Abstract
In this work, the ability of artificial neural nets was investigated for the online biomass prediction of the simulated growth of a strain of Bacillus thuringiensis in fedbatch mode. For this purpose, multilayered backpropagation nets with sigmoid nodes were trained. The patterns were composed of input data on current values of biomass concentration, limiting substrate concentration and dilution rate, and output data on prediction of biomass concentration for the following step. The dilution rate was disturbed by a PRBS input, and simulations were conducted using a phenomenological experimentally validated model. The nets were able to predict the biomass concentration for different feeding techniques, and they were also compared with the variable estimation technique using the extended Kalman filter.
neural networks; extended Kalman filter; fedbatch; Bacillus thuringiensis
COMPARISON OF BIOMASS ESTIMATION TECHNIQUES FOR A Bacillus thuringiensis FEDBATCH CULTURE
^{1}C.C.F.Cunha and ^{2}M.B.Souza Júnior^{*} * To whom correspondence should be addressed
^{1} Department of Chemical and Process Engineering, University of Newcastle upon Tyne Merz Court,
Newcastle upon Tyne NE1 7RU, Phone: +44 191 2225332/ Fax: +44 191 2225292 U. K.
Email: C.C.F.Cunha@newcastle.ac.uk
^{2} Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Bloco E, Escola de Química, Ilha do Fundão
CEP. 21949900, Phone: (0xx21) 5627636/ Fax: (0xx21) 5627567, Rio de Janeiro  RJ, Brasil
Email: mbsj@h2o.eq.ufrj.br
(Received: September 20, 1998 ; Accepted: October 23, 2000)
Abstract In this work, the ability of artificial neural nets was investigated for the online biomass prediction of the simulated growth of a strain of Bacillus thuringiensis in fedbatch mode. For this purpose, multilayered backpropagation nets with sigmoid nodes were trained. The patterns were composed of input data on current values of biomass concentration, limiting substrate concentration and dilution rate, and output data on prediction of biomass concentration for the following step. The dilution rate was disturbed by a PRBS input, and simulations were conducted using a phenomenological experimentally validated model. The nets were able to predict the biomass concentration for different feeding techniques, and they were also compared with the variable estimation technique using the extended Kalman filter.
Keywords: neural networks, extended Kalman filter; fedbatch, Bacillus thuringiensis.
INTRODUCTION
Software sensors are sophisticated monitoring systems that can relate the less accessible state variables and the variables that can be measured during the process. These tools are useful in fermentative process control applications, for which state variables are not always available. Control of these processes around the optimum increases yield and reduces operational costs. However, this is a difficult task, because it involves a number of complex biochemical reactions (Zhang et al., 1994).
For this research, the growth phase of a strain of Bacillus thuringiensis  a bioinsecticide producer  was chosen. Fedbatch simulations were then carried out, because of their flexibility in monitoring the growth phase.
The justification of this choice is that biological insecticides are more efficient than chemical ones, and are not harmful to human beings and nontarget animals and plants. The importance of the optimization of such a process is related to the tropical climate, which is characterized by greater vulnerability to insect pests, both in agriculture and in urban centers.
METHODS
Software sensors are programs that apply process models and estimation algorithms to estimate variables and parameters that are not easily measured or that are available only after a significant time delay from the online measured data (Bastin and Dochain, 1990).
In biological processes, the main variables (such as biomass concentration, substrate concentration and metabolites) are usually determined by laboratory analysis. The cost and the time required for these analyses limit sampling frequency. Software sensors use online data to estimate these variables. For this purpose empirical models, such as artificial neural nets (ANNs) and Kalman filters (or observers), which can be combined with structured or nonstructured models, are applied.
Modeling of the Process under Study
In this work, a nonsegregated and nonstructured perspective and the presence of only one limiting substrate were assumed.
A fedbatch process can be described by an overall balance, where the culture volume (V) variation with time (t) is equal to the volumetric feed rate (f), according to the equation below (Dunn and Mor, 1975):
The mass balance of cells and substrate can be described by equations (2) and (3), respectively:
Biomass
Substrate
where s_{F} is the limiting substrate concentration in the feed and Y_{x/s} is the yield coefficient.
The dilution rate  defined as the rate between the volumetric feed rate, f, of the limiting substrate and the culture volume, V (equation 4)  was used as the control variable.
In this case, overall balance is not considered for problem solution, because the biomass and substrate equations do not depend explicitly on volume, as the equations (6) and (7) below show:
Overall Balance
Biomass
Substrate
The specific growth rate, m, was determined by the Monod (Monod, 1942) and Hump (used by Agrawal et al., 1982) kinetic models.
