Eva Balsa-Canto

Abstract: Mathematical optimization is at the core of many problems in systems biology: (1) as the underlying hypothesis for model development, (2) in model identification, or (3) in the computation of optimal stimulation procedures to synthetically achieve a desired biological behavior. These problems are usually formulated as nonlinear programing problems (NLPs) with dynamic and algebraic constraints. However the nonlinear and highly constrained nature of systems biology models, together with the usually large number of decision variables, can make their solution a daunting task, therefore calling for efficient and robust optimization techniques. Here, we present novel global optimization methods and software tools such as cooperative enhanced scatter search (eSS), AMIGO, or DOTcvpSB, and illustrate their possibilities in the context of modeling including model identification and stimulation design in systems biology.

Abstract: Background Systems biology allows the analysis of biological systems behavior under different conditions through in silico experimentation. The possibility of perturbing biological systems in different manners calls for the design of perturbations to achieve particular goals. Examples would include, the design of a chemical stimulation to maximize the amplitude of a given cellular signal or to achieve a desired pattern in pattern formation systems, etc. Such design problems can be mathematically formulated as dynamic optimization problems which are particularly challenging when the system is described by partial differential equations. This work addresses the numerical solution of such dynamic optimization problems for spatially distributed biological systems. The usual nonlinear and large scale nature of the mathematical models related to this class of systems and the presence of constraints on the optimization problems, impose a number of difficulties, such as the presence of suboptimal solutions, which call for robust and efficient numerical techniques. Results Here, the use of a control vector parameterization approach combined with efficient and robust hybrid global optimization methods and a reduced order model methodology is proposed. The capabilities of this strategy are illustrated considering the solution of a two challenging problems: bacterial chemotaxis and the FitzHugh-Nagumo model. Conclusions In the process of chemotaxis the objective was to efficiently compute the time-varying optimal concentration of chemotractant in one of the spatial boundaries in order to achieve predefined cell distribution profiles. Results are in agreement with those previously published in the literature. The FitzHugh-Nagumo problem is also efficiently solved and it illustrates very well how dynamic optimization may be used to force a system to evolve from an undesired to a desired pattern with a reduced number of actuators. The presented methodology can be used for the efficient dynamic optimization of generic distributed biological systems.

Abstract: Analysing the properties of a biological system through in silico experimentation requires a satisfactory mathematical representation of the system including accurate values of the model parameters. Fortunately, modern experimental techniques allow obtaining time-series data of appropriate quality which may then be used to estimate unknown parameters. However, in many cases, a subset of those parameters may not be uniquely estimated, independently of the experimental data available or the numerical techniques used for estimation. This lack of identifiability is related to the structure of the model, i.e. the system dynamics plus the observation function. Despite the interest in knowing a priori whether there is any chance of uniquely estimating all model unknown parameters, the structural identifiability analysis for general non-linear dynamic models is still an open question. There is no method amenable to every model, thus at some point we have to face the selection of one of the possibilities. This work presents a critical comparison of the currently available techniques. To this end, we perform the structural identifiability analysis of a collection of biological models. The results reveal that the generating series approach, in combination with identifiability tableaus, offers the most advantageous compromise among range of applicability, computational complexity and information provided.

Abstract: Motivation: Mathematical models of complex biological systems usually consist of sets of differential equations which depend on several parameters which are not accessible to experimentation. These parameters must be estimated by fitting the model to experimental data. This estimation problem is very challenging due to the non-linear character of the dynamics, the large number of parameters and the frequently poor information content of the experimental data (poor practical identifiability). The design of optimal (more informative) experiments is an associated problem of the highest interest. Results: This work presents AMIGO, a toolbox which facilitates parametric identification by means of advanced numerical techniques which cover the full iterative identification procedure putting especial emphasis on robust methods for parameter estimation and practical identifiability analyses, plus flexible capabilities for optimal experimental design.

