Biotechnol. frog. 1994, IO, 480-488
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Optimal Experimental Design for Parameter Estimation in Unstructured Growth Models M. Baltes, R. Schneider, C. Sturm, and M. Reuss' Institut fiir Bioverfahrenstechnik, University of Stuttgart, Allmandring 31, D-70569 Stuttgart, Germany
A method of optimal experimental design for parameter estimation in unstructured growth models is presented. The approach is based on a method suggested by Munack (1991) for application in fed-batch processes. In a critical analysis of this method, special emphasis is given to the model validity, because unstructured growth models often are not valid under transient conditions. In consequence, a combined object function has been introduced, which considers model validity and the accuracy of the kinetic parameters to be estimated. The application of this method for fed-batch processes leads to satisfactory results. Investigations of different fed-batch strategies regarding model validity and the quality of parameter estimation are presented. I n addition, a n experimental verification has been performed with fermentations of the yeast Trichosporon cutaneum.
Introduction The estimation of kinetic parameters in unstructured growth models with a high parameter accuracy is essential for successful model validation. The application of optimal experimental design for fed-batch processes will be illustrated for Monod growth kinetics:
from which the saturation constant K,and the maximal growth rate p,, have to be identified. Parameter estimation in unstructured growth models is often performed with the aid of continuous fermentations. These experiments are time-consumingand complex in nature. In contrast, more simple experiments can be achieved with batch fermentations. However, the kinetic parameters of the Monod model cannot be identified from these experiments, as has been pointed out by Nihtila and Virkkunen (1977). From the application of sensitivity analysis, Holmberg (1981) demonstrated that the behavior of the sensitivity functions is strongly correlated. From the nearly linear dependency between the parameters p,, and K,, it follows that a variation in the parameter p- may be compensated by a corresponding variation in K,. Hence, the parameters may not be uniquely determined in practice. Munack (1989) showed that an identification error functional (output leastsquares error criterion) is obtained that is extremely flat in one direction. To illustrate the situation, if the values of the saturation constant and the maximal growth rate are k e d at K,= 60 mg/L and p,, = 0.39 lh,a flat valley in the direction of AK$Ap,, = 10 occurs (Figure la), causing a relative error in the determination of K,of 58% compared to a 1%relative error in pmm. This clearly indicates that no sufficient parameter estimation of K, can be obtained from batch experiments. Another approach to identify kinetic parameters has been suggested by Esner and Roels (1981) and by Turner and Ramkrishna (19891,who performed fed-batch experiments for the estimation of the maintenance constants in Monod-type kinetics or cybernetic models. Following
* Author to whom all correspondence should be addressed.
0.08-
0.07-
0.06-
0.05-
0.384
0.386
0.388
0.390
Pmax
0.392
0.394
0.396
[l/hI
Figure 1. Plots of the identification functional J for a batch process (a) and an optimized fed-batch process with a constant feeding rate (b).
this idea, Munack (1989)introduced a method for optimal experimental design of the fed-batch experiments to increase the quality of parameter estimation of pmaxand K,. This approach, based on the Fisher information matrix, was f i s t addressed by Goodwin and Payne (1973) for predicting optimal input functions in the time domain. The Fisher information matrix contains information about parameter sensitivities and measurement errors and, thus, permits a quantification of the quality of parameter estimation. The aim is to calculate an optimal feeding rate related to this functional. Munack (1989) obtained an optimal feeding profile for the estimation of parameters pmmand K,.The suggested feeding program,
8756-7938/94/3010-0480$04.50/00 1994 American Chemical Society and American Institute of Chemical Engineers
