Monoclonal antibodies (mAbs) are at present one of the fastest growing products of pharmaceutical industry, with widespread applications in biochemistry, biology, and medicine. The operation of mAbs production processes is predominantly based on empirical knowledge, the improvements being achieved by using trialanderror experiments and precedent practices. The nonlinearity of these processes and the absence of suitable instrumentation require an enhanced modelling effort and modern kinetic parameter estimation strategies. The present work is dedicated to nonlinear dynamic modelling and parameter estimation for a mammalian cell culture process used for mAb production. By using a dynamical model of such kind of processes, an optimizationbased technique for estimation of kinetic parameters in the model of mammalian cell culture process is developed. The estimation is achieved as a result of minimizing an error function by a particle swarm optimization (PSO) algorithm. The proposed estimation approach is analyzed in this work by using a particular model of mammalian cell culture, as a case study, but is generic for this class of bioprocesses. The presented case study shows that the proposed parameter estimation technique provides a more accurate simulation of the experimentally observed process behaviour than reported in previous studies.
As the market demand for monoclonal antibodies is increasing, there is significant interest in developing proper models for mammalian cell culture processes, due to the fact that these are commonly used as production platforms for mAbs, which are the fastest growing segment of the biopharmaceutical industry [
Typically, the industrial operation for mammalian cell culture mAb platforms relies on empirical knowledge [
In order to surmount the abovementioned limitations of trialanderror process development, the socalled predictive models for mammalian cell culture processes are quite attractive [
In this paper, which is an extended work of [
Concerning the applications of PSO for identification of biological systems, some results were reported for the process of glycerol fermentation by
During the last decade, PSO algorithms have gained much attention and wide applications in different fields due to their effectiveness in performing difficult optimization issues, as well as simplicity of implementation and ability of fast converge to a reasonably good solution. PSO is a populationbased heuristic global optimization technique, first introduced by Kennedy and Eberhart [
This paper proposes a multistep PSO version that uses timevarying acceleration coefficients [
The proposed nonlinear modelling and estimation approaches are analyzed in this work by using a particular model of mammalian cell culture, as a case study, but they are generic for this class of bioprocesses. A previously published dynamic model of mammalian cell culture by Gao et al. in [
In order to model the mammalian cell culture processes, first it is necessary to analyze the reconstruction of metabolic activities. However, the reconstruction generally includes only a subset of the highly active metabolic units found in proliferating mammalian cells [
Therefore, if all of the interaction of metabolites and cell physiology are included in the modelling process, then the size of the obtained model is very large and it is not appropriate for modelbased optimization and control purposes. The usual solution is to select a priori the elementary reaction schemes and to relate the major macroscopic species such as biomass, essential substrates, and products by a set of socalled macroreactions [
Next, a particular model of mammalian cell culture, published by Gao et al. [
Macroreactions of the mAb production process [
Reaction number  Macroreaction scheme 

1  GLC → 2LAC 
2  GLC + 2GLU → 2ALA + 2LAC 
3  GLC + 2GLU → 2ASP + 2LAC 
4  GLU → PRO 
5  ASN → ASP + NH_{3} 
6  GLN + ASP → ASN + GLU 
7  0.0508GLC + 0.0577GLN + 0.0133ALA + 0.006ASN + 0.0201ASP + 0.0016GLU + 0.081PRO → BM 
8  0.0104GLN + 0.011ALA + 0.072ASN + 0.082ASP + 0.0107GLU + 0.0148PRO → MAb 
9  GLN → GLU + NH_{3} 
The dynamical model of a generic bioprocess inside a bioreactor can be obtained by using the mass balance of the component and it is given by the following set of differential equations [
Model (
Typically, in a batch process the reactor is filled with the reactant mixture: substrates and microorganisms. Then, the reactions occur inside the reactor for a time period; after that, the products are removed from the tank. Because the studied bioprocess takes place inside a batch reactor, model (
For the mAb production process, the concentrations of the 11 extracellular metabolites (given in the reaction scheme from Table
However, in order to complete the model of the mAb production process, it is necessary to add the evolution of the viable cell concentrations of the culture, because the metabolite mass balances depend on the amount of viable cells. Gao et al. [
To be exact, for the mAb production process the exchange with environment is zero except the CO_{2} gaseous flow, but this flow is not measured and CO_{2} is not predicted in the final model, as it is considered in [
In the following, the dynamical model (
The dynamical model of the form (
Model (
The nonlinear dynamical model (
The most difficult modelling problem for the system of differential equations (
In the kinetic rates expression (
Kinetics expressions for the macroreactions [
Reaction number  Kinetic rate 