Abstract: This work presents a methodology which exploits the underlying biochemical structure of bioprocesses to estimate concentrations in aerobic fermenters from oxygen measurements. Although a number of estimators have been proposed over the years in the literature, the methodology proposed in this work is able to operate in transient conditions while does not require the knowledge of the growth kinetics. In addition, it can be also applied to fermenters where the spatial distribution of the concentrations is relevant. In this case, we propose a systematic approach to optimally locate the sensors based on the use of reduced order models. This method allows the reconstruction of the oxygen concentrations from a limited number of sensors. Finally, the methodology proposed will be illustrated on a horizontal tubular reactor for the production of gluconic acid by free-growth of Aspergillus niger. (C) 2010 Elsevier Ltd. All rights reserved.

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Abstract: A new approach for parameter estimation in chemical kinetics has been recently proposed (Ross et al. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 12777). It makes use of an optimization criterion based on a Generalized Fisher Equation (GFE). Its utility has been demonstrated with two reaction mechanisms, the chlorite-iodide and Oregonator, which are computationally stiff systems. In this Article, the performance of the GFE-based algorithm is compared to that obtained from minimization of the squared distances between the observed and predicted concentrations obtained by solving the corresponding initial value problem (we call this latter approach �traditional� for simplicity). Comparison of the proposed GFE-based optimization method with the �traditional� one has revealed their differences in performance. This difference can be seen as a trade-off between speed (which favors GFE) and accuracy (which favors the traditional method). The chlorite-iodide and Oregonator systems are again chosen as case studies. An identifiability analysis is performed for both of them, followed by an optimal experimental design based on the Fisher Information Matrix (FIM). This allows to identify and overcome most of the previously encountered identifiability issues, improving the estimation accuracy. With the new data, obtained from optimally designed experiments, it is now possible to estimate effectively more parameters than with the previous data This result, which holds for both GFE-based and traditional methods, stresses the importance of an appropriate experimental design. Finally, a new hybrid method that combines advantages from the GFE and traditional approaches is presented.

Abstract: Mathematical modeling has a key role in systems biology. Model building is often regarded as an iterative loop involving several tasks, among which the estimation of unknown parameters of the model from a certain set of experimental data is of central importance. This problem of parameter estimation has many possible pitfalls, and modelers should be very careful to avoid them. Many of such difficulties arise from a fundamental (yet often overlooked) property: the so-called structural (or a prion) identifiability, which considers the uniqueness of the estimated parameters. Obviously, the structural identifiability of any tentative model should be checked at the beginning of the model building loop. However, checking this property for arbitrary non-linear dynamic models is not an easy task. Here we present a software toolbox, GenSSI (Generating Series for testing Structural Identifiability), which enables non-expert users to carry out such analysis. The toolbox runs under the popular MATLAB environment and is accompanied by detailed documentation and relevant examples.

Abstract: In the present work, a set of generic parameters was proposed for a pharmacokinetic model, with the objective of predicting Cd concentration in the tissues of diverse fish species under different environmental conditions. Cd concentrations in a number of tissues of Oncorhynchus mykiss and Cyprinus carpio were estimated by a structurally identifiable multicompartmental model (unique solution). The 13 generic parameters of the model comprised exchange rates, tissue-blood partition coefficients, and weight-corrected elimination rate constants accounting for the routes of water respiration, excretion and egestion. On the other hand, absorption efficiencies from water and food were considered to be condition-specific and estimated for each experiment. These two parameters reflected the differences in fish exposure to diet (food type and metal concentration) or water (water chemistry and bioavailable metal concentration). A data set of 27 experiments of Cd bioaccumulation in fish tissues was compiled for model calibration. The selected dynamics on trout and carp were performed under very different experimental conditions, involving water and/or food exposure, different fish weights and exposure concentrations and the presence/absence of depuration periods. Model predicted, for most compartments and experiments, the tendency of Cd dynamics. However, accumulation in liver and kidney was underestimated in approximately a half of the experiments, due mainly to a rapid metallothionein (MT) sequestration phenomena and subsequent saturation on liver and kidney produced under high exposure concentrations. On the other hand, both generic and condition-specific parameter values were in accordance with the values reported in literature when available. Therefore, the results obtained in this work are an initial step indicating that a generic global input parameter set could be applied to physiology-based pharmacokinetic (PBPK) models for estimating Cd accumulation in fish in different types of scenarios. (C) 2010 Published by Elsevier Ltd.

