Sensitivity Analysis for the Purposes of Parameter Identification

Sensitivity Analysis for the Purposes of Parameter Identification of a S. cerevisiae Fed-batch Cultivation Angelova M. , T. Pencheva maria. angelova@clbme. bas. bg, tania. pencheva@clbme. bas. bg

Fermentation processes Biotechnological and particularly fermentation processes (FP) are widely used in different branches of industry – in the production of pharmaceuticals, chemicals and enzymes, yeast, foods and beverages. Fermentation processes are: • characterized as complex, nonlinear, dynamic systems with interdependent and time-varying process variables; • described by non-linear models with a very complex structure, often with unidentifiable parameters involved. That is why high accuracy in model parameters estimation is essential for the purposes of adequate models development. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

A purpose of the investigation • In order to provide a precisely and accurate model parameter identification a sensitivity analysis to be applied. • A stepwise parameter identification procedure to be proposed. • Developed procedure to be demonstrated on S. cerevisiae fed-batch cultivation described by fifth order nonlinear model, determined the interdependence between main state variables – biomass, substrate, ethanol and dissolved oxygen. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Sensitivity analysis can provide strong indications as to which parameters are the most difficult to be identified either because of their limited influence on the total system behavior, or due to the fact that some parameters compensate for the effects of others. High sensitivity to a parameter suggests that the system's performance can drastically change with small variations in the parameter. Vice versa, low sensitivity suggests little change in the performance. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Sensitivity analysis when applied to FP • Sensitivity analysis applied for biotechnological processes including FP allows the influence of the models’ parameters to be investigated. • Sensitivity analysis also can help in solving the parameter estimation problem. • Obtained knowledge through sensitivity analysis is a powerful tool for elucidation a system’s behavior due to variations in the parameters that affect the system dynamics. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

FP model description using sensitivity functions (1) A general mathematical model of fed-batch cultivation can be given as: where are state variables and (1) are model parameters. By definition sensitivity functions are the partial derivatives of the state variables with respect to the parameters. Sensitivity functions are defined as follows: (2) where are the sensitivity function of i-th parameter according j-th variable. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

FP model description using sensitivity functions (2) Differentiation of Eq. (1) regarding p leads to (3) The derivatives equation: are obtained by solving the sensitivity (4) Thus sensitivity model of considered system is formed from the mathematical model (1) and the sensitivity equations (4). 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Mathematical model of S. cerevisiae fed-batch cultivation (5) – (9) where X, S, E, O 2 and O 2* are concentrations of biomass, substrate (glucose), ethanol, [g. l-1], oxygen and dissolved oxygen saturation, [%]; F – feeding rate, [l. h-1]; V – volume of bioreactor, [l]; – volumetric oxygen transfer coefficient, [h-1]; Sin – glucose concentration in the feeding solution, [g. l-1]; , q. S, q. E and are respectively specific rates of growth, substrate utilization, ethanol production and dissolved oxygen consumption, [h-1]. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Application of Monod kinetics in FP model (10) where max is the maximum growth rates, [1/h]; k. S – saturation constant, [g/l]; Yij – yield coefficients, [g/g]. For the purposes of sensitivity analysis application to considered model (5)-(9) the state variable vector is presented as x = [X, S, E, O 2], while the model parameters set consists of p = [ max ks YS/X YE/S YO/X]. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Simulation curves from sensitivity analysis when Monod kinetics is used (1) 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Simulation curves from sensitivity analysis when Monod kinetics is used (2) The results from the sensitivity analysis for considered model (5)-(9) with Monod kinetics can be summarized as follows: the highest sensitivity is featured to yield coefficients YS/X, YE/S, YO/X, followed by maximum growth rate and saturation constant k. S. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Parameter identification procedure Presented investigation based on the sensitivity analysis can result in the following 3 -step parameter identification procedure: 1. estimation of parameters max, ks and YS/X based on experimental data set for dynamics of X and S. The system (5), (6) and (9) with vector parameter p = [ max, ks, YS/X] is considered. 2. estimation of parameter YE/S based on an experimental data set for dynamics of E. 3. estimation of parameter YO/X based on an experimental data set for dynamics of O 2. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Experimental data and model prediction of S. cerevisiae fed-batch cultivation applying Monod kinetics could not describe dynamics of yeast growth process – neither for biomass growth and substrate utilization, moreover for ethanol production and dissolved oxygen consumption. Even the fact that proposed stepwise procedure for parameter identification leads to reducing the number of simultaneous estimated parameters, it is obvious that this is not always a precondition for obtaining of an adequate model. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Functional state modeling approach applied to FP In order to be obtained an adequate mathematical description of considered yeast cultivation specific rates presented by Zhang et al. have been applied. According to the rules for functional states recognition, first ethanol production state is identified for considered S. cerevisiae fed-batch cultivation. Local models in Eqs. (5)-(9) have structures originally presented by Zhang et al. : (11) where the previously mentioned symbols keep their meaning, while q. Scrit is the value of q. S at the critical value of Scrit. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Simulation curves from sensitivity analysis when functional state modeling approach is used (1) 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Simulation curves from sensitivity analysis when functional state modeling approach is used (2) Results from the sensitivity analysis in this case are more closely grouped with no prominent parameter. Moreover, attempts to be applied the proposed stepwise parameter identification procedure using mentioned above conventional optimization methods, do not lead to satisfactory results. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Experimental data and model prediction of S. cerevisiae fed-batch cultivation applying functional state modeling approach The idea of using genetic algorithms as an alternative stochastic technique, intuitively appeared. Applying genetic algorithms for a parameter identification of considered here S. cerevisiae fed-batch cultivation, described with model (5)-(9) applying local models of Zhang et al. the following parameter values have been obtained: p = [0. 24 0. 28 0. 09 1. 48 1600]. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Analysis and conclusions • In this investigation the sensitivity analysis of a fifth order non linear mathematical model of a S. cerevisiae fed-batch cultivation process is studied. • Two different structures of functions described specific rates are examined. • Sensitivity analysis is firstly applied in respect to state variables and model parameters in case when Monod kinetics is generally used in a global model of considered cultivation process. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Analysis and conclusions • 3 -step parameter identification procedure is proposed. Despite the fact that such procedure leads to reducing the number of simultaneously estimated parameters, this is not always a precondition for obtaining of an adequate model. • In considered here cultivation process Monod kinetics could not describe adequately the process dynamics. • Sensitivity analysis has been further analytically worked out to state variables and model parameters when specific growth rates have been described according to functional state modeling approach. • In this case model parameters are more closely grouped with no prominent parameter. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Analysis and conclusions • When proposed stepwise parameter identification procedure has been applied using conventional optimization methods, no satisfactory results have been obtained. That provokes using of genetic algorithms as an alternative stochastic technique. • Presented here methodology could be implemented also for another fermentation process described by nonlinear mathematical model with lower or higher order. Corresponding to the parameter identification purposes, sensitivity analysis can be applied to other specific growth rates, as well as to different optimization criteria. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Acknowledgements This work is partially supported by the European Social Fund and Bulgarian Ministry of Education, Youth and Science under Operative Program “Human Resources Development”, grant BG 051 PO 001 -3. 3. 04/40 and National Scientific Fund of Bulgaria, grant number DID 02 -29. 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol

Thank you for your attention 8 th International Conference on "Large-Scale Scientific Computations", June 6 -10, 2011, Sozopol