakis and Behie (1997) that relates the apparent k La to operational and bioreactor
design parameters. The raw data from a typical run are sho w n in Table 17.12 .
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342 Chapter 17
In th e absence of viable cells in the bioreactor, an effective mass transfer
coeff icient can be obtained from
= kLa[C*02-C02] (17 .17)
where C 02 is th e dissolved oxygen concentration in the bubble-free region and
€02 is the dissolved oxygen concentrat ion in equi l ibr ium with th e oxygen in the
air supply at the operating pressure and temperature of the system. Since th e dis-
solved oxygen concentrat ion is measured as percent of saturation, DO (%), th e
above equat ion can be rewrit ten as~=kLa[D0100o/0-DO] (17 .18 )dt
where th e fo l lowing l inear calibration curve has been assumed for the probe
C02 D0-D00%
€02 DO 1 0 0 % ~D0
0%
(17.19)
T he constants DO 10o% and DO 0% are the DO values measured by the probeat 1 0 0 % or 0% saturation respectively. With th i s formulat ion one can readily a n a -
lyze th e operation with oxygen enriched a i r .
U po n integration of Equation 17.18 we o btain
D 0 ( t ) = D0100o/0-[D0100%-DO(t0)]e-kL
at(17 .20 )
Using th e data shown in Table 17.12 you are asked to determine the effec-
tive k L a of the bioreactor employ ing different assumptions :
(a) Plot the data (DO(t) versus t ime) and determine DO 1 0 0 % , i.e., th e
steady state value of the DO t ransient . Generate th e differences
[D O 10o%-DO(t)] and plot them in a semi-log scale with respect to t ime.
In this case k L a can be s imply obtained as the slope the l ine formed by
th e t ransformed data points .
(b) B esides k L a consider DO 10o% as an additional parameter and estimate
simultaneous both parameters us ing nonlinear least squares estima-
tion. In this case assume that DO(tO) is k now n precisely and its valueis given in the first data entry of Table 17.12.
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Parameter Estimation in Biochemical Engineering 343
(c) Redo part (b) above, however, consider that DO(to) is also an un-
k n o w n parameter.
Finally , based on your f indings discuss th e reliability of each solution ap-
proach.
Table 17.12 Bubble-free Oxyg enation: Experim ental Data From
a Typical Gassing-in Experiment'
Time(min)
12
34
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
2223
24
25
26
2 7
D O
(%)
-0.41
0.41
1 . 2 12.10
3.00
3.71
4.60
5.31
6.10
6.81
7.50
8.20
8.81
9.40
9.90
10.40
11.00
11.40
11.90
12.40
12.81
13.2013.61
14.00
14.31
14.61
15.00
Time(min)
28
29
3031
32
33
34
35
36
37
38
39
40
4 1
4 2
4 3
4 4
4 5
4 6
4 7
4 8
4950
51
52
53
54
D O
(% )
15.31
15.61
15.901 6 . 1 1
16.40
16.71
16.90
17.21
17.40
17.61
17.80
18.00
18.11
18.40
18.50
18.71
18.90
19.00
19.21
19.30
19.50
19.6119.71
19.80
20.00
20.00
20.11
Time
(min)
55
56
5758
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
7677
78
79
80
D O
(% )
20.30
20.40
20.5020.50
20.61
20.71
20.80
20.90
20.90
21.00
21 .11
21 .11
21.21
21.21
21.21
21.21
21.30
21.40
21.40
21.50
21.61
21.6121.61
21.61
21.71
21 .71
Bioreactor N o. 1, Work ing vo lume 500 L , 4 0 RPM, A ir flow 0.08 wm.
Source: Kalogerak is and Behie (1997).
Copyright © 2001 by Taylor & Francis Group LLC
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344 Chap ter 17
17.3 O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N M O D E L S
17.3.1 Contact Inhibition in Microcarrier Cultures of MRC-5 Cells
Contact inhibition is a characteristic of the growth of anchorage dependent
cells grown on microcarriers as a monolayer . Hawbold t et al. (1994) reported data
on M R C 5 cells grown on Cytodex I I microcarriers and they are reproduced here in
Table 17.13.