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however, shows enormous gradients in the feeding rate. I t is then questionable whether Monod-type kinetics are still valid under these dynamic conditions. The question is justifiable because this kinetic approach is found to be applicable only under steady-state or slowly changing environmental conditions. An essential requirement for the validity of the model is balanced growth, a biological state during which the intracellular metabolic reaction network is operating at steady-state conditions. If an unstructured model is applied to dynamic conditions, changes in the cell environment must be slow enough to guarantee a sequence of steady states in the metabolic network. This is the case in batch cultures during exponential growth or in continuous fermentation processes. Mor and Fiechter (1968) demonstrated that shift experiments from one steady state to another could not be described by Monod kinetics. This clearly indicates that dynamic conditions can only be simulated with the application of structured models. A class of potentially useful models are, for instance, compartment models (Harder and Roels, 1982), cybernetic models (Ramkrishna et al., 1992), and timedelay models (Reuss and Goetz, 1992). Given this context, fed-batch experiments are restricted to only small rates of change of substrate concentration that have been obtained with constant or exponential feeding rates mamane and Shimizu, 1984). Dynamic steady states, which have been termed “quasi steady states” (Pirt, 19741, will be achieved for constant flow rates when the specific growth rate is maintained exactly equal to the resulting dilution rate. This state is characterized by a constant value of biomass concentration and a slowly decreasing substrate concentration. The other state, however, where dc$dt = 0, can be realized only in exponential fed-batch cultures. The concept of Munack (19911, who suggested piecewise linear functions with a finite number of node points for the unknown time profile of the feeding rate, is being varied in the present paper in that simple functions, like u = ut b, have been chosen. By predicting the parameters of the function, which optimize the functional of parameter accuracy, optimal feeding rates can be obtained. One advantage of this method is a remarkable reduction in computing time. An important disadvantage of this method, however, is the fact that optimal prediction depends on the predefined function type. A critical analysis of the special problem, an optimization of the feeding rate for unstructured growth models, will be able to answer the following two questions: (1) What kind of functions have to be chosen to guarantee satisfactory results in parameter accuracy? (2) Does the method of optimal experimental design guarantee model validity? In this context, an experimental verification of different feeding strategies with the model system Trichosporon cutuneum will be shown.
+
Theoretical Background The Model. The material balance equations for biomass and substrate in a batch process can be written as
-dC_S - - p1c
,
- mscx
(3)
dt Yxls where p denotes the specific growth rate, for instance, the Monod equation (eq 1). Then m,is the maintenance constant (Pirt, 1965) and yd, is the yield biomass coef-
Table 1. Initial Conditions and Model Parameters 0.54 g of celldg of substrate Yd8 Pmax 0.39l/h KS 0.06 giL m, 0.037l/h 9gn 15 L 0.5 g L 6.0 giL
CBf
vo
c,(t=O) c,(t=O)
ficient. For a fed-batch process, we may define a timevarying dilution of the reactor contents: (4)
where u(t) is the time-varying feeding rate and U t ) is the reactor volume. The following balance equations for the fed-batch processes can then be formulated as
-dcx - pc, dt
- D(t)c,
The feeding concentration, which is held constant, is denoted as csf. Initial conditions along with the process and kinetic parameters used for optimal experimental design are listed in Table 1. Optimal Experimental Design. The problem to be solved is the design of an optimal experiment for a fedbatch process with regard to kinetic parameters p = and states x = (cx,c,)*:
= f(x,u,t,p)
(7)
y = g(x(t,u,p)) (8) where u denotes the time-varying feeding rate, which has to be optimized. For identification of the unknown parameters p, the difference Ay of the time-discrete model output y [in the case of an unstructured growth and measurements yMis weighted model: y = (cx(t,),cS(tJ)?l in a quadratic criterion: N
(9)
where Ayi = y(ti,u,p) - yM(ti),fi is the vector of estimated parameter, and Q is the positive definite symmetric weighting matrix. For the optimal estimate fi = fi*, the functional J reaches its minimum. A zero value of the identification functional cannot be expected because in practical situations the states are not measurable without noise, causing a disturbance E of the “true” measuring output
Y:
+
(10) yM(tJ = y(tJ €(ti) It is assumed that ci represents uncorrelated white noise processes with zero mean E(c(ti)) = 0, i = 1, ..., N
(11)
E(c(ti)&tj)) = Gijc(ti), i , j = 1,...,N
(12)
and covariance where C is the measurement covariance matrix, which
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can be formulated to the corresponding problem as loooo
L
