K.J Versyck

Shelf life of modified atmosphere packed cooked meat products: addition of Na-lactate as a fourth shelf life determinative factor in a model and product validation.

Cooked meat products are often post-contaminated because of a packaging and/or slicing step after the pasteurisation process. The shelf life is therefore limited and can be extended by adding Na-lactate. A previously developed model for the spoilage of gas packed cooked meat products, including temperature, water activity and dissolved CO2 as independent variables, was extended with a fourth factor: the Na-lactate concentration in the aqueous phase of the meat product. Models were developed for the maximum specific growth rate mu(max) and the lag phase lambda of the specific spoilage organism Lactobacillus sake subsp. carnosum. Quadratic response surface equations were compared with extended Ratkowsky models. In general, response surface equations fitted the experimental data best but in the case of mu(max) the response surface model predicted illogical growth behaviour at low water activities and high Na-lactate concentrations. A extensive product validation of the mathematical models was performed by means of inoculated as well as naturally contaminated industrially prepared cooked meat products. The deviations of the experimentally determined versus predicted growth parameters in inoculated cooked meat products were in general small. Both types of models were also able to predict the shelf life of naturally contaminated cooked meat products, except for pâté where an under-estimation of the shelf life was predicted by the response surface equations. The validation studies revealed higher accuracy of the extended Ratkowsky models in comparison to the response surface equations. A significant shelf life extending effect of Na-lactate was predicted, which was more pronounced at low refrigerated temperatures. A synergistic effect has also been noticed between Na-lactate and carbon dioxide which, at least partly, could be explained by the pH-decreasing effect of CO2.

Optimal temperature input design for estimation of the Square Root model parameters: parameter accuracy and model validity restrictions

As part of the model building process, parameter estimation is of great importance in view of accurate prediction making. Confidence limits on the predicted model output are largely determined by the parameter estimation accuracy that is reflected by its parameter estimation covariance matrix. In view of the accurate estimation of the Square Root model parameters, Bernaerts et al. have successfully applied the techniques of optimal experiment design for parameter estimation [Int. J. Food Microbiol. 54 (1-2) (2000) 27]. Simulation-based results have proved that dynamic (i.e., time-varying) temperature conditions characterised by a large abrupt temperature increase yield highly informative cell density data enabling precise estimation of the Square Root model parameters. In this study, it is shown by bioreactor experiments with detailed and precise sampling that extreme temperature shifts disturb the exponential growth of Escherichia coli K12. A too large shift results in an intermediate lag phase. Because common growth models lack the ability to model this intermediate lag phase, temperature conditions should be designed such that exponential growth persist even though the temperature may be changing. The current publication presents (i) the design of an optimal temperature input guaranteeing model validity yet yielding accurate Square Root model parameters, and (ii) the experimental implementation of the optimal input in a computer-controlled bioreactor. Starting values for the experiment design are generated by a traditional two-step procedure based on static experiments. Opposed to the single step temperature profile, the novel temperature input comprises a sequence of smaller temperature increments. The structural development of the temperature input is extensively explained. High quality data of E. coli K12 under optimally varying temperature conditions realised in a computer-controlled bioreactor yield accurate estimates for the Square Root model parameters. The latter is illustrated by means of the individual confidence intervals and the joint confidence region.

Introducing optimal experimental design in predictive modeling : A motivating example

