René Schenkendorf

Optimal Experimental Design with the Sigma Point method

Using mathematical models for a quantitative description of dynamical systems requires the identification of uncertain parameters by minimising the difference between simulation and measurement. Owing to the measurement noise also, the estimated parameters possess an uncertainty expressed by their variances. To obtain highly predictive models, very precise parameters are needed. The optimal experimental design (OED) as a numerical optimisation method is used to reduce the parameter uncertainty by minimising the parameter variances iteratively. A frequently applied method to define a cost function for OED is based on the inverse of the Fisher information matrix. The application of this traditional method has at least two shortcomings for models that are nonlinear in their parameters: (i) it gives only a lower bound of the parameter variances and (ii) the bias of the estimator is neglected. Here, the authors show that by applying the sigma point (SP) method a better approximation of characteristic values of the parameter statistics can be obtained, which has a direct benefit on OED. An additional advantage of the SP method is that it can also be used to investigate the influence of the parameter uncertainties on the simulation results. The SP method is demonstrated for the example of a widely used biological model.

Qualitative and Quantitative Optimal Experimental Design for Parameter Identification of a MAP Kinase Model

Abstract Mathematical models ensuring a highly predictive power are of inestimable value in systems biology. Their application ranges from investigations of basic processes in living organisms up to model based drug design in the field of pharmacology. For this purpose simulation results have to be consistent with the real process, i.e, suitable model parameters have to be identified minimizing the difference between the model outcome and measurement data. In this work graph based methods are used to figure out if conditions of parameter identifiability are fulfilled. In combination with network centralities, the structural representation of the underlying mathematical model provides a first guess of informative output configurations. As at least the most influential parameters should be identifiable and to reduce the complexity of the parameter identification process further a parameter ranking is done by Sobol’ indices. The calculation of these indices goes along with a highly computational effort, hence monomial cubature rules are used as an efficient approach of numerical integration. All methods are demonstrated for a well known motif in signaling pathways, the MAP kinase cascade.

Parameter Identification for Ordinary and Delay Differential Equations by Using Flat Inputs

The concept of differential flatness has been widely used for nonlinear controller design. In this contribution, it is shown that flatness may also be a very useful property for parameter identification. An identification method based on flat inputs is introduced. The treatment of noisy measurements and the extension of the method to delay differential equations are discussed. The method is illustrated by two case studies: the well-known FitzHugh-Nagumo equations and a virus replication model with delays.

A systematic reactor design approach for the synthesis of active pharmaceutical ingredients

Graphical abstract Figure. No Caption available. ABSTRACT Today’s highly competitive pharmaceutical industry is in dire need of an accelerated transition from the drug development phase to the drug production phase. At the heart of this transition are chemical reactors that facilitate the synthesis of active pharmaceutical ingredients (APIs) and whose design can affect subsequent processing steps. Inspired by this challenge, we present a model‐based approach for systematic reactor design. The proposed concept is based on the elementary process functions (EPF) methodology to select an optimal reactor configuration from existing state‐of‐the‐art reactor types or can possibly lead to the design of novel reactors. As a conceptual study, this work summarizes the essential steps in adapting the EPF approach to optimal reactor design problems in the field of API syntheses. Practically, the nucleophilic aromatic substitution of 2,4‐difluoronitrobenzene was analyzed as a case study of pharmaceutical relevance. Here, a small‐scale tubular coil reactor with controlled heating was identified as the optimal set‐up reducing the residence time by 33% in comparison to literature values.

Supporting the shift towards continuous pharmaceutical manufacturing by condition monitoring

Over the last decade there has been an increased interest in the pharmaceutical industry to shift the manufacturing process of drugs from batch to continuous operation. The continuous manufacturing of pharmaceuticals provides significant benefits, e.g. savings in cost, time and materials - to name but a few. The implementation of a continuous manufacturing strategy, however, is challenging. To gain profit from a continuous process one has to ensure its proper operation, i.e. a long time-span until the next prospective unscheduled downtime. Thus, the installed operation units have to be: 1) robust against disturbances by engineering design principles and by advanced fault tolerant control schemes, respectively; and 2) the condition of the operation units has to be monitored reliably to trigger, in case of need, appropriate intervention strategies in a timely manner. In this paper, the focus is on the monitoring aspect. Here, a model-based fault detection and identification framework is implemented, which selects the most data-supported model candidate from a set of predefined model hypotheses including the reference model (normal behavior) as well as failure models. In addition, to enable an improved diagnosis the system under study can be steered deliberately based on the proposed concept resulting into an active fault diagnosis framework. Preliminary results are demonstrated by an academic three-tank system.

