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, Méthodes statistiques en agronomie, 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.

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