Growth inhibition on microcarriers cultures can be best quantif ied by cellu-
la r automata (Hawboldt et al., 1994; Zygourak is e t al., 1991) but simpler models
have also been proposed. For example, Frame and Hu (1988) proposed th e fol-
lowing model
— =j imax l - e r p - C — — — — — j | x x(0)=x 0 (17 .21)
where x is the average cell density in the culture and u.max, and C are adjustable
parameters. The constant xm represents th e maximum cell density that can be
achieved at confluence. For microcarrier cultures this constant depends on the
inoculation level since different inocula result in different percentages of beadswith no cel ls attached. Th i s port ion of the beads does not reach conf luence since at
a low bead loading there is no cell transfer from bead to bead (Forestell et al.,
1992; Hawboldt et al., 1994). T he maximum specific growth rate, umax
and the
constant C are expected to be the sam e for the two experiments presented here as
the only difference is the inoculation level. The other parameter, X M , depends on
the inoculation level (Forestell et al., 1992) and hence it cannot be considered the
same. As initial condition w e shall consider the first measurement of each data set.
A rather good estimate of the maximum growth rate, umax , can be readily
obtained from the first few data points in each experiment as the slope of /«(x)
versus time. During th e early measurements th e contact inhibition effects are
rather negligible and hence u.max is estimated satisfactorily. In particular for the
first experiment inoculated with 2.26 cells/bead, um ax w as found to be 0.026 (K1)
whereas for the second experiment inoculated with 4.76 cells/bead a, w as esti-
mated to be 0.0253 (/?"'). Estimates of the unk no w n parameter, X M can be obtained
directly from th e data as the ma x i mum cell density achieved in each experiment;
namely, 1.8 and 2.1 (106
cells /mL) . To aid convergence and simplify programming
w e have used l / x O T rather than X O T as the parameter to be estimated.Let us consider first the regression of the data from the first experiment (in-
oculation level=2.26). Using as an initial guess [l/x T O C, umax](0 )
= [0.55, 5, 0.026],the Gauss-Newton method converged to the opt imum within 13 iterations. The
optimal parameter values were [0.6142 ± 2.8%, 2.86 ± 41.9%, 0.0280 ± 12.1%]corresponding to a LS objective function value of 0.091621. The raw data together
with th e model calculated values are shown in Figure 17.14 .
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Parameter Estimation in Biochemical Engineering 345
Table 17.13: Growth o f MRC5 Cells: Cell Density versus T i m e fo r
MRC5 Cells Grown on Cytodex 1 Microcarriers in 250 m L
Spinner Flasks Inoculated at 2.26 and 4 .76 cells/bead
Inoculation level: 2.26
(cells/bead)Time
( h )
21.8
24.2
30.2
41.7
48.4
66.5
73.8
91.9
99.2
1 1 1 . 3
118.5
134.3
144.0
158.5
166.9182.7
205.6
215.3
239.5
Cell Density
( I O6
cel ls /ml)
0.06
0.06
0.07
0.08
0.13
0.09
0.15
0.34
0.39
0.65
0.67
0.92
1.24
1.47
1.361.56
1.52
1.61
1.78
Inoculation level: 4.76
(cells/bead)Time
( h )
23.0
48.4
58.1
73.8
94.4
101.6
1 3 3 . 1
150.0
166.9
1 8 1 . 5
196.0
Cell Density
( 1 06
cells/ m l)
0.16
0.37
0.47
0.58
0.99
1.33
1.88
1.78
1.89
1.78
2.09
Source: Hawboldt et al. (1994).
Next using the converged parameter values as an init ial guess for the second
set of data (inoculation level=4.76) w e encounter convergence problems as theparameter estimation problem is severly ill-conditioned. Convergence was only
achieved with a nonzero value fo r Marquardt 's parameter (10~5). Two of the
parameters, xm and u.max , were estimated quite accurately; however, parameter C
w as very poorly estimated. The best parameter values w ere [0.5308 ± 2.5%, 119.7
+ 4 . 1x l08%, 0.0265 ± 3 .1%] corresponding to a LS objective function value of
0.079807. The raw data together with th e model calculated values are shown in
Figure 17.15. As seen, the overall match is quite acceptable.