To illustrate the effect of measurement noise, a series of identical experiments has to be performed. The measured outputs will be different due to the measurement noise and will lead to different identified parameters P*. A very important task of optimization of the measurements is therefore posed by the requirement that the variance of the estimates has to be as small as possible, which may be formulated as
-
A(E(P*- pxP* - plT) min (14) where A is a functional used to weight the covariance. Munack (1988) suggested a quantification of parameter accuracy from the Fisher information matrix F. It is assumed that the process may be linearized with respect to the nominal parameters P, which have been estimated from preliminary experiments along the nominal trajectory xo: y(t,u,p+dp)= y(t,u,p)+
*I
dp = y(t,u,p)+ ap XO>U,P yp(t)dp(15)
where yp denotes the output sensitivities, which are obtained from the total differential
Batch
u=a
ucaebbl
u=atbAt u=a sin(b At)-c cos(d At)
Figure 2. Comparison of different types of feeding rates regarding the functional A. Table 2. Conditions for ODtimal Emerimental DesiPn sampling frequency 15 l/min t,r* 295 min 540 min duration of process consequence, A should be as near as possible to 1to yield
a functional whose shape should look like a funnel. Then a high quality of parameter accuracy will have been achieved. Time-varying feeding rates have to be optimized to give the best shape of the functional, which results in the following optimization problem:
-
A (F) min
u(t)
+
and xpis the state sensitivities: (17)
Inserting this linearization into the identification functional J (eq 9), and taking the expectation of the functional with assumptions (eqs 11 and 121, one obtains N
E(J(u,p+Gp))= dpTICy,T(u,ti,p)Q~~(u,ti,~)Idfi -tc i=l
(21)
(18)
with c being a constant. The Fisher information matrix F of the estimation problem (Ljung, 1987) can then be obtained by choosing the weighting matrix Q as the inverse of the covariance matrix C:
For feeding rates, simple functions, like u = a bAt with At = t - tsf, have been chosen whose parameters are optimized with the simplex algorithm (Nelder and Mead, 1964). The fed-batch process starts at t = t,$. The feed concentration c,f is held constant. Conditions of optimal experimental design are shown in Table 2. Determination of the Error in Glucose and Biomass Measurements. For the application of optimal experimental design, the quantification of errors in measurements is essential. The covariances a, and a, were determined from measurements of different glucose and biomass concentrations, from which 10 samples of equal concentration were analyzed. The resulting correlations can be formulated as
: o = (3.9
N
Comparison of Different Feeding Rates. Figure This matrix is the inverse of the parameter estimation error covariance matrix and permits quantification of the quality of the parameter estimation. As a consequence of eq 18, F has t o be positive definite. If this very basic condition is fulfilled, then the corresponding experiment is said to be informative (Goodwin, 1987). A quantification of the quality of parameter estimation, A, has been suggested by Munack (1989): (20)
where denotes the largest and ni,& the smallest eigenvalue of the Fisher information matrix F. The functional A may be used as a measure for the shape of the functional J, because the lines of constant functional values J form ellipses whose axes are the eigenvectors of F. The lengths are proportional to the inverses of the square roots of the corresponding eigenvalues A. As a
2 summarizes the values of functional A, which have
been obtained from different assumptions regarding the type of function u = f f t ) . In agreement with Munack (19891, a batch experiment delivers the highest value of functional A. The situation can be improved by choosing constant or time-varying feeding rates. It is interesting to notice, however, that the differences in the absolute values of functional A are rather small, indicating that very simple functions are sufficient for improvements in parameter accuracy. This can also be seen from the functional plot corresponding to the simplest function, u = const, as shown in Figure lb. The resulting shape is very satisfactory. The flat valley from a batch experiment (Figure l a ) can be changed to almost a cone.
Materials and Methods Microorganism and Growth Conditions. A n experimental verification of different feeding strategies was
483
Biotechnol. Prog., 1994, Vol. 10, No. 5 Table 3. Medium Composition per 1 L of Medium in Batch Culture 8.5 g/L (experiment 1) 7.0 g/L (experiment 2) 17.9 gfL 5.3 g/L 3.7 giL 0.3 g/L CaC03 0.2 giL KC1 mc03 4.6 g/L MgSOc7H2O 3.1 g/L FeC13 50.0 mg/L CuS04.5HzO 8.0 mg/L 35.0 mg/L MnS04eH20 30.0 mg/L ZnSO4 biotin 0.1 mg/L calcium panthothenate 100.0 mg/L myo-inositol 200.0 mg/L pyridoxine 5.0 mgL thiamine 25.0 mg/L
rBatch
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6
8
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s 0.020.00.
