Biotechnol. Prog. 1093, 9, 174-178
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Biomass Estimation in Plant Cell Cultures Using an Extended Kalman Filter Joan Albiol; Jordi Robust6 Carles Casas, and Mane1 Poch Unitat d'Enginyeria Quimica, Institut d'Enginyeria Bioquimica UAB-CSIC, Universitat Autbnoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Use of carbohydrate concentration measurements linked t o a digital filter was studied for biomass estimation in plant cell cultures. Due t o the high complexity of the biological system, the strategy followed has been t o work with a simple model and t o rely on the filter t o adapt the values of the model parameters t o changes in the system behavior. Results obtained with this approach have shown the method t o be efficient for biomass estimation, both in Erlenmeyer flasks and in bioreactors with changing operational conditions.
Introduction Higher plants have been an important source of bioactive compounds for a long time. Over the last decade, major progress has been made toward the use of plant cell cultures as a source of economically valuable biochemicals such as enzymes, pigments, flavonoids, or alkaloids, and a number of process systems are in commercial operation. Success in achieving a high yield of the product is largely dependent on the knowledge of the behavior of the system. Thus, it is indispensable to understand the growth kfnetics of plant cells, as well as to carefully monitor it, in order to obtain the most fundamental information for the design, optimization, and control of plant cell cultures. Different methods have been used in order to estimate cell growth, such as oxygen consumption, protein or nucleic acid content, fresh or dry cell weight, mitotic index, pH, or conductivity (Taya, 1989), measurements which are more or less directly linked with biomass evolution. A comparative study can be found in Ryu et al. (1990). Plant cells, which are larger than microorganisms, grow in aggregates with a very slow growth rate and usually adhere to the bioreactor walls. These facts make it difficult to obtain a homogeneous, low-volume sample and to monitor cell growth carefully with conventional methods (such as volumetric determinations, gravimetric measurements on a wet or dry basis, or cell counting after protoplast-forming treatment). These methods require a great number of samples and/or large volumes to obtain the desired accuracy. The total volume of samples should be kept small in relation to the total volume of the culture in order to maintain the same culture conditions. Therefore, the number of samples becomes limited, especially on a laboratory scale. Moreover, increasing the sampling frequency increases the risk of microbial contamination, which is especially dangerous in these systems due to slow growth rates. In other cases, such as hairy root cultures or immobilized cells, direct measurement of biomass is not possible until the end of the culture. In recent years there has been development of the socalled software sensors. A software sensor is an algorithm for the on-line estimation of state variables and the parameters which are not measurable in real time, based on related measurements which are more easily accessible (Bastin, 1990). Biomass identification through the measurement of related variables may be achieved using a digital filter, provided a mathematical model description 8756-7938/93/3009-0174$04.00/0
is available, or using an adaptive inferential estimation in those cases in which a mathematical description is not available. One of the digital filters most used in nonlinear systems is the extended Kalman filter (EKF), which has been used for the identification of biomass growth in other fermentation systems (San, 1984; Valero, 1990). The utilization of EKF assumes that the model and the initial values of its parameters are k n o w . Also, a certain knowledge of the variability of the system expressed as a noise matrix is assumed. The work presented reports on the utility of using the time course of the change in sugar concentration in the medium, linked to a digital filter, in order to follow the evolution of a plant cell culture without having to measure biomass too frequently, therefore diminishing the volume of samples withdrawn from the system. Moreover, the evolution of kinetic parameters of the model is used in assessing the system behavior and troubleshooting identification.
Materials and Methods Cell Line and Culture Methods. Daucus carota cells were obtained from a root explant in solid medium made with a Murashigue Skoog basal salt mixture supplemented with 0.1 g L-' myoinositol, 0.5 mg L-l nicotinic acid, 0.5 mg L-' pyridoxine-HCl, 0.5 mg L-' thiamine-HCl, 2 mg L-l glycine, 1pM (2,4-dichlorophenoxy)aceticacid, 1pM kinetine, 30 g L-' sucrose or glucose, and 10 g L-' agar. Suspension cultures were obtained by placing some calluses in the medium described above without agar in a reciprocal shaker a t 100 oscillations min-l and subcultured every 2 weeks for more than a year. All cultures were incubated in the dark at 30 "C. Experiments in Erlenmeyer flasks were performed as follows: several 100-mL erlenmeyers, with 20 mL of the culture medium, were inoculated a t the same time with 5 mL of a 7-day-old culture. Each flask was used to obtain one experimental point. Cultivation in a bioreactor was performed using either a l - L reactor (Biolab Braun-Melsungen) or a 4-L reactor with a low shear stirring system (Celligen, New Brunswick Scientific, Edison, NJ). Operational conditions were 30 "C, 100 rpm, and 60% dissolved oxygen without using a pH controller. The culture medium used was the same as above, and bioreactors were inoculated with 50 mL of a previous culture.