Influence of non-linearity to the Optimal Experimental Design demonstrated by a biological system

A precise estimation of parameters is essential to generate mathematical models with a highly predictive power. A framework that attempts to reduce parameter uncertainties caused by measurement errors is known as Optimal Experimental Design (OED). The Fisher Information Matrix (FIM), which is commonly used to define a cost function for OED, provides at the best only a lower bound of parameter uncertainties for models that are non-linear in their parameters. In this work, the Sigma Point method is used instead, because it enables a more reliable approximation of the parameter statistics accompanied by a manageable computational effort. Moreover, it is shown that Sigma Points can also be used to define design criteria for OED that incorporate the influence of parameter uncertainties on the simulated model states, i.e. mean square error of prediction. To reduce the computational effort of OED further, the Kriging Interpolation approach is applied leading to an easily evaluable surrogate cost function. The advantages of the Sigma Point method combined with the Kriging Interpolation in the framework of OED are demonstrated for the example of a biological two-substrate uptake model.

Model-based optimization of biopharmaceutical manufacturing in Pichia pastoris based on dynamic flux balance analysis

Abstract Biologic drugs are promising therapeutics, and their efficient production is essential for a competitive pharma industry. Dynamic flux balance analysis (dFBA) enables the dynamic simulation of the extracellular bioreactor environment and intracellular fluxes in microorganisms, but it is rarely used for model-based optimization of biopharmaceutical manufacturing in Pichia pastoris. To bridge this gap, we present a model-based optimization approach based on dFBA to produce biologics in P. pastoris that combines ideas from bilevel optimization, penalization schemes, and direct dynamic optimization. As a case study, we consider the production of recombinant erythropoietin in P. pastoris growing on glucose, and predict a 66% improvement in the productivity of erythropoietin. We show that this improvement could be obtained by implementing an almost constant optimal feeding strategy which is different from typical exponential feeding strategies and that a high activity of most pathways in the central carbon metabolism is crucial for a high productivity.

Robust Design of Chemical Processes Based on a One-Shot Sparse Polynomial Chaos Expansion Concept

Abstract The application of robust model-based design concepts for complex chemical processes is limited due to the repeated cpu-intensive uncertainty quantification step for any new tested process design configuration. Therefore, an efficient One-Shot Sparse Polynomial Chaos Expansion (OS 2 -PCE) based process design framework is introduced in this work. The key idea is to define the process design variables as uncertain quantities as well and, in consequence, they become an integral part of the robust optimization routine. Moreover, by utilizing the sparsity feature of the PCE approach, the implementation of a least angle regression (LAR) concept leads to a significant reduction in computational costs. The overall performance of the novel OS 2 -PCE approach is illustrated by a robust process design study of a jacketed tubular reactor. In comparison to state-of-the-art concepts, the proposed framework shows promising results in terms of efficiency and robustness.

Model-based optimization of the recombinant protein production in Pichia pastoris based on dynamic flux balance analysis and elementary process functions

Abstract Pichia pastoris is an important host cell for the heterologous expression of recombinant proteins. In order to understand and design optimal biopharmaceutical processes with P. pastoris, unstructured and flux balance analysis (FBA) models have been developed which consider either extracellular fluxes or intracellular fluxes, respectively. In an at-tempt to predict both intracellular and extracellular fluxes, structured models have been introduced, but existing structured models compartmentalize pathways and do not con-sider the metabolic networks in detail. In this paper, we present a model-based optimiza-tion approach for the production of recombinant proteins in P. pastoris. Our approach is based on dynamic flux balance analysis (dFBA) and elementary process functions (EPF). We show that our concept is able to predict both dynamic extracellular concentrations and time-dependent intracellular fluxes in detail with no need for compartmentalization. We also present an efficient solution strategy for our approach.

Parameter Identification of Time-Delay Systems: A Flatness Based Approach

Abstract The use of mathematical models is widely established in various fields of application. To name but a few of their major applications, mathematical models can improve the controller design of complex technical systems or are able to facilitate the understanding of highly complex biochemical systems. No matter what mathematical models are used for, however, they fail to perform the intended task if they are badly parameterized. In general, during the process of parameterization one tries to make differences between simulation results and measurement data as small as possible. Under the assumption of a suitable model candidate this is done by choosing optimal model parameters. Unfortunately, the majority of used models cannot be solved analytically. For example, many dynamical processes are described by systems of ordinary differential equations (ODEs). Usually, analytical solutions do not exist. Although quite efficient numerical routines are available they usually slow down the parameterization process dramatically. The situation is even more demanding if one has to deal with processes that are described by delay differential equations (DDEs). Commonly, standard DDE solvers show a lack of efficiency as well as of robustness, i.e., they are likely to fail to solve the underlying DDE system. Consequently, it would be of great benefit to eliminate any need of numerical ODE/DDE solvers. Here, the concept of flat inputs comes into play. The key aspect is to transform the DDE system into an algebraic input/output representation, i.e., the inputs of the system are expressed analytically by the outputs and derivatives thereof. Now, the objective of parameterization is to minimize differences between these flat inputs and the physical inputs of the related process. As no numerical DDE solver is involved there is a significant speedup of the parameter identification step. In addition, the presented approach is closely linked to optimal experimental design for parameter identification. In particular, the reformulation of the cost function also affects parameter sensitivities. Using the same measurement data it is possible that previously insensitive model parameters become sensitive. To check this, global parameter sensitivities are determined by Sobol’ Indices of first order. All results are demonstrated for the example of a mathematical model of the influenza A virus production.