performed with the model system Trichosporon cutaneum (DSM 70698) maintained on H+S-agar slants at 4 "C. The yeast was grown in batch and fed-batch cultures at 30 "C in a stirred tank reactor (Bioengineering, Wald, Switzerland). The air flow was maintained at 12 l/min, while the pH was controlled automatically at 5.0 by the addition of 2 N NaOH and 2 N HC1. To prevent foam formation, small quantities of Contraspum 210 (Zschimmer & Schwarz, Lahnstein, Germany) were added. The medium composition is given in Table 3. Determination of Biomass Concentration. For determination of the biomass concentration, 5 mL of suspension was centrifuged for 10 min at 4000 rpm (Biofuge,Heraeus, Hanau, Germany) in previously tared centrifuge tubes. The sediment was washed twice with 0.9% NaCl solution and then dried to a constant weight at 105 "C. Determination of Glucose Concentration. Glucose was determined enzymatically with the glucose dehydrogenase reaction using a test kit (Granutest 250, Merck, Darmstadt, Germany) after rapid filtration (0.45pm filter, ME 25, Schleicher & Schiill, Dassel, Germany). Rapid Sampling and Sample Preparation. Rapid samples were taken aseptically with vacuum-sealed, precooled (-10 "C) glass tubes (Pyrex, France) through a specially developed sampling device (Theobald et al., 1991). The high-grade steel capillary has an inside diameter of 0.7 mm, resulting in a sampling time of less than 200 ms from the mixing zone of the fermentor to the valve outlet. The glass tubes were filled with 2 mL of 2 N K~HPOI.The probe volume was determined after difference weighing. Fed-Batch Experiments. Two fed-batch experiments were performed with different feeding rates optimized in regard to the functional A: experiment 1, constant feeding rate; experiment 2, time-variable feeding rate of type u = aAt. The experimental conditions are shown in Table 4. The fed-batch processes were started if a substrate limitation was recognized (look at optimal verification of the switching point). The feeding inlet concentration was held constant. The feeding rate of the pump (Watson-Marlow, Cornwall, England) was controlled through an IBM-compatible computer (CD Computer, Freiburg, Germany) with an on-board industrial interface (Burr-Brown, Tucson, AZ). Experimental Verification of Different Feeding Strategies. Figure 3 shows the comparison between model prediction and experimental observation of a fedbatch experiment with a constant feeding rate. A reasonable estimation of kinetic parameters from this experiment is possible, resulting in satisfactory agree-
"
'
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u "
v
"
'
e-
P .-
Y
n 3.
100 150 200 250 Time [min] Figure 3. Experimental verification of a fed-batch process with a constant feeding rate.
0
50
Table 4. Conditions for Fed-Batch Experiments expt 1 expt 2 u = u(t - taf)with feeding rate u = uo with a = 0.8 l/h2 uo = 2.6 1/h Caf 9.2 giL 52.6 g/L vo 10.5 L 14 L Table 5. Kinetic Parameters, Identified from a Fed-Batch Experiment with a Constant Feeding Rate Pmau 0.41 l/h KS 12 mg/L ma 0.02 l/h 0.58 g of celldg of substrate Y ds
ment between measurements and simulations. The kinetic parameters, which were identified with a simplex algorithm (Nelder and Mead, 19641, are shown in Table 5. As a consequence of nearly quasi-steady-state conditions @ x D),only small rates of change of specific growth rate p are obtained. Only after the fed-batch process has been started can a steep decrease in p be observed. In contrast, remarkable deviations between model predictions and measured data are observed in an experiment with a time-varying feeding rate of the type u = a(t - t 3 , as shown in Figure 4. Drastic rates of change of p can be observed. These deviations from quasi-steady-state conditions cause a violation of Monod kinetics. As a consequence of the violation, it is impossible to estimate a reasonable set of parameters. This clearly indicates that the optimal experimental design method should be applied with caution to estimate parameters in unstructured growth kinetics, which is usually valid only for quasi-stationary conditions.
Improvements in the Method of Optimal Experimental Design Criteria of Model Validity. As a consequence of the limited validity of unstructured growth models, the functional A has to be extended in order to guarantee model validity.