(1) Monod model:
Relates the specific growth rate, m, with the limiting substrate concentration, s, as follows:
where parameters m_{max} and k_{s} are the maximum specific growth rate and the saturation constant, respectively.
(2) Hump model:
Considers the possibility of growth inhibition in the presence of determined concentrations of the limiting substrate and presents only two parameters, k_{1} and k_{2}, as follows:
The simulations were carried out under the following conditions for a strain of Bacillus thuringiensis var. israelensis in a modified GYS medium (Kang et al., 1992), where glucose was considered the limiting substrate: m_{max} = 0.672 h^{1}; k_{s} = 0.331 g/L; k_{1} = 0.398 L/gh; k_{2} = 4.535 g/L; x_{0} = 0.31 g/L; s_{0} = 6.05 g/L; Y_{x/s} = 0.643 g(cell)/ g(subst.); V_{0} = 5.0 L; V_{max} = 10.0 L; f_{max} = 2.5 L/h; s_{F} = 20.0 g/L (Cunha et al., 1998).
The dilution rate was varied between its minimum (D = 0.0 h^{1}) and maximum (D = 0.5 h^{1}) values by a pseudorandom binary signal (PRBS). The PRBS changes randomly between 1 and 1, and the dilution rate was calculated by the following equation:
Artificial Neural Nets (ANNs)
Neural networks have been proved to be suitable for the development of prediction models (e.g. Glassey et al., 1994; Wilkinson and Yuksel, 1997; Hajmeer et al, 1998).
ANNs technology was inspired by simplified theories about how the human brain works. The ANNs are a computational paradigm in which a dense distribution of simple processing elements (neurons), connected by weights (synapses), process input/output data by a mechanism called "training", providing a nonlinear relation (De Souza Jr. et al., 1996).
The neuron can receive a variety of input signals, s_{pi,k} (i = 1, ..., n_{k}), which correspond to the output (or activation) of the n_{k} neurons of the previous layer, but give only one output signal, s_{pj,k+1}. The input signals are modulated by weights, w_{jik}, associated with each information channel. Subsequently the neuron sums up all its inputs weighted by the weights to an internal threshold (bias, q_{j,k+1}) (Zhang et al., 1994). Mathematically, neuron behavior can be expressed by (De Souza Jr. et al., 1996)
The activation of individual neurons can be modeled as a sigmoid function (equation 12), which is the most frequently used for the multilayered nets and is used in this work (De Souza Jr. et al., 1996).
(1) Training ANNs
Training is a procedure for obtaining the neural net parameters, based on the minimization of an objective function that gives the difference between the true values (experimental or simulated) and those predicted by the net. In this process, the weights and the biases are modified.
One of the most commonly used training methods is the backpropagation. The nets trained by this technique are multilayered, without connections between neurons of the same layer, but with feedforward signals. A multilayered net is composed of (Zhang et al., 1994): an input layer (which receives the information from external signal sources and can be composed of various receptors); an output layer (which is composed of various transmitters and gives the output information); and one or more hidden layers (which connect the receptors and transmitters with different connection strengths).
In backpropagation nets, the training process is conducted by a feedback error method. An error signal is obtained by comparison between the predicted and the desired outputs. This error is backpropagated through the net and used to adjust the weights. This procedure is repeated until the desired convergence is achieved. This is equivalent to a minimization of the weights of the objective function, which gives the error between the predicted outputs, s, and the desired outputs, t, for N_{p} patterns (De Souza Jr. et al., 1996) as follows:
where
At a minimum (local or overall):
(2) Application to the process under study:
In this work, training was carried out using a backpropagation net with feedforward signals, with data obtained from the fedbatch simulation of the process described previously. These data were used to train the net by the application of an algorithm that uses the conjugated gradient method (Leonard e Kramer, 1990), which minimizes the objective function of equation (13).
The nets used here  called (a) and (b)  were composed of three layers, with three variables for the input data (substrate concentration, biomass concentration, and dilution rate taken at sampling time K) and one variable in the output patterns (biomass concentration taken at sampling time K+1). In this way, the nets provide a discrete seriesparallel model of the kind x(K+1) = F[s(K), x(K), D(K)], where F is a function representing the overall nonlinearities. The outputs of the nets (predictions) were compared with the simulated process. Net (a) had two hidden neurons and net (b), three.