1 

2 

3 

4 

5 

6 

7 

8 

9 

In conclusion, the full dynamical model of mAb production process is given by (
Experimental concentration measurements [
Time  0 h  28 h  54 h  76 h  101 h  124 h  147 h 

GLC 
3.59 ± 0.04  2.59 ± 0.09  1.88 ± 0.15  1.77 ± 0.05  1.70 ± 0.03  1.67 ± 0.04  1.68 ± 0.05 
GLN 
2.85 ± 0.04  1.27 ± 0.30  0.42 ± 0.31  0.11 ± 0.10  0.00 ± 0.00  0.00 ± 0.00  0.00 ± 0.00 
ASN 
0.46 ± 0.00  0.39 ± 0.02  0.35 ± 0.02  0.32 ± 0.03  0.28 ± 0.02  0.25 ± 0.02  0.22 ± 0.02 
ASP 
0.27 ± 0.02  0.18 ± 0.02  0.10 ± 0.04  0.07 ± 0.04  0.04 ± 0.04  0.03 ± 0.04  0.03 ± 0.04 
LAC 
0.51 ± 0.01  1.33 ± 0.06  1.66 ± 0.19  1.74 ± 0.01  1.71 ± 0.01  1.72 ± 0.01  1.73 ± 0.05 
ALA 
0.33 ± 0.03  0.72 ± 0.11  1.15 ± 0.17  1.32 ± 0.12  1.46 ± 0.07  1.48 ± 0.07  1.51 ± 0.08 
PRO 
0.30 ± 0.01  0.27 ± 0.02  0.42 ± 0.04  0.53 ± 0.02  0.56 ± 0.02  0.60 ± 0.01  0.60 ± 0.01 
MAB 
0.34 ± 0.12  1.02 ± 0.06  1.58 ± 0.16  2.31 ± 0.24  2.66 ± 0.41  3.09 ± 0.60  3.41 ± 0.75 
BM 
2.01 ± 0.20  11.61 ± 0.46  16.51 ± 0.85  17.98 ± 0.84  19.41 ± 2.21  18.67 ± 2.49  17.97 ± 1.33 

0.09 ± 0.01  0.58 ± 0.02  0.79 ± 0.05  0.72 ± 0.01  0.47 ± 0.06  0.17 ± 0.03  0.06 ± 0.02 