0 1993 American Chemical Society and American Instltute of Chemical Englneers
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Carbohydrate Analysis. Sucrose, glucose, and fructose levels were analyzed using an HPLC system (HewlettPackard 1090) coupled to a refractive index detector (Hewlett-Packard 1037A)with a Biorad HPX-87N column. The running conditions were as follows: eluent, 15 mM NaZS04; column temperature, 80 "C; flow rate, 0.5 mL min-'. Biomass Determination. For cell mass determination, 20-mL culture samples from the bioreactors or the contents of one Erlenmeyer flask were filtered using a preweighted glass fiber filter (Nucleopore D49). Dry cell weight was measured after drying the filter a t 80 "C until constant weight was obtained. Cell mass concentration was calculated over filtered medium volume instead of total volume, as explained below. Mathematical Methods. The EKF is an extension of the Kalman filter linear approach (Kalman, 1960; Kalman and Bucy, 1961) to nonlinear ordinary differential equations. The EKF methods optimally tries to estimate the state of the system by assuming that the behavior of the system is described by a nonlinear model and that the mean and the covariances of the measurement errors are known. The nonlinear model can be expressed in a general form by dx = f(x,t) + € ( t )
dt where x is the state vector, a set of dependent variables which adequately describe the situation in the bioreactor, including some parameters of the model. The function c(t) represents the random disturbances of the system. The observations are related to the state vector by Y = h(x) + t ( t )
(2)
where &t) represents the noise of the measurements. It is assumed that the system and the measurement noises are independent random Gaussian white noises with zero mean and covariance matrices Q and S. The EKF works in two steps: (1) A prediction step between observations in which the uncertainty in the estimates increases with time; the state vector and the variance are calculated by f(t/t,-,) = f(f,t) (3) P(t/t,-,) = f,(f)P + PfTW + Q
(4)
and (2) a correction step that takes place when observations occur. Then, the estimate is corrected by a term that is proportional to the difference between the actual measurement and the prediction:
K(t,) = P(t,/t~_l)h,T[6xP(t,/t~-l)~xT + SI-' (7)
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Figure 1. D. cnrota cell line batch culture. Initial values for the variables were C, = 1.05 g L-l, C, = 33 mM, C, = 13 mM, Cf = 28 mM, V = 0.025 L. Dots are experimental data; continuous
lines are the computer simulation.
To develop the model, several experiments were carried out in Erlenmeyer flasks. Some of the results obtained can be seen in Figures 1and 2. Figure 1shows that sucrose is hydrolyzed, obtaining glucose and fructose which are consumed by cells to yield an increase in dry weight. Glucose is consumed with preference over fructose, and cells take up water during growth. Figure 2 shows the behavior of the system when only glucose is used. It must be mentioned that a low level of fructose appears in this culture which comes from the inoculum. It will be assumed that cell cultures can be characterized by the dry weight as well as carbohydrate concentration in the culture medium, namely sucrose, glucose, and fructose. As the sugar concentration obtained from the analytical method is referenced to the filtered culture medium, biomass concentration is calculated over this volume to simplify the system. Also, the variation of culture medium volume with time is taken into account. Variation of the filtered culture medium volume is assumed to be mainly due to water evaporation and water uptake by the cells. Other effects are considered negligible. Glucose and fructose appear from sucrose on an equimolecular basis and are consumed to give dry weight. Also, part of the dry weight is assumed to be consumed for maintenance purposes. The specific growth rate is assumed to be a function of the sugar concentration with Monod type kinetics. The preferential consumption of glucose over fructose is taken into account using competitive type kinetics. Sucrose invertase is assumed to be located on the cell wall (Fowler, 1982), and so its concentration is taken as proportional to the dry cell weight. Taking all of these assumptions into account, the following equations result:
(8)
where K is the gain matrix.
Results and Discussion Model Development. Due to the high complexity of biological systems, the strategy followed has been to work with a simple model and to rely on the filter in order to adapt the values of the model parameters to changes in the behavior of the system. This strategy gives us valuable information about the system besides biomass estimation.
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In order to obtain the parameter values of the model, numerical integration of the equations was carried out using a fourth-order Runge-Kutta algorithm followed by minimization of the sum of weighted square differences from experimental data obtained in flask cultures, using a Davidon-Fletcher-Powell algorithm (Himmelblau, 1972). The mean values of the parameters obtained from different data fittings were used for biomass estimation of two different experiments with different initial sugar and biomass concentrations. The results, obtained by integration of the equations using initial concentrations as the values of the variables a t t = 0, are shown in Figures 1and 2 as continuous lines. The values of the parameters used in each experiment are shown in Table I.