Extended Kalman Filter (EKF)
The fermentation process under study in this work can be represented by the following overall equation (Bastin and Dochain, 1990):
where is the state variable matrix; K is the yield coefficient matrix; F is the process input matrix; and is the reaction rate as a function of the state variables. In this work, the reaction rate is directly proportional to the specific growth rate for:
Monod model:
Hump model:
It is assumed that the model is completely known (the reactions kinetic structure and all the coefficients are known). It is also assumed that the dilution rate, D, the inputs, F, and a subset of variables are determined online. The measured variables vector, , is given by
where L is the q´N matrix, which selects the measured components of ; q is the number of measured variables and N is the total number of variables.
The theory behind the observer, as given by Bastin and Dochain (1990), is summarized in the ^{appendix} appendix .
(1) Application to the process under study:
In this work, process model equations (6) and (7) can be rewritten as
where the substrate concentration, s, is the only measured variable; N = 2; q = 1; = s; L = [1 0]; and
with
The observability matrix is
The observer equations are written as
The gain matrix is calculated by equation (A7), as shown below:
where:
and:
Equations (20), (25), (26), (29), (30) and (31) were used in the simulation algorithm of the EKF.
The initial values for R_{1}, R_{2} and R_{3} were (Bastin and Dochain, 1990):
Filter (a): R_{1}(0) = R_{2}(0) = 500, R_{3}(0) = 0
Filter (b): R_{1}(0) = R_{2}(0) = 100, R_{3}(0) = 0
Filter (c): R_{1}(0) = R_{2}(0) = 20, R_{3}(0) = 0
Filters (a), (b) and (c) described above were tested for three sets of initial conditions of biomass and substrate concentrations:
i) Initial values of x and s for the observer were equal to the initial values of the process (x_{0} =0.31 g/L and s_{0} = 6.05 g/L);
ii) Initial values of x and s for the observer were near the initial values of the process (x_{0} =0.5 g/L and s_{0} = 5 g/L);
iii) Initial values of x and s for the observer were different from the initial values of the process (x_{0} =1 g/L and s_{0} = 1 g/L).
RESULTS AND DISCUSSION
In this section, the simulations of the growth of Bacillus thuringiensis var. israelensis in a fedbatch mode are presented and compared, using the initial conditions and parameters described previously. The techniques for variable estimation by artificial neural nets and the extended Kalman filter were applied.
The simulated process was implemented with a disturbance in the dilution rate, as shown in equation (10). The feed was interrupted when the bioreactor volume reached its maximum value.
In the following items, the results for the neural nets and the extended Kalman filter are presented, followed by a comparison between the methods.
Artificial Neural Nets
Nets (a) and (b) were tested. The training patterns used were obtained by simulations with the Monod and the Hump models and the dilution rate randomly given by equation (10). The results are presented in Figure 1.
The applied nets were able to represent the increase in biomass concentration over time. Although they showed a small oscillation at the beginning, the nets achieved the final cells concentration. Net (b) presented a small advantage over Net (a), because it oscillated a bit less and predicted the final concentration obtained from the simulation more accurately than the Net (a).
In the case of the Hump model, the performances of Nets (a) and (b) were similar to those in the case of the Monod model.
Input values given by a fedbatch operation using different feed regimes (Cunha et al., 1998) were presented to Net (b) for validation. Its outputs were compared with those from the real process. Different feeding regimes, namely constant, exponential and optimized feed were used. For this test, the Monod model was chosen, because of its simplicity.
(1) Constant feed: In this regime the feed rate is kept constant. Figure 2 shows the biomass concentration over time for the data obtained from the fermentation simulation and the net output.
(2) Exponential feed: In this regime, the feed rate depends exponentially on the specific growth rate and the time. Figure 3 illustrates the performance of the net for the patterns obtained from the simulation of this technique.
(3) Optimized feed: This regime is based on the determination of the best possible feed rate to increase production and reduce costs. There are several optimized feed regimes. For this work, singular control and "bangbang" control were chosen. Figure 4 shows the curves for biomass concentration in both cases.
In all the cases, Net (b) was able to predict the values for biomass concentration.
Extended Kalman Filter
The extended Kalman filter was also applied using the Monod and Hump models. Here the simulation results from filters (a), (b) and (c) are compared for each set of initial conditions presented in the "Methods" section.
i) x_{0} = 0.31 g/L; s_{0} = 6.05 g/L
Figure 5 shows that the curves obtained from the estimator overlaid those resulting from fermentation, with both Monod and Hump models.
ii) x_{0} = 0.5 g/L; s_{0} = 5 g/L
In this case, the ability of the estimator can be confirmed by looking at Figure 6. Note that the filter is able to adjust its outputs to the values from the fermentation simulation.
iii) x_{0} = 1 g/L; s_{0} = 1 g/L
This last case is represented in Figure 7. From this figure, it is possible to see that Filter (a) (with the largest initial larger values of R_{1} and R_{2}) presents the best estimation of the biomass concentration over time. Filter (c) (with the smallest initial values of R_{1} and R_{2}) fails even for the final time results.
