On the trade-off between experimental effort and information content in optimal experimental design for calibrating a predictive microbiology model

Telen, Dries; Logist, Filip; Van Derlinden, Eva; Van Impe, Jan

Journal de la société française de statistique, Tome 154 (2013) no. 3, pp. 95-112.

Aujourd’hui, dans le domaine de la microbiologie prédictive, la croissance et la décroissance des populations bactériennes sont décrites par des modèles mathématiques dynamiques, dits primaires. Les paramètres de ces modèles primaires sont eux-mêmes liés à des facteurs environnementaux par des modèles, dits secondaires, comprenant plusieurs paramètres secondaires (typiquement des températures et des pH minimaux, maximaux et optimaux). A cette complexité hiérarchique des modèles s’ajoutent inéluctablement de nombreuses sources d’erreurs de mesure compte tenu de la nature biologique des objets expérimentaux, erreurs qui affectent les observations issues des expériences de microbiologie. A l’évidence, il est donc particulièrement crucial de disposer de protocoles expérimentaux performants, qui prennent en compte à la fois ces structures de modèles et les erreurs, pour conduire in fine à des estimations de qualité (faible biais, faible variance, ...) des paramètres secondaires, et aussi à des prédictions fiables avec les modèles estimés. Même s’il existe plusieurs critères d’optimalité pour construire de tels protocoles performants, le coût de leur mise en oeuvre reste toutefois souvent prohibitif de par le nombre d’essais expérimentaux à réaliser. Dans cet article, deux nouvelles approches sont proposées dans un cadre de critères d’optimalité de plans d’expériences bien connus : a) un schéma d’échantillonnage temporel formalisé par une fonction de pondération de type « tout ou rien » ajoutée dans la forme dynamique de la fonction d’information de Fisher ; cette fonction va permettre à l’expérimentateur de décider si on doit échantillonner ou non à un temps donné optimisé (effort expérimental), et b) l’élaboration d’un compromis entre l’effort expérimental à fournir, commandé par le protocole optimal, et le contenu de l’information obtenue grâce à ce protocole ; cette élaboration est basée sur l’utilisation de techniques d’optimisation multi-objectifs. Usuellement, quand ces techniques sont utilisées pour déterminer un protocole optimal, l’effort expérimental n’est pas pris en compte. Enfin, à partir d’un modèle bien connu des microbiologistes, des simulations ont été menées à la fois pour évaluer la faisabilité des protocoles calculés et déterminer des intervalles de confiance pour les estimations des paramètres.

In predictive microbiology, dynamic mathematical models are developed to describe microbial evolution under time-varying environmental conditions. Next to an acceptable model structure, reliable parameter values are necessary to obtain valid model predictions. To obtain these accurate estimates of the model parameters, labor-and cost-intensive experiments have to be performed. Optimal experimental design techniques for parameter estimation are beneficial to limit the experimental burden. An important issue in optimal experimental design, included in this work, is the sampling scheme. Recent work illustrates that identifying sampling decisions results in bang-bang control of the weighting function in the Fisher information matrix. A second point addressed in this work is the trade-off between the amount of time an experimenter has available for measurements on the one hand, and information content on the other hand. Recently, multi-objective optimization is applied to several different optimal experimental design criteria, whereas in this paper the workload expressed as when to sample, is considered. The procedure is illustrated through simulations with a case study for the Cardinal Temperature Model with Inflection. The viability of the obtained experiments is assessed by calculating the confidence regions with two different methods: the Fisher information matrix approach and the Monte-Carlo method approach.

Keywords: optimal experimental design, multi-objective optimization, parameter estimation, Monte-Carlo simulation
Mot clés : plan d’expériences optimal, optimisation multi-objectif, l’estimation paramétrique, méthode de Monte-Carlo

[1] Balsa-Canto, E.; Alonso, A.A.; J.R., Banga Computing optimal dynamic experiments for model calibration in predictive microbiology, Journal of Food Process Engineering, Volume 31 (2007), pp. 186-206

[2] Biegler, L.T. An overview of simultaneous strategies for dynamic optimization, Chemical Engineering and Processing: Process Intensification, Volume 46 (2007), pp. 1043-1053

[3] Biegler, L.T. Nonlinear Programming: Concepts, Algorithms and Applications to Chemical Engineering, SIAM, 2010 | MR | Zbl

[4] Biegler, L.T. Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation, Computers and Chemical Engineering, Volume 8 (1984), pp. 243-248

[5] Bock, H.G.; Plitt, K.J. A multiple shooting algorithm for direct solution of optimal control problems, Proceedings of the 9th IFAC world congress, Budapest, Pergamon Press (1984)

[6] Baranyi, J.; Roberts, T.A. A dynamic approach to predicting bacterial growth in food, International Journal of Food Microbiology, Volume 23 (1994), pp. 277-294

[7] Das, I.; Dennis, J.E. A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems, Structural Optimization, Volume 14 (1997), pp. 63-69

[8] Das, I.; Dennis, J.E. Normal-Boundary Intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems, SIAM Journal on Optimization, Volume 8 (1998), pp. 631-657 | MR | Zbl

[9] Fedorov, V.V. Optimal design with bounded density: optimization algorithms of the exchange type, Journal of Statistical Planning and Inference, Volume 22 (1989), pp. 1-13 | MR | Zbl