Biotechnol. Prog., 1994, Vol. 10, No. 5
484
1
-
J
Batch
Fed-Batch
I
I-
i
-0
F r
20
0.25t I I
\
K=5
0.20-
I
0
16-
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- sir
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260
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7 Y
"
'
300
"
"
'
340
'
'
'
'
'
380 420 Time [min]
460
500
540
Figure 5. Time courses of specific growth rate resulting from feeding rates of the type u = a + b(t - t,f*). A (eq 20) has to be combined with @:
% 0.2 0.0
300 400 500 600 Time [min] Figure 4. Experimental verification of a fed-batchexperiment with a linear feeding rate of the type u = d t - t,d.
0
100
200
The restriction that balanced growth must exist for relations of unstructured growth models to be valid puts a strong limitation on the rate of change. To design fedbatch experiments, the rates of change of substrate concentration in the fermentor have to be minimal. A critical biological criterion with regard to balanced growth is the gradient of specific growth rate with time, which is shown here for Monod kinetics:
&="( dt dt pmaxKs A) + c,
(22)
and
(23) By combining these equations with eqs 3 and 6, growth rate equations for batch and fed-batch processes are obtained:
Changes in specific growth rates are obtained under substrate limitation, which may occur at the end of batch processes and in fed-batch processes. Similar to an output least-squares error functional, the deviation from steady-state conditions can be quantified as (26) Large deviations from steady-state conditions cause high values of the functional @ and a decrease in model validity. In order to create the criterion for optimal experimental design for unstructured growth models, the functional
where K denotes a factor penalizing violations of model validity. The ,aim is to optimize time-varying feeding rates, u(t), regarding improvements in the quality of parameter estimation with only minimal violations of model validity, which may be formulated as min (28)
-
u(t)
Influence of the Criterion of Model Validity. To illustrate the effect of consideration of model validity, K has been varied by choosing a feeding rate of the type u = a b(t - t,f*). Simulations show that an increase in K leads to remarkable improvements in model validity. It is interesting to note, however, that K rarely influences the criterion of parameter accuracy, which is some indication that the problem of optimal experimental design is constrained by model validity. An interpretation of improvements in model validity can be given from the plot of time courses of specific growth rate. Figure 5 shows that smooth time courses of p can be obtained if the criterion of model validity has been sufficiently considered. In contrast, drastic gradients of p are observed for small values of K. As a consequence, it seems reasonable to perform optimizations with the extended criterion 6 (eq 271, from which K has been chosen as K = 50.
+
Optimal Verification of the SwitchingPoint. For the experimental verification of the simulated process conditions, a definite switching from the batch to fedbatch process is essential. A successful strategy can be performed by starting the fed-batch process at the beginning of substrate limitation. Appropriate system variables that can be used for an experimental verification of this point are dissolved oxygen concentration a n d or oxygen concentration in the exhaust air. Both variables increase with time if substrate limitation occurs, as shown in Figure 6 . The calculation of the switching point can be obtained from the solution of the material balances of named states: dc, - ~ u2 - = kla(c*02 - co,) - p 1 c , (29) dt Yxlo,
Biotecbnol. frog., 1994,Vol. lo, No. 5
4% 21 .o
90
20
switch point
0 calculation according eq. 31 and 32
15-
80 .
s
I
:
8 75:
10-
:
'0
3 :
0 70-
'3
65 yo2
n "
0
Figure 6. Experimental observations of the switching point. I s
I
10
20
40
30
50
, l.o
251
- - - - - - - - - - -- 20
_*-----------
30
c i -
f
3
-
8
15
-----_____ -------_-____ -,'
10.4
I
10
lo
Y 7
15-5
5t
e
-10 -.2A
II
I
Figure 8. Influence of substrate inlet concentration. Table 6. Parameters of Gas Balances 0.03 mmoVg 900 l/h 300 l/h 43800 bar 0.15
dY"0 2= dt
(d
- l)[&ao, V(t>
- ywo2)- kLaVn(c*o,- c0J]
(30) where kLa is the volumetric gas-liquid mass-transfer coefficient, EG is the gas hold-up, c*o2 is the saturation concentration of dissolved oxygen, CO, is the dissolved oxygen concentration, ydo2 is the yield coefficient, VG is the gas flow rate, and V,, is the gas volume under standard conditions. From these equations, a minimum in the gas concentration CO, or yWozindicates a substrate limitation at t = tsf*. Process and kinetic parameters used for the simulations are listed in Table 6. Investigations of this method show that a rather robust design can be obtained from starting fed-batch processes at the defined switching point, t,f*. Figure 7 summarizes the results of A and CP regarding a constant feeding strategy. A minimum of named functionals can be observed, nearly corresponding to the beginning of substrate limitation at t = &*. Earlier switching causes a significant decay in parameter accuracy. Later switching leads to a drastic violation of model validity, because substrate is then consumed and will increase with time after the fed-batch process has been started.