Comparison
In this section, Net (b) and Filter (a)  case x_{0} = 0.5 g/L and s_{0} = 5 g/L  are compared with the real values for fermentation (Figure 8).
Notice that both the extended Kalman filter and the neural networks are able to predict satisfactorily the data for the biomass concentration. The absence of oscillations in the filter simulation can be considered an advantage. However, it should be taken into account that the filter uses a known model while the net works only with data from the process. On the other hand, the filter did not use the biomass concentration as a known input variable, in opposition to the nets.
In order to make a stricter comparison between the estimators, it is necessary to exclude the biomass concentration from the input data of the nets. Some topologies were tested, still using three inputs, namely dilution rate and substrate concentration at time K, and substrate concentration at time K1. However, satisfactory results were not obtained (data not shown). Such results possibly happened due to the lower correlation between the variables used as inputs to the network and the variable to be predicted (biomass).
Since simulated data is being used, it is difficult to draw a definitive conclusion about which technique performs better. In a practical situation, for example, data such as optical density and other growth related variables (such as carbon dioxide evolution rate or oxygen uptake rate) would be easier to measure online than the biomass concentration itself. The fact that the Kalman filter works with a preset model may also rise the need for a more complex model that describes the process in more details. This could mean hard and time consuming work, and in this case, a data based model could be an advantage. Further studies on this subject may show different results, i.e., that the neural networks are more suitable for the development of a prediction model for the selected process.
CONCLUSIONS
In this work, the ability of ANNs in online prediction of the biomass concentration in a simulated fedbatch process for the growth of Bacillus thuringiensis var. israelensis was tested. The nets were able to predict biomass concentration for different feed regimes and were compared with the variable estimation technique using the extended Kalman filter. It was shown that the superiority of the extended Kalman filter in the present study can be related to the fact that the filter works with a known model of the process, and the nets depend solely on given data. When the nets studied do not use biomass values as inputs, the output presents oscillations for low values of biomass concentration.
ACKNOWLEDGEMENT
The first author receives financial support from the Brazilian Reasearch Council  Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). Finantial support from FACEPE is also gratefully acknowledged.
NOMENCLATURE
Process
D dilution rate, h^{1} f feed rate, m^{3}/h k_{1} Hump model parameter, m^{3}/kg.h k_{2} Hump model parameter, kg/m^{3} k_{s} saturation constant, kg/m^{3} s substrate concentration, kg/m^{3} t time, h V volume of culture broth, m^{3} x cell concentration, kg/m^{3} Y_{x/s} yield coefficient, kg(cell)/kg(substrate) m specific growth rate, h^{1}Subscript
F in the feedArtificial Neural Networks
E error s input w weight l behaviour of a neuron q biasExtended Kalman Filter
L matrix that selects the measured components of x R matrix generated by Riccati equation j (x ) reaction rate W (x ) gain matrix x measured variablesSuperscripts
T transpose ^ estimated valueAppendix
A state observer or estimator is an algorithm projected to build the nonmeasurable variables from the measurable ones. An overall equation for the state observers can be given by (Bastin and Dochain, 1990)
where is the online estimation of and is the N´q gain matrix proportional to the observation error for the measured part of the state, which disappears in the case of perfect estimation.
The state observer project is resumed by the choice of that gain matrix. To solve this problem, the following observation error is introduced:
where:
This particular form of observation error indicates that it is possible to have a convergence rate, , in a random and quick way (exponential) for its real value, , if we allocate the eigenvalues of the matrix, , by an appropriate choice of . When this possibility exists, the system can be called "exponentially observable" and the observation scheme is called the "exponential observer."
It is possible to verify whether a system is exponentially observable using the following property:
post of o = post of
where is the observability matrix.
The extended Kalman filter (EKF), also called the extended Kalman observer, consists of the choice of correction matrix, , to minimize the observation error, given by (Bastin and Dochain, 1990):
The solution is given by
where is a symmetric square matrix, N´N, generated by the Riccati equation as follows:
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appendix
Publication Dates

Publication in this collection
11 Oct 2001 
Date of issue
Mar 2001
History

Accepted
23 Oct 2000 
Received
20 Sept 1998