Predictive microbiology emerges more and more as a rational quantitative framework for predicting and understanding microbial evolution in food products. During the mathematical modeling of microbial growth and/or inactivation, great, but not always efficient, effort is spent on the determination of the model parameters from experimental data. In order to optimize experimental conditions with respect to parameter estimation, experimental design has been extensively studied since the 1980s in the field of bioreactor engineering. The so-called methodology of optimal experimental design established in this research area enabled the reliable estimation of model parameters from data collected in well-designed fed-batch reactor experiments. In this paper, we introduce the optimal experimental design methodology for parameter estimation in the field of predictive microbiology. This study points out that optimal design of dynamic input signals is necessary to maximize the information content contained within the resulting experimental data. It is shown that from few dynamic experiments, more pertinent information can be extracted than from the classical static experiments. By introducing optimal experimental design into the field of predictive microbiology, a new promising frame for maximization of the information content of experimental data with respect to parameter estimation is provided. As a case study, the design of an optimal temperature profile for estimation of the parameters D(ref) and z of an Arrhenius-type model for the maximum inactivation rate kmax as a function of the temperature, T, was considered. Microbial inactivation by heating is described using the model of Geeraerd et al. (1999). The need for dynamic temperature profiles in experiments aimed at the simultaneous estimation of the model parameters from measurements of the microbial population density is clearly illustrated by analytical elaboration of the mathematical expressions involved on the one hand, and by numerical simulations on the other.

Practical Identification of Unstructured Growth Kinetics by Application of Optimal Experimental Design

This paper deals with the practical identification of the parameters of unstructured growth kinetics during growth of biomass on one limiting substrate in a fed‐batch bioreactor. We consider kinetic models in which the specific growth rate is a function of the substrate concentration only. Two classes of models are distinguished: non‐monotonickinetics (with the Haldanemodel as a prototype) and monotonickinetics (with the Monodmodel as a prototype) . The information content of several simulation experiments, each with a different volumetric feed rate profile is evaluated by using the modified E‐criterion for optimal experimental design (i.e., ratio of the largest to the smallest eigenvalue of the Fisher information matrix) . The main contribution of this paper is to provide theoretical evidence and to present illustrative examples for the following conjecture: A feed rate strategy which is optimal in the sense of process performance, is an excellent starting point for feed rate optimization with respect to estimation of those parameters with large influence upon process performance. For a two degrees of freedom optimization of the feed rate profile, we obtain for the first time a modified E‐criterion value equal to 1, which is the optimal value for this criterion.

On the design of optimal dynamic experiments for parameter estimation of a Ratkowsky-type growth kinetics at suboptimal temperatures

It is generally known that accurate model building, i.e., proper model structure selection and reliable parameter estimation, constitutes an essential matter in the field of predictive microbiology, in particular, when integrating these predictive models in food safety systems. In this context, Versyck et al. (1999) have introduced the methodology of optimal experimental design techniques for parameter estimation within the field. Optimal experimental design focuses on the development of optimal input profiles such that the resulting rich (i.e., highly informative) experimental data enable unique model parameter estimation. As a case study, Versyck et al. (1999) [Versyck, K., Bernaerts, K., Geeraerd, A.H., Van Impe, J.F., 1999. Introducing optimal experimental design in predictive modeling: a motivating example. Int. J. Food Microbiol., 51(1), 39-51] have elaborated the estimation of Bigelow inactivation kinetics parameters (in a numerical way). Opposed to the classic (static) experimental approach in predictive modelling, an optimal dynamic experimental setup is presented. In this paper, the methodology of optimal experimental design or parameter estimation is applied to obtain uncorrelated estimates of the square root model parameters [Ratkowsky, D.A., Olley, J., McMeekin, T.A., Ball, A., 1982. Relationship between temperature and growth rate of bacterial cultures. J. Bacteriol. 149, 1-5] describing the effect of suboptimal growth temperatures on the maximum specific growth rate of microorganisms. These estimates are the direct result of fitting a primary growth model to cell density measurements as a function of time. Apart from the design of an optimal time-varying temperature profile based on a sensitivity study of the model output, an important contribution of this publication is a first experimental validation of this innovative dynamic experimental approach for uncorrelated parameter identification. An optimal step temperature profile, within the range of model validity and practical feasibility, is developed for Escherichia coli K12 and successfully applied in practice. The presented experimental validation result illustrates the large potential of the dynamic experimental approach in the context of uncorrelated parameter estimation. Based on the experimental validation result, additional remarks are formulated related to future research in the field of optimal experimental design.