Results The aim of this project was to look for simple and robust feeding strategies that allow satisfactory results
of parameter accuracy of kinetic parameters from unstructured growth models. The introduction of the extended objective function (eq 27) and the experimental verification of the switching point are the bases for this method. The task of the user is to look for a suitable solution regarding the type of function. Helpful criteria to evaluate this problem are (1)quality of the parameter estimation, (2) deviation to steady-state conditions, (3) robustness of the feeding strategy, and (4) experimental effort. Strategy with Constant Feeding Rate. Because fed-batch experiments with constant feeding rates are regarded as the simplest systems, it seems reasonable to perform investigations of this strategy. Depending on the substrate inlet concentration, c.f, only one parameter, the constant feeding rate, ug, has to be optimized. It can be seen from Figure 8 that only small effects are obtained. This may be somewhat of an advantage for experiments with small reactor volumes, because the feeding rate can be decreased by choosing high concentrations of csf. Another possibility to design fed-batch processes with constant feeding rates can be performed by the assumption of quasi-steady-state conditions, p = D. Dunn and Mor (1975) suggested different possibilities to approximate the required condition. Satisfactory results regarding deviations of p from D are obtained by making use of material balance eqs 5 and 6. With the assumptions of Wdt = dD/dt and dc$dt = 0, it is possible to calculate the following equations for prediction of the substrate inlet concentration c.f and the constant feeding rate uo:
(32) The results from calculations of uo and c,f are shown in
Biofechnol. frog., 1994,Vol. 10, No. 5 0.401
I
h!-+ Fed batch
I u=a+bt
I
0.251
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u= a eM
I
\
0.15 o.20
,
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300
,
,
340
,
,
380
,
,
,
420
,
460
,
,
,
500
1 540
Time [min]
Figure 9. Time courses of specific growth rate, p , regarding optimizations with the extended objective function, 6.
4n _."
i
t
50
0
u=uo
u=atbAt
u=ae
bbt
Figure 10. Robustness characteristics of different feeding strategies with regard to a single parameter perturbation in pof about *lo%.
Figure 8. I t is interesting to note that good agreement with corresponding conditions of optimal experimental design can be obtained. This leads to a very simple possibility of optimal experimental design for this type of unstructured growth model. Comparison to More Complex Feeding Strategies. Functions of the polynomial and exponential type have been chosen to show the effects of more complex functions. The results clearly indicate that remarkable improvements in model validity can be attained from time-varying feeding rates. An interpretation can be
Figure 11. Influence of feeding parameters on the functional A.
given from Figure 9, showing time courses of specific growth rates p corresponding to the optimized feeding rates. Very smooth time courses of p are obtained by choosing exponential or polynomial feeding rates. In contrast to a constant feeding strategy, much smaller rates of change are observed in the transition time after the fed-batch process has been started. For realization of this method, robust characteristics are essential for experimental verification. Deviations from parameters used for optimal experimental design may cause a significant decay of parameter accuracy or model validity in the experiment. From our own investigations (Sturm, 1991), perturbations of the parameter pmax mostly influence functional A. Therefore, pmaxhas been chosen as a control parameter for the comparison of different feeding strategies to determine their robustness characteristics. Figure 10 shows relative deviations of A and Q, after a perturbation ofpm, of about 110%.A significant influence from the chosen type of function can be obtained. Only small deviations in named criteria can be obtained for a constant feeding rate, indicating rather robust characteristics for this feeding strategy. For this type of kinetics, a satisfactory compromise may be attained by the application of a constant feeding rate. The disadvantage is that deviations from steady-state conditions occur in the transition time after the fed-batch process has been started, which can be compensated only by the application of more complex functions. Indeed, the experimental verification of a constant feeding strategy (Figure 3) shows satisfactory results, indicating that these perturbations can be compensated by Monod kinetics. An important advantage of this strategy is the remarkably robust characteristics, which may also be seen from variations in the feeding conditions. The threedimensional plot (Figure 11) shows that a flat valley exists in which only small changes in functional A can be observed.