Finally, we consider the simulatneous regression of both data sets. U sing the
converged parameter values from the first data set as an initial guess convergencew as obtained in 1 1 iterations. In this case param eters C and u.max were common to
both data sets; however , tw o different values fo r X O T were used - one for each data
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346 Chapter 17
set. T he best parameter values were l / x O T = 0.6310 ± 2.9% for the first data set and
l/xM = 0.5233 ± 2.6% for the second one. Parameters C and umax were found to
have values equal to 4.23 ± 40.38% and 0.0265 ± 6.5% respectively. The value of
th e L S object ive function w as found to be equal to 0.21890. T he model calculated
values using th e parameters from th e simultaneous regression of both data sets arealso shown in Figures 17.14 and 17.15.
2.5
2.00)oC
1 . 5
oO
1.0
0.5
0.0
Figure 17.14:
Inoculation Level = 2.26 cells/bead
Both Data Sets
Regressed
Only this Data
Set Regressed
50 100 150
Time (h)
200 250
Growth of MRC5 Cells: Measurements and mode l calculatedvalues of cell density versus t im e for MRC5 cells grow n on Cy-
todex I I microcarriers . Inoculation level was 2.26 cells/bead.
—2-5
_i£J 2 2" 5 3o
I 1.5
toI0.5"3O
Figure 17.15:
Inoculat ion Level = 4.76 cel ls /beadA
Both Data S ets
Regressed
50 20 0 25000 150
Time (h )Growth of MRC5 Cells: Measurements and model calculated
values of cell density versus t ime for MRC5 cells grown on Cy-
todex I I microcarriers . Inoculation level was 4 .76 cells/bead.
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Parameter Estimation in Biochemical Engineering 347
17 .4 P R O B L E M S W I T H O D E M O D E L S
17.4.1 Vero Cells Grown on Microcarriers (Contact Inhibition)
Hawboldt e t al. (1994) have also reported data on anchorage dependent
Vero cells grown on Cytodex I microcarriers in 250 m L spinner f lasks and they are
reproduced here in Table 17.14.
You are asked to determine the adjustable parameters in the growth model
proposed by Frame and Hu (1988)
dx X o o - xx(0)=x 0 (17 .22)
Table 17.14: Growth of Vero Cells: Cell Density versus Time for
Vero Cells Grown on Cytodex I Microcarriers in 250
mL Spinner Flasks at Two Inoculat ion Leve ls (2.00 and
9.72 cells/bead).
T im e
( h )
0.0
9.9
24.1
36.8
51.0
76.5
85.0
99.2
108.4
119.0144.5
155.9
168.6
212.6
223.9
236.6
260.7
286.9
Cell Density (106
cel l s /ml )
Inoculation level:
2.00 (cells/bead)
0.07
0.12
0.18
0.21
0.24
0.46
0.54
0.74
0.99
1.241.98
2.19
2.64
3.68
3.43
3.80
3.97
3.99
Inoculation level:
9.72 (cells/bead)
0.33
0.37
0.52
0.56
0.65
1.15
1.49
1.84
2.07
2.603.64
3.53
3.64
4.36
4.42
4 .48
4.77
4.63
Source: H aw boldt et al. (1994)
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348 Chapter 17
T he parameter est imation should be performed for the fo l lowing tw o cases:
(1) x ( 0 ) is know n precisely and it is equal to x 0 (i.e., the value given
below at t ime t = 0 ) .
(2 ) x ( 0 ) is not k n o w n and hence, x 0 is to be c onsidered ju st as an ad-
ditional measurement .
Furthermore, you are asked to determine:
(3 ) What is the effect of assuming that x 0 is u n k n o w n on the standard
error of the model parameters?
(4) Is parameter C independent of the inoculat ion level?