0.02 ± 0.01  0.05 ± 0.01  0.11 ± 0.01  0.25 ± 0.05  0.58 ± 0.05  0.85 ± 0.12  0.91 ± 0.07 
The state variables within the dynamical model (
The problem that remains to be solved now is related to the estimation of the unknown (inaccessible) kinetic parameters of the dynamical model (
At the beginning of parameter estimation, the input and output data are known and the real system parameters are assumed as unknown. The identification problem is formulated in terms of an optimization problem in which the error between an actual physical measured response of the system and the simulated response of a parameterized model is minimized. The estimation of the system parameters is achieved as a result of minimizing the error function by the PSO algorithm.
Consider that the nonlinear system (
To estimate the unknown parameters in (
The objective function defined as the mean squared errors between real and estimated responses for a number
This objective function is a function difficult to minimize because there are many local minima and the global minimum has a very narrow domain of attraction. Our goal is to determine the system parameters, using particle swarm optimization algorithms in such a way that the value of
Mathematical description of basic PSO and some important variants is presented in the following.
Candidate solutions of a population called particles coexist and evolve simultaneously based on knowledge sharing with neighbouring particles. Each particle represents a potential solution to the optimization problem and it has a fitness value decided by optimal function. Supposing search space is
Unfortunately, this simple form of PSO suffers from the premature convergence problem, which is particularly true in complex problems since the interacted information among particles in PSO is too simple to encourage a global search. Many efforts have been made to avoid the premature convergence. One solution is the use of a constriction factor to insure convergence of the PSO, introduced in [
In this variant of the PSO algorithm,
It is shown that a larger inertia weight tends to facilitate the global exploration and a smaller inertia weight achieves the local exploration to finetune the current search area. The best performance could be obtained by initially setting
On the other hand, in [
The dynamical model of mAb production process given by the relations (
In the following, a multistep PSObased version that uses timevarying acceleration coefficients is implemented and an optimal set of kinetic parameter values of the mAb production process is obtained.
In order to implement the PSObased technique, the model of mAb production process (
The experimental concentration values for all the involved extracellular metabolites are provided in the work of Gao et al. [
To facilitate the application of the proposed PSObased parameter estimation strategy, the time derivatives of the states from model (
From mathematical point of view, a discussion about the interpolation technique can be done. Many authors use the linear interpolation, with advantages related to rapidity and simple implementation [
The time derivatives of the states are approximated using forward differences:
Because a 23dimensional optimization problem that must be solved for simultaneously estimation of all unknown parameters requires great computational resources, a multistep approach was used. So, the problem was split in nine simpler problems that are solved sequentially until all 23 parameters are found. These problems are noted with
The subproblems solved by using the multistep PSObased approach.
Subproblem  Parameters 

P1 

P2 

P3 

P4 

P5 

P6 

P7 

P8 

P9 

The flowchart of the multistep PSO algorithm.
For example, the problem
The optimization problem formulated in the previous section is nonlinear and nonconvex with many local minima. The estimated parameters of one subproblem are then considered known in the subsequent equations. In order to be clear, the already estimated parameters are not updated between solutions of subproblems. For example, in the frame of problem
Kinetic parameter estimates obtained via the multistep PSO approach.
Kinetic parameters  Estimated values 


8.443 

2.481 

3.968 

1.090 

7.283 

3.337 

3.977 

6.697 

3.261 

8.989 

6.495 

3.723 

7.076 

2.782 

0.019 

2.719 

9.324 

0.537 

6.683 

1.920 

4.488 

0.043 

0.067 
The partition of the multidimensional optimization problem proposed within our PSO algorithm not only ensures the decrease of necessary computational resources by comparison with Gao et al. [
Another important problem approached and solved by using the proposed PSO method is related to the expressions of reaction kinetics. To simplify the optimization problem, Gao et al. [
Simulation results, profile of the biomass concentration.
Simulation results, profiles of substrate concentrations: GLC, GLN, ASN, and ASP.
Simulation results, profiles of concentrations: LAC, PRO, ALA, and MAb.
Simulation results, profiles of concentrations: GLU, NH_{3}, X, and Xd.
There are necessary some comments concerning the overfitting problems. Overfitting arises when a statistical model describes noise instead of the underlying relationship; it usually occurs when a model is very complex, such as having many parameters relative to the number of observations. Even though the approached mAb production model is quite complex, it is not a statistical model. Also, even if the number of measured concentration samples is relatively small by comparison with the number of parameters, the PSO technique is an optimization procedure, which is much less sensitive to overfitting than the methods that are based on model training, such as neural network techniques (see Tetko et al. [
Some comparisons of the proposed PSO approach with other PSO applications to bioprocesses can be done. Most of the reported works were focused on the process of glycerol fermentation by
The particle swarmbased multistep nonlinear optimization algorithm proposed in the present work was used for the estimation of 23 parameters of an elevendimensional nonlinear system (the pathways identification was not considered). By using the multistep approach, the computational effort was quite small (about 20 min. on a computer with Intel Core i5, 64bit, 3.3 GHz processor). The obtained results and the statistical analysis show a good accuracy of the identification results.
The performance of the proposed estimation technique was analyzed by using numerical simulations. All these simulations are achieved by using the development, programming, and simulation environment MATLAB (registered trademark of The MathWorks, Inc., USA). For comparison, the simulated profiles based on the kinetic parameters obtained via PSO technique (Table
Kinetic parameter estimates obtained by Gao et al. [
Kinetic parameters  Values (Gao et al. [ 
Values (Gao et al. [ 
Values (Baughman et al. [ 


