Engineers (Chemical Industries) by Peter Englezos [Cap 17]

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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

http://dk5985_ch04.pdf/





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



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.







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



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

Copyright © 2001 by Taylor & Francis Group, LLC



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 .

Copyright © 2001 by Taylor & Francis Group LLC



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-





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-

http://dk5985_appendix2.pdf/





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




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.



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.




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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f _ g ~ s^ ts- C t3Oa5TO

(_ Oo NOo GOO -OO Ooo Gi o

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———

————

————

————

———

—

——

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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 .




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 .





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 .






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)




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) .



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:




(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 .



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.




(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).




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 .




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



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.




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)




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).




Table 17.16: Growth ofBmS Cells: Growth Data taken from a Batch


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).




350 Chapter 1 7


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).



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}



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




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 .

Engineers (Chemical Industries) by Peter Englezos [Cap 17]

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