Abstract: Background: Mathematical models provide abstract representations of the information gained from experimental observations on the structure and function of a particular biological system. Conferring a predictive character on a given mathematical formulation often relies on determining a number of non-measurable parameters that largely condition the model's response. These parameters can be identified by fitting the model to experimental data. However, this fit can only be accomplished when identifiability can be guaranteed. Results: We propose a novel iterative identification procedure for detecting and dealing with the lack of identifiability. The procedure involves the following steps: 1) performing a structural identifiability analysis to detect identifiable parameters; 2) globally ranking the parameters to assist in the selection of the most relevant parameters; 3) calibrating the model using global optimization methods; 4) conducting a practical identifiability analysis consisting of two (a priori and a posteriori) phases aimed at evaluating the quality of given experimental designs and of the parameter estimates, respectively and 5) optimal experimental design so as to compute the scheme of experiments that maximizes the quality and quantity of information for fitting the model. Conclusions: The presented procedure was used to iteratively identify a mathematical model that describes the NF-kappa B regulatory module involving several unknown parameters. We demonstrated the lack of identifiability of the model under typical experimental conditions and computed optimal dynamic experiments that largely improved identifiability properties.

Abstract: Metal bioaccumulation in fish is influenced by factors specific to the chemical and environmental conditions, the exposure route and the species. Fora better understanding of the main interactions among these factors, models are needed to capture the basic principles driving the dynamics of metal bioaccumulation in fish, taking into account different exposure routes and the distribution among representative organs. There is a significant amount of data in the literature concerning metal bioaccumulation experiments in different species of fish. Quantitative information about rate constants of the processes involved in bioaccumulation (diffusion, uptake and elimination) can be obtained from these data by means of dynamic models, that, once validated, can be used for predictive purposes. In this work, a compartmental model structure is developed aiming, in the first instance, to obtain the maximum amount of information from published experimental data. Once calibrated, the model can be further used to predict metal bioaccumulation under different scenarios. The model structure is able to reproduce the experimental behaviour for those species-metal pairs tested and, in addition, is demonstrated to be robust and identifiable. Then, the complete set of parameters can be estimated uniquely, for a specific species-metal pair by using concentration measures in a reduced number of organs. In this way, the optimal parameter sets obtained for different pairs can be compared, and the parameter specificity with respect to the metal or the species can be investigated. (C) 2009 Elsevier Ltd. All rights reserved.

Abstract: Background: Mathematical models provide abstract representations of the information gained from experimental observations on the structure and function of a particular biological system. Conferring a predictive character on a given mathematical formulation often relies on determining a number of non-measurable parameters that largely condition the model�s response. These parameters can be identified by fitting the model to experimental data. However, this fit can only be accomplished when identifiability can be guaranteed. Results: We propose a novel iterative identification procedure for detecting and dealing with the lack of identifiability. The procedure involves the following steps: 1) performing a structural identifiability analysis to detect identifiable parameters; 2) globally ranking the parameters to assist in the selection of the most relevant parameters; 3) calibrating the model using global optimization methods; 4) conducting a practical identifiability analysis consisting of two (a priori and a posteriori) phases aimed at evaluating the quality of given experimental designs and of the parameter estimates, respectively and 5) optimal experimental design so as to compute the scheme of experiments that maximizes the quality and quantity of information for fitting the model. Conclusions: The presented procedure was used to iteratively identify a mathematical model that describes the NF-kappa B regulatory module involving several unknown parameters. We demonstrated the lack of identifiability of the model under typical experimental conditions and computed optimal dynamic experiments that largely improved identifiability properties.