Biomass Estimation. Biomass estimation in plant cell cultures can be done using a set of differential equations expressing the time rate of change of the state variables as function of the operating and culture parameters. This is what we see in Figures 1and 2 where the integration of the differential equations with the appropriate initial conditions gives the evolution of the culture variables along with time. However, any event that occurred during the fermentation time having an effect on the above-mentioned model parameters (and not taken into account by the model) will result in the failure of the description of the
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culture evolution and, in this case, of the biomass estimation. Also, any experimental error in the determination of the values of the variables used as initial conditions is not taken into account by this method. In this work, the general approach proposed by Stephanopoulos (1984) was investigated for its application to the biomass estimation in plant cell cultures. This approach is based on the extension of the state vector to include, in addition to the state variables, the culture parameters which are not desirably expressed by models. However, in this case instead of the estimation of a growth parameter such as p, a model parameter such as pmm or rmaxis estimated in order to easily identify any disagreement from the model prediction. That is to say, those parameters are considered to be constant and changes reflect either a change in the behavior or a malfunction. To predict biomass using EKF, the filter was applied to the data of experiments shown in Figures 1and 2. The c h o s e n s t a t e v e c t o r of t h e s y s t e m w a s (C,,Cg,Cf,C,,V,rmax,pmax) and the measurement vector was (Cs,Cg,Cf). The values in the state vector not present in the measurement vector are corrected by the filter a t each step, according to the evolution of variables in the measurement vector. System noise and measurement noise covariance matrices were determined as in Valero (19901, although different approximations exist (Shimizu et al., 1988). The system noise covariance matrix and measurement noise covariance matrix are taken as two diagonal matrices with determined diagonal elements: [1.E-4,1.E-4,1.E-4,1.E-15,1.E-15,1.E-5,1.E-2] and [ 1.E3,1.E-3,1.E-3]. The application of the EKF to the case in which model parameters do not change with time can be done using the data of Figures 1 and 2. The results, given in Figures 3 and 4, show that the corrected values given by EKF adequately follow the biomass evolution without changing the parameter values. This is what is expected since the model has been fit to these experimental data. Figure 4 shows the values given by the filter in a system where only glucose is used instead of sucrose. Similar experiments performed with fructose alone give similar results, showing that the method can be used regardless of the carbohydrate used.
Parameter Evolution. With the inclusion in the state vector of the parameters rmax and pmax as well as x , it is expected to give not only a reliable, noise-free biomass estimation but also information about the evolution of culture parameters. Any unexpected change in the parameter values will be due to other effects not considered in the mathematical model; therefore, useful information about process malfunctions is given, greatly facilitating the study of the performance of plant cell cultures. The other utility of the filter lies in its ability to estimate biomass even when changes in the behavior of the system occur during the fermentation time, also taking into account the possible random variations that occur in sampling. Neither of these effects could be accomplished simply by integrating the model equations using initial values. To test this ability, cultures with behavior different from that observed in the shake flasks, such as a lag phase, slow
Table I. Values of the Parameters Used in the Simulation/Estimation of Each Figure 1,3 2,4
5 6
0.03 0.03 0.03 0.03
0.03 0.03 0.03 0.03
0.2 0.2 0.2 0.2
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0.025 0.025 0.025 0.025
0.02 0.02 0.02 0.02
57 57 57 57
0.004 0.004 5 x 10-5 2 x 10-5
0.014 0.014 0.014 0.014
0.06 0.06 0.06 0.06
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Figure 3. Experimental data from Figure 1. Continuous lines are EKF estimations;dots are experimentaldata. Initial values were the same as in Figure 1.
Figure 5. D.carota cell line culture ina Braun Biolab bioreactor. Dots are experimental data; lines are EKF estimations. Initial values for the variables were C, = 0.58 g L-I, C, = 60 mM, C, = 26 mM,Cf= 25 mM, V = 1.05 L.
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Bioreactor. Dots are experimental data; lines are EKF estimations. Initial values were C, = 0.58 g L-I, C, = 79 mM, C, = 6 mM, Cf= 5 mM, V = 4.05 L. was hydrolyzed to free hexoses and, although the concentration of monomers was increasing, growth did not start until day 25. This effect could be due to the low initial cell concentration, which extends the time needed by the culture to adapt to the medium. This change in the biomass behavior is correctly identified by the filter in both cases, and the value of the biomass given by the filter closely approximates the measurements. Also in both cultures, a growth limitation appeared before the sugars were exhausted: in this case a failure in the air supply system caused the culture to stop growing. The reduction
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in the amount of sugar being consumed by the cells is correctly identified as a reduction in the growth rate. In addition to biomass prediction, the evolution of model parameters can also give us valuable information about the process. In our case, the slow growth rate during the lag phase causes a reduction in the pmaxparameter, which informs us that the reduced growth rate is not due to the low hexose concentration and therefore must be due to another unknown factor. Later, when the cells have adapted to the culture medium, this parameter again reaches its maximum value for that system. The rmax parameter is also reduced to take into account the reduced sucrose hydrolysis rate observed in the system, and a change in its value can be interpreted as a variation in its concentration related to the biomass concentration.