Conclusions The problem of the identification of kinetic parameters in a Monod-type model is considered. Investigations clearly indicate that the method of optimal experimental design for unstructured growth models should be applied with caution regarding model validity. Experimental verification of different feeding strategies performed with the model system Trichosporon cutaneum
Biotechnol. Prog., 1994,Vol. 10, No. 5
clearly shows that only fed-batch processes with small rates of change in substrate concentration allow a reasonable estimation of the kinetic parameters. In the case of high violations of model validity, fed-batch experiments cannot be described by the Monod equation. As a consequence of the limited model validity of unstructured growth models, the functional of optimal experimental design has been extended to an objective function considering model validity, from which remarkable improvements may be obtained. Otherwise, the dynamic response may be described only with more complex models, for instance, compartment models (Harder and Roels, 198l), cybernetic approaches (Ramkrishna et al., 19921, or structured models considering the dynamics of ribosome synthesis (Reuss and Goetz, 1992). Investigations indicate that the method of optimal experimental design for unstructured growth models can be performed with very simple types of functions. A rather robust design can be obtained from the application of a constant feeding rate. Improvements with regard to deviations from steady-state conditions can be obtained by the application of time-varying feeding rates. The application of this method shows that the optimization of feeding strategies is constrained to model validity. This kind of problem is also particularly important for the trajectory optimization of fed-batch processes, such as the control of substrate feeding of penicillin fermentations, because the substrate control is linear in the Hamiltonian, resulting in bang-bang control policies (Reuss, 1986). Any abrupt change in the feeding of substrate will probably not only drift the culture from its predicted optimal course but may also result in the production of undesirable byproducts and other unforeseen changes in the intracellular metabolism. The situation is further complicated in those situations in which a feedback control based on an on-line estimation of the system’s state is to be superimposed on the feed-forward control. Due to the fact that the system parameters required for the indirect estimation can change drastically in the case of the variations mentioned in the cellular metabolism, the system becomes difficult to control. Thus, it may happen that the optimal feeding strategy totally fails, and the fermentations altogether then have no significance.
487
t*f* U
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vo VG
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optimal switching point, min time-varying feeding rate, l/h constant feeding rate, l/h reactor volume, L initiai condition of reactor volume, L gas flow rate, l/h gas volume under standard conditions, Ummol state vector vector of state sensitivities vector of model output vector of measuring output inlet volume fraction of oxygen volume fraction of oxygen in the exhaust of fermentor vector of output sensitivities yield coefficient of oxygen, mmoVg yield coefficient of substrate
Greek Symbols parameter perturbation time difference between fed batch startup and process time, min error of measurements gas hold-up functional of quality of parameter estimation minimal eigenvalue maximal eigenvalue specific growth rate, l/h maximal growth rate, l/h extended objective function of unstructured growth models variance of measurements of substrate, (g/LI2 variance of measurements of biomass, (g/LI2 functional of model validity, l/h3
Literature Cited Dunn, I. J.; Mor, J. R. Variable-volume continuous cultivation. Biotechnol. Bioeng. 1976,17, 1805-1822.
Esener, A. A.; Roels, J. A.; Kossen, N. W. F. Biotechnol. Bioeng. 1981,23.
Notation measurement covariance matrix, (g/LI2 substrate concentration, g/L substrate inlet concentration, giL dissolved oxygen concentration, mmoVL saturation concentration of dissolved oxygen, mmoVL biomass concentration, giL dilution rate, l/h Fischer information matrix identification functional Henry constant of oxygen, bar penalty factor for model violation volumetric gas-liquid mass-transfer coefficient,
Goodwin, G. C. Identification: Experiment Design. In Systems and Control Encyclopedia; Pergamon: Oxford, U.K., 1987; Vol. 4,pp 2257-2267. Goodwin, G. C.; Payne, R. L. Design and characterization of optimal test signals for linear SISO parameter estimation.
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Abstract published in Advance ACS Abstracts, June 1, 1994.