0.008  −0.0033  8.85 

0.0191  0.0058  1.12 

0.0023  −0.0014  1.1 

0.0081  0.0057  1.97 

−0.01  0.0056  4.95 

−0.011  0.0029  1.34 

0.6429  0.0573  1.36 

0.0046  0.0077  1 

0.0731  0.0113  1.83 

0.01  0.01  1.63 

0.01  0.01  1.08 

0.01  0.01  1.44 

0.001  0.001  9.64 

0.001  0.001  1.04 

0.001  0.001  5.42 

0.01  0.01  3.03 

0.01  0.01  6.39 

0.01  0.01  3.73 

0.01  0.01  3.23 

0.001  0.001  7.42 

0.001  0.001  4.45 

0.0399  0.0399  3.22 

0.06  0.06  4.99 
The simulated concentration profiles are presented in Figures
Since in industrial practice the measured data are affected by various disturbances, one explored the extent to which noisy measurements affects the estimated parameter values. For this reason, a Monte Carlo simulation approach was used. First, normal (Gaussian) distributions were constructed for every measured data set in Table
A set of 150 simulated measurement sets were generated. Finally, using each randomized data set, a new cubic interpolation of the data was generated for our standard condition and the parameter estimation problems (
Outlying solutions were identified and excluded using a basic quartile classification method. The quartile values are chosen in the following manner. First, use the median to divide the ordered data set into two halves. The median is not included into the halves. Then, the lower quartile value is the median of the lower half of the data (
The standard deviations were then converted to percentages of their associated mean value. These means and standard deviations are listed in Table
Monte Carlo parameter means and standard deviation over complete measurement set.
Kinetic parameters  Mean value  Standard deviation 


8.127 
44 

2.868 
51 

2.441 
63 

1.330 
71 

7.873  23 

3.923 
35 

3.157 
93 

4.067 
24 

2.991 
36 

8.936 
17 

7.030 
29 

3.684 
5 

7.31 
29 

2.565 
14 

0.026  13 

2.295 
87 

9.273 
51 

0.649  116 

6.224 
39 

1.814 
44 

4.657 
75 

0.048  2 

0.070  0.73 
Certain parameter estimates are much more susceptible to variability induced through perturbations in measured data than are others. It can be seen that certain parameters (
The proposed modelling and parameter estimation method can be applied to cellular processes described by the general form (
In order to develop accurate models for mammalian cell culture processes and to overcome some of the specific problems of mAb production processes such as the nonlinearity, the absence of instrumentation, and the kinetics uncertainties, a multistep nonlinear particle swarm optimizationbased technique for the estimation of experimentally unavailable kinetic parameters was designed and implemented. The proposed approach was tested by using a particular dynamical model of mammalian cell culture, as a case study, but is generic for this class of bioprocesses. We have established the capability of proposed technique to identify model parameters that provide an accurate simulation of experimentally observed mAb production process behaviour. The performed statistical analysis demonstrates that the proposed estimation method is robust against normal distributed noisy measurements. The simulations showed that the PSO parameter estimation technique provides more accurate results than those reported in previous studies.
The obtained dynamical model of the mAb production process is accurate and can contribute to the development of modelbased applications, which lead to high productivity and better quality products. The performed simulations represent one of the possibilities of model validation. The results show that the proposed model offers good predictions not only of the cell culture, for instance predictions of concentrations of energy sources such as glutamine and glucose, but also of the main amino acids and products. The proposed estimation approach can be also applied to other bioprocesses belonging to the nonlinear class considered in the present study.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by UEFISCDI, Project PACBIO no. 701/2013 (FrenchRomanian project) and Project ADCOSBIO no. 211/2014, PNIIPTPCCA201340544.