This document contains the post-print pdf-version of the refereed paper: "Optimal experiment design for dynamic bioprocesses: a multiobjective approach" by Dries Telen, Filip Logist, Eva Van Derlinden, Ignace Tack and Jan Van Impe which has been archived in the university repository Lirias (https://lirias.kuleuven.be/) of the Katholieke Universiteit Leuven. The content is identical to the content of the published paper, but without the final typesetting by the publisher.

BackgroundMicro-organisms play an important role in various industrial sectors (including biochemical, food and pharmaceutical industries). A profound insight in the biochemical reactions inside micro-organisms enables an improved biochemical process control. Biological networks are an important tool in systems biology for incorporating microscopic level knowledge. Biochemical processes are typically dynamic and the cells have often more than one objective which are typically conflicting, e.g., minimizing the energy consumption while maximizing the production of a specific metabolite. Therefore multi-objective optimization is needed to compute trade-offs between those conflicting objectives. In model-based optimization, one of the inherent problems is the presence of uncertainty. In biological processes, this uncertainty can be present due to, e.g., inherent biological variability. Not taking this uncertainty into account, possibly leads to the violation of constraints and erroneous estimates of the actual objective function(s). To account for the variance in model predictions and compute a prediction interval, this uncertainty should be taken into account during process optimization. This leads to a challenging optimization problem under uncertainty, which requires a robustified solution.ResultsThree techniques for uncertainty propagation: linearization, sigma points and polynomial chaos expansion, are compared for the dynamic optimization of biological networks under parametric uncertainty. These approaches are compared in two case studies: (i) a three-step linear pathway model in which the accumulation of intermediate metabolites has to be minimized and (ii) a glycolysis inspired network model in which a multi-objective optimization problem is considered, being the minimization of the enzymatic cost and the minimization of the end time before reaching a minimum extracellular metabolite concentration. A Monte Carlo simulation procedure has been applied for the assessment of the constraint violations. For the multi-objective case study one Pareto point has been considered for the assessment of the constraint violations. However, this analysis can be performed for any Pareto point.ConclusionsThe different uncertainty propagation strategies each offer a robustified solution under parametric uncertainty. When making the trade-off between computation time and the robustness of the obtained profiles, the sigma points and polynomial chaos expansion strategies score better in reducing the percentage of constraint violations. This has been investigated for a normal and a uniform parametric uncertainty distribution. The polynomial chaos expansion approach allows to directly take prior knowledge of the parametric uncertainty distribution into account.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-016-0328-6) contains supplementary material, which is available to authorized users.

a b s t r a c tDynamic optimization techniques for complex nonlinear systems can provide the process industry with sustainable and efficient operating regimes. The problem with these regimes is that they usually lie close to the limits of the process. It is therefore paramount that these operating conditions are robust with respect to the parameter uncertainties and to the process noise such that critical constraints are not violated. Besides the uncertainty in the constraints, also the uncertainty in the objective function needs to be taken into account. However, including robustness in an optimization problem typically leads to semi-infinite optimization problems that are challenging to solve in practice. In the current manuscript several computationally tractable methods are exploited to approximately solve the robust dynamic optimization problem. These methods allow the use of fast deterministic gradient based optimization techniques. The first type of methods are based on a linearization approach while the second method exploits the unscented transformation to construct an estimation of the uncertainty propagation. Both types provide the user with an approximation of the variance-covariance matrix of the critical constraints and of the objective function. This allows the user to easily take them into account in the dynamic optimization routine in a stochastic setting without the need of using computationally expensive Monte Carlo simulations in the optimization procedure. Moreover, an iterative scheme is mentioned to evaluate the approximate results and to improve them if necessary. Two illustrative case studies are discussed, a jacketed tubular reactor and the Williams-Otto reactor.

In this paper, we present a numerical method for optimal experiment design of nonlinear dynamic processes. Here, we suggest to optimize an approximation of the predicted variance-covariance matrix of the parameter estimates, which can be computed as the solution of a Riccati differential equation. In contrast to existing approaches, the proposed method allows us to take process noise into account and requires less derivative states to be computed compared to the traditional Fisher information matrix based approach. This process noise is assumed to be a time-varying random disturbance which is not known at the time when the experiment is designed. We illustrate the technique by solving an optimal experiment design problem for a fed-batch bioreactor benchmark case study. Here, we concentrate on how the optimal input design and associated accuracy of the parameter identification is influenced when process noise is present.

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