17.4.2 Effect of Temperature on Insect Cell Growth Kinetics
Andersen et al. (1996) and Andersen (1995) have studied th e effect of t e m -
perature on the recombinant protein production using a baulovirus/ insect cell ex-
pression system. In Tables 17.15, 17.16, 17.17, 17.18 and 17.19 w e reproduce th e
growth data obtained in sp inner f l ask s (batch cul tures) us ing B o m b y x mori (Bm5 )
cells adapted to serum-free media (Ex-Cel l 4 0 0 ) . T he work ing volume was 125
m L and samples were taken twice daily. The cultures were carried out at six d i f -
ferent incubation temperatures ( 2 2 , 26, 28, 30 and 32 <€).
Table 17.15: Growth o fBmS Cells: Growth Data taken from a Batch
Culture o fBmS Cells Incubated at 22 ° C
t
0.0
2 4 . 0
4 8 . 0
6 9 . 3
8 9 . 61 1 3 . 6
1 4 2 . 9
1 6 5 . 4
1 8 8 . 9
2 1 7 . 0
2 4 1 . 8
264.8
290.0338.0
Xy
2 . 0 9
2 . 4 3
2 . 7 0
3 . 3 2
3 . 6 64 . 1 6
7 . 4 1
1 0 . 9 5
1 7 . 2 9
1 8 . 5 0
1 9 . 0 4
2 0 . 1 5
1 9 . 3 01 8 . 4 0
Xj
2 . 1 8
2 . 6 6
2 . 9 8
3 . 7 4
4 . 0 24 . 5 5
8 . 0 3
11.44
1 8 . 4 3
1 9 . 5 0
2 1 . 0 0
23.00
2 2 . 0 122.00
Gls
2.267
2 . 1 7 0
2 . 1 3 0
2.044
1 . 9 6 71 . 7 5 3
1 . 4 0 0
1 . 2 4 0
0.927
0.487
0.267
0 . 1 0 0
L ac
0.040
0.020
0.040
0.040
0.0280.020
0.027
0.040
0.020
0.040
0.027
0.027
Source: Andersen et al. (1996) & Andersen (1995).
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Parameter Estimation in Biochemical Engineering 349
Table 17.16: Growth ofBmS Cells: Growth Data taken from a Batch
Culture o fBmS Cells Incubated at 26 ° C
t
0.0
2 4 . 0
4 8 . 0
7 0 . 5
9 0 . 5
1 1 5 . 0
1 3 7 . 8
1 6 1 . 8
1 9 0 . 8
2 1 7 . 8
234 .0
257.0
283.0 _,
x v
2 . 6 5
2 . 7 3
4 . 0 4
6 . 2 1
8 . 9 0
1 3 . 1 8
1 7 . 1 0
2 1 . 6 9
28.80
2 9 . 4 1
28.72
29.00
25.20
Xj
2 . 8 5
3 . 0 7
4 . 5 3
7 . 0 8
9 . 3 0
1 3 . 6 1
1 7 . 5 5
2 2 . 1 3
29.80
3 0 . 1 7
3 0 . 1 3
30.40
29.05
Gls
2 . 2 3
2 . 1 7
2 . 0 5
1 . 8 5
1 . 5 5
1 . 1 4
0 . 6 7
0 . 2 0
0 . 0 6
0 . 0 8
0 . 1 0
0 . 1 4
0 . 0 4
L ac
0.027
0.020
0.030
0.020
0.020
0.020
0.033
0.087
0.040
0.040
0.033
0.040
0.020
Source: Andersen et al. (1996) & Andersen (1995)
Table 17.17: Growth ofBmS Cells: Growth Data taken from a BatchCulture ofBmS Cells Incubated at 28 ° C
t
0.0
2 4 . 0
4 8 . 0
7 0 . 5
9 0 . 5
1 1 4 . 0
1 3 7 . 0
1 6 1 . 0
1 9 0 . 0
2 1 6 . 7
237.0
259.0
Xv
2 . 1 4 2
2 . 9 8 1
4.369
7.583
10.51316.416
20.677
27.458
29.685
30.452
25.890
18.900
Xj
2.292
3.300
4 . 8 1 3
8.066
10.72517.020
21.468
28.040
30.438
32.030
33.000
31.500
Gls
2.320
2 . 1 3 0
1 . 9 9 0
1 . 6 6 7
1 . 3 0 0
0.700
0.220
0.060
0.087
0.080
0.080
L ac
0.020
0.040
0.020
0.027
0.0300.020
0.080
0.053
0.020
0.027
0.040
Source: Andersen et al. (1996) & A ndersen (1995).