[10] Franceschini, G.; Macchietto, S. Model-based design of experiments for parameter precision: State of the art, Chemical Engineering Science, Volume 63 (2008), pp. 4846-4872

[11] Houska, B.; Ferreau, H.J.; Diehl, M. ACADO Toolkit - An Open-Source Framework for Automatic Control and Dynamic Optimization, Optimal Control Applications and Methods, Volume 32 (2011), pp. 298-312 | MR | Zbl

[12] Heine, T.; Kawohl, M.; King, R. Derivative-free optimal experimental design, Chemical Engineering Science, Volume 63 (2008), pp. 4873-4880

[13] Haimes, Y.Y.; Lasdon, L.S.; Wismer, D.A. On a bicriterion formulation of the problems of integrated system identification and system optimization, IEEE Transactions on Systems, Man, and Cybernetics, Volume SMC-1 (1971), pp. 296-297 | MR | Zbl

[14] Leineweber, D.B.; Bauer, I.; Bock, H.G.; Schlöder, J.P. An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part I: theoretical aspects, Computers and Chemical Engineering, Volume 27 (2003), pp. 157-166

[15] Logist, F.; Houska, B.; Diehl, M.; Van Impe, J. Fast Pareto set generation for nonlinear optimal control problems with multiple objectives, Structural and Multidisciplinary Optimization, Volume 42 (2010), pp. 591-603 | Zbl

[16] Ljung, L. System Identification: Theory for the User, Prentice Hall, 1999 | Zbl

[17] Logist, F.; Telen, D.; Houska, B.; Diehl, M.; Van Impe, J. Multi-objective optimal control of dynamic bioprocesses using ACADO Toolkit, Bioprocess and Biosystems Engineering, Volume 36 (2013), pp. 151-164

[18] Logist, F.; Vallerio, M.; Houska, B.; Diehl, M.; Van Impe, J. Multi-objective optimal control of chemical processes using ACADO toolkit, Computers and Chemical Engineering, Volume 37 (2012), pp. 191-199

[19] Logist, F.; Van Impe, J. Novel insights for multi-objective optimisation in engineering using Normal Boundary Intersection and (Enhanced) Normalised Normal Constraint, Structural and Multidisciplinary Optimization, Volume 45 (2012), pp. 417-431 | MR | Zbl

[20] Miettinen, K. Nonlinear multiobjective optimization, Kluwer Academic Publishers, Boston, 1999 | MR | Zbl

[21] Messac, A.; Ismail-Yahaya, A.; Mattson, C.A. The normalized normal constraint method for generating the Pareto frontier, Structural & Multidisciplinary Optimization, Volume 25 (2003), pp. 86-98 | MR | Zbl

[22] Pontryagin, L.; Boltyanskiy, V.; Gamkrelidze, R.; Mishchenko, Y. The mathematical theory of optimal processes, Wiley - Interscience, New York, 1962 | MR

[23] Pukelsheim, F. Optimal design of Experiments, John Wiley & Sons, Inc., New York, 1993 | MR | Zbl

[24] Rosso, L.; Lobry, J.R.; Flandrois, J.P. An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model, Journal of Theoretical Biology, Volume 162 (1993), pp. 447-463

[25] Sager, S. Sampling Decisions in Optimum Experimental Design in the Light of Pontryagin’s Maximum Principle, SIAM Journal on Control and Optimization (2012) http://www.optimization-online.org/DB_HTML/2011/05/3037.html (submitted) | MR | Zbl

[26] Schenkendorf, R.; Kremling, A.; Mangold, M. Optimal experiment design with the sigma point method, Systems Biology, Volume 3 (2009), pp. 10-23

[27] Sargent, R.W.H.; Sullivan, G.R. The development of an efficient optimal control package, Proceedings of the 8th IFIP Conference on Optimization Techniques (Stoer, J., ed.), Springer-Verlag, Heidelberg (1978), pp. 158-168 | Zbl

[28] Sonneborn, L.; Van Vleck, F. The Bang-Bang Principle for Linear Control Systems, J. SIAM Control, Ser. A, Volume 2 (1965), pp. 151-159 | MR | Zbl

[29] Telen, D.; Logist, F.; Vanderlinden, E.; Van Impe, J. Optimal experiment design for dynamic bioprocesses: a multi-objective approach, Chemical Engineering Science, Volume 78 (2012), pp. 82-97

[30] Van Derlinden, E. Quantifying microbial dynamics as a function of temperature: towards an optimal trade-off between biological and model complexity (2009) (Ph. D. Thesis)

[31] Van Derlinden, E.; Bernaerts, K.; Van Impe, J. Simultaneous versus sequential optimal experiment design for the identification of multi-parameter microbial growth kinetics as a function of temperature, Journal of Theoretical Biology, Volume 264 (2010), pp. 347-355 | MR | Zbl

[32] Vassiliadis, V.S.; Sargent, R.W.H.; Pantelides, C.C. Solution of a class of multistage dynamic optimization problems. 1. Problems without path constraints, Industrial and Engineering Chemistry Research, Volume 33 (1994), pp. 2111-2122

[33] Vassiliadis, V.S.; Sargent, R.W.H.; Pantelides, C.C. Solution of a class of multistage dynamic optimization problems. 2. Problems with path constraints, Industrial and Engineering Chemistry Research, Volume 33 (1994), pp. 2123-2133

[34] Walter, E.; Pronzato, L. Identification of Parametric Models from Experimental Data, Springer, Paris, 1997 | MR | Zbl