Optimal control theory: A generic tool for identification and control of (bio-)chemical reactors

Abstract In this review paper the potential of optimal control theory for optimization in the time as well as in the space domain is highlighted. Various case studies in the area of (bio-)chemical reactors are discussed ranging from the dual problem of performance optimization and accurate parameter identification (time domain) to plug flow reactor optimization (space domain). Furthermore, it is illustrated that application of the Minimum Principle of Pontryagin to distributed parameter systems leads to extremal control profile structures (in the space domain) which are very similar to those obtained during optimization (in the time domain) of well mixed bioreactors. The analogy is reflected at various levels during analytical optimal control computations.

FEED RATE OPTIMIZATION FOR FED-BATCH BIOREACTORS: FROM OPTIMAL PROCESS PERFORMANCE TO OPTIMAL PARAMETER ESTIMATION

In this paper the optimization of a cost criterion combining both a performance cost and a parameter identification cost is discussed for the growth of biomass on one limiting substrate in a fed-batch bioreactor. Both monotonic and non-monotonic unstructured kinetic models in which the specific growth rate is a function of the substrate concentration only are considered. The parametric optimization of the feed rate profile in the fed-batch bioreactor with respect to a novel combined cost function yields a continuous transformation of the optimal performance feed rate profile towards the optimal identification profile as more weight is assigned to the identification purpose. As such we provide support for the conjecture that a feed rate strategy which is optimal in the sense of process performance, is an excellent starting point for feed rate optimization with respect to estimation of those parameters with large influence upon process performance.

Solving Dual Optimization Problems in Identification and Performance of Fed-Batch Bioreactors

Abstract In this paper some aspects of the duality found within the following two tightly interconnected problems are discussed: (i) (model based) process performance optimization, and (ii) information content optimization for model identification. In the two case studies presented, the volumetric substrate feed rate into a fedbatch bioreactor in which one biomass grows on one limiting substrate is the control input to be optimized. Unstructured kinetic models in which the specific growth rate is a function of the substrate concentration only are considered. In case study #1 the optimization of a cost criterion combining both a performance cost and a parameter identification cost is discussed. Case study #2 is entirely devoted to information content optimization. A novel combined input design criterion for parameter estimation is constructed. For specific values of the weighting factor, it reduces to the modified E-criterion, the E-criterion, or the D-criterion.

Practical Implementation of the Optimal Experiment Design Methodology for Estimation of Microbial Heat Resistance Parameters

Abstract In this paper, the design and the implementation of an experiment aimed at optimal estimation of the microbial kinetic parameters in a thermal inactivation model is discussed. Traditionally, the parameters are obtained by applying (linear or nonlinear) regression procedures on data which are collected in a time-consuming and labor-intensive series of inactivation experiments at constant temperatures (so-called static experiments). Opposed to the static approach, in this contribution, the temperature is considered as a time-varying (dynamic) control input for identification experiments. The temperature-time profile is optimized with respect to a Fisher information matrix based objective function. During this design on simulation level, parameter values estimated from preliminar experiments are considered as the nominal values. The optimal dynamic temperature profile obtained as such has been implemented in practice. The quality of the resulting parameter estimates is discussed.

Optimal design of a closed loop controller for estimation of parameter couples of microbial growth kinetics

This paper deals with the design of a feedback controller for the decorrelated estimation of a parameter couple appearing in an unstructured model for growth of one biomass on one limiting substrate in a fed-batch bioreactor. By optimal experimental design, feed rate strategies for decorrelated parameter estimation have been constructed (as reported elsewhere). These profiles have in common a feeding phase in which the substrate concentration is kept constant at a setpoint value by a feedforward controller. On simulation level this controller can be applied as an open loop controller. However, in practice these feed rate strategies are to be implemented in closed loop. Since during experimental design on simulation level a nominal parameter set is required, an iterative approach is needed to end up with a truly optimal experiment for estimation of the unknown parameters. Such an iterative scheme is proposed and its convergence properties are discussed