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350 Chapter 1 7
Table 17.18: Growth o fBmS Cells: Growth Data taken from a Batch
Culture o f B m S Cells Incubated at30°C
t
0.024.0
48.0
70.5
91.5
114.0
137.0
161.0
190.5
216.0
262.0
x v
2.8003.500
4.800
6.775
11.925
16.713
20.958
24.500
2 4 . 41 6
24.290
15.000
Xj
2.9504.066
5.296
7.700
12.499
17.500
22.208
26.354
25.833
26.708
26.600
Gls
2.502.40
1.88
1.42
0.98
0.44
0.07
0.12
0.06
0.08
0.11
L ac
0.0300.020
0.040
0.020
0.047
0.040
0.040
0.040
0.027
0.053
0.073
Source: Andersen et al. (1996) & Andersen (1995).
Table 17.19: Growth o fBmS Cells: Growth Data taken from a Batch
Culture o fBmS Cells Incubated at 32 ° C
t0.0
24.0
40.0
67.0
91.0
113.0
137.0
166.8192.3
212 .3
235.0
261.0
x v
1.98
3.12
3.64
7.19
11.57
15.55
20.50
23.6324.06
24.31
22.75
12.00
xd
2.18
3.45
4 . 1 1
7.58
11.95
16.10
21.09
24.9226.03
26.83
28.13
26.63
Gls2.260
2.100
1.950
1.610
1.060
0.527
0.087
0.0930.090
0.140
0.140
0.126
L ac0.020
0.020
0.020
0.020
0.020
0.040
0.073
0.0200.040
0.020
0.040
0.220
Source: Andersen et al. (1996) & Andersen (1995) .
In th e above tables the following notation has been used:
t Elapsed Time (h )
Viable Cell Density (10 6 cells /mL)
No nviable Cell Density (106
cel ls /ml}
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Parameter Estimation in Biochemic al Engin eering 351
Gls Glucose (g /L)
L ac Eactate (g /L)
At each temperature the s imple Monod kinetic model can be used that can
be combined with material balances to arrive at the fo l lowing unstructured m odel
=( u - k d ) - x v (17.23a)dt
dxd—5- = k d - x v (17 .23b)
dt
— =-— - x v (17.23c)dt Y v
where
= ^ m a xs
(17.23d)K S + S
T he l imit ing substrate (glucose) concentration is denoted by S. There are
four parameters: um ax i s the maximum specific growth rate, K s is the saturation
constant for S, k d is the specific death rate and Y is the average yield coefficient
(assumed constant).In this problem you are asked to :
(1) Estimate th e parameters (um ax , K s, k j and Y) for each operating tempera-
ture. Use the portion of the data where glucose is above th e threshold
value of 0.1 g/L that corresponds approximately to the exponential
growth period of the batch cultures .
(2 ) Examine whether any of the estimated parameters follow an Arrhenius-
type relat ionship. If they do , re-estimate these parameters s imul taneously .A better way to numerical ly evaluate Arrhenius type constants is through
th e use of a reference value. For example, if we consider th e death rate,
k d as a function of tem perature we have
kd = A - e R T (17 .24 )
At th e reference temperature, T 0 (usually taken close to the mean operat-
ing temperature), the p revious equation becomes
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352 Chap ter 17
__E_
kdo = A - eRT
° (17 .25)
B y div iding the tw o expressions, w e arrive at
E T j . _ , -
kd = kd ( reRV
T° ^ (17 .26)
Equat ion 17.26behaves m u c h better numer ical ly than the standard A r-
rhenius equation and it is particularly suited for parameter estimation
and/or s imulation purposes. In this case instead of A and E/R we estimate
k d o and E / R . In this example you may choose T 0 = 28Q
C .