Conclusion Using a digital filter it is possible to monitor the biomass evolution in plant cells with very low volume samples. It is also possible to check any significant change in the behavior of the system reflected in the change of the estimated parameters. Moreover, as carbohydrate concentrations can be easily monitored on-line, the procedure followed in this article can be implemented in on-line monitoring of plant cell culture.
Notation fructose concentration in culture medium (mol L-1) glucose concentration in culture medium (mol L-I) sucrose concentration in culture medium (mol L-1) biomass dry weight concentration (g L-1) evaporation coefficient (day') function for the state dynamics measurement equation filter gain matrix dry weight decay due to endogenous metabolism (day') glucose inhibition coefficient on fructose consumption (mol L-l) saturation constant for invertase (mol L-1) saturation constant for monosaccharide uptake (mol L-') state estimation covariance matrix covariance matrix of the process noise intensity maximum invertase activity (mol day1 g-1) sucrose conversion rate (mol day1 gl) covariance matrix of the measurement noise intensity time (days) total medium volume (L) biomass water uptake coefficient (L gl) biomass water uptake coefficient in stationary phase (L g-1) general state vector observation vector biomass yield by substrate consumption (g mol-')
Greek Letters random disturbances of the system dt) specific growth rate due to fructose consumption Pf (day') specific growth rate due to glucose consumption Pg (day') maximum specific growth rate (day') P,,, total specific growth rate (day') C(T .$(t) noise of the measurement Superscripts and Subscripts estimated values partial differentiation of the indicated function [ I, respect to x
Acknowledgment This work received financial support from the CIRITGeneralitat de Catalunya and the Ma Francisca Roviralta Foundation. J.A. was supported by a FPI grant from the Ministerio de Educaci6n y Ciencia, Spain. Literature Cited Bastin, G.; Dochain,D. On-line Estimation and Adaptive Control of Bioreactors; Elsevier: Amsterdam, 1990. Bond, P. A.; Fowler,M. W.; Scragg,A. H. Growth of Catharanthus roseus Cell Suspensions in Bioreactors: On-line Analysis of Oxygen and Carbon Dioxide Levels in Inlet and Outlet Gas Streams. Biotechnol. Lett. 1988, IO, 713-718. Fowler, M. W. Substrate Utilisation by Plant-Cell Cultures. J . Chem. Technol. Biotechnol. 1982, 32, 338-346.
Himmelblau, D. M. Applied Nonlinear Programming; McGraw-Hill: New York, 1972; pp 111-123. Kalman, R. E. A New Approach to Linear Filtering and Prediction Problems. Trans. ASME: J . Basic Eng. 1960, 82, 33-45. Kalman, R. E.; Bucy, R. S. New Results in Linear Filtering and Prediction Theory. Trans. ASME: J . Basic Eng. 1961, 83, 95-108.
Ryu, D. D. Y.; Lee, S. 0.;Romani, R. J. Determination of Growth Rate for Plant Cell Cultures: Comparative Studies. Biotechnol. Bioeng. 1989,35, 305-311.
San, K.; Stephanopoulos, G. Studies on On-Line Bioreactor Identification. 11. Numerical and Experimental Results. Biotechnol. Bioeng. 1984, 26, 1189-1197.
Shimizu, H.; Takamatsu, T.;Shioya, S.; Suga, K. An Algorithmic Approach to Constructing the On-line Estimation System for the Specific Growth Rate. Biotechnol. Bioeng. 1989,33,354364.
Stephanopoulos, G.; San, K. Studies on On-Line Bioreactor Identification. I. Theory. Biotechnol. Bioeng. 1984,26,11761188.
Taya, M.; Hegglin, M.; Prenosil, J. E.; Bourne, J. R. On-line Monitoring of Cell Growth in Plant Tissue Cultures by Conductometry. Enzyme Microb. Technol. 1989, 11, 170176.
Valero, F.; Lafuente, J.; Poch, M.; Soli, C. Biomass Estimation Using On-LineGlucose Monitoring by Flow InjectionAnalysis. Appl. Riochem. Biotechnol. 1990, 24, 591-601.
Accepted December 7, 1992.