Abstract: The investigation of the structure and dynamics of signal transduction systems through data-based mathematical models in ordinary differential equations or other paradigms has proven to be a successful approach in recent times. Extending this concept, we here analysed the use of kinetic models based on power-law terms with non-integer kinetic orders in the validation of hypotheses concerning regulatory structures in signalling systems. We integrated pre-existent biological knowledge, hypotheses and experimental quantitative data into a power-law model to validate the existence of certain regulatory loops in the Ras/Raf-1/MEK/ERK pathway, a MAPK pathway involved in the transduction of mitogenic and differentiation signals. Towards this end, samples of a human mammary epithelial cell line (MCF-10A) were used to obtain time-series data, characterising the behaviour of the system after epidermal growth factor stimulation in different scenarios of expression for the critical players of the system regarding the investigated loops (e. g., the inhibitory protein RKIP). The mathematical model was calibrated using a computational procedure that included: analysis of structural identifiability, global ranking of parameters to detect the most sensitivity ones towards the experimental setup, model calibration using global optimization methods to find the parameter values that better fit the data, and practical identifiability analysis to estimate the confidence in the estimated values for the parameters. The obtained model was used to perform computational simulations concerning the role of the investigated regulatory loops in the time response of the signalling pathway. Our findings suggest that the special regularity in the structure of the power-law terms make them suitable for a data-based validation of regulatory loops in signalling pathways. The model-based analysis performed identified RKIP as an actual inhibitor of the activation of the ERK pathway, but also suggested the existence of an intense feedback-loop control of the pathway by the activated ERK that maybe responsible for the damped oscillations we saw in the fraction of activated MEK both in the experiments and simulations. In addition, the model analysis suggested that phosphorylation/deactivation of RKIP during the transient stimulation may have a significant effect on the signalling peaks of both MEK and ERK. This later result suggests that dynamic modulation of signal inhibitors during stimulation may be a regulatory mechanism in ERK signalling and other pathways.

Abstract: Metal bioaccumulation in fish is influenced by factors specific to the chemical and environmental conditions, the exposure route and the species. Fora better understanding of the main interactions among these factors, models are needed to capture the basic principles driving the dynamics of metal bioaccumulation in fish, taking into account different exposure routes and the distribution among representative organs. There is a significant amount of data in the literature concerning metal bioaccumulation experiments in different species of fish. Quantitative information about rate constants of the processes involved in bioaccumulation (diffusion, uptake and elimination) can be obtained from these data by means of dynamic models, that, once validated, can be used for predictive purposes. In this work, a compartmental model structure is developed aiming, in the first instance, to obtain the maximum amount of information from published experimental data. Once calibrated, the model can be further used to predict metal bioaccumulation under different scenarios. The model structure is able to reproduce the experimental behaviour for those species-metal pairs tested and, in addition, is demonstrated to be robust and identifiable. Then, the complete set of parameters can be estimated uniquely, for a specific species-metal pair by using concentration measures in a reduced number of organs. In this way, the optimal parameter sets obtained for different pairs can be compared, and the parameter specificity with respect to the metal or the species can be investigated. (C) 2009 Elsevier Ltd. All rights reserved.

Abstract: In the present work, a set of generic parameters was proposed for a pharmacokinetic model, with the objective of predicting Cd concentration in the tissues of diverse fish species under different environmental conditions. Cd concentrations in a number of tissues of Oncorhynchus mykiss and Cyprinus carpio were estimated by a structurally identifiable multicompartmental model (unique solution). The 13 generic parameters of the model comprised exchange rates, tissue-blood partition coefficients, and weight-corrected elimination rate constants accounting for the routes of water respiration, excretion and egestion. On the other hand, absorption efficiencies from water and food were considered to be condition-specific and estimated for each experiment. These two parameters reflected the differences in fish exposure to diet (food type and metal concentration) or water (water chemistry and bioavailable metal concentration). A data set of 27 experiments of Cd bioaccumulation in fish tissues was compiled for model calibration. The selected dynamics on trout and carp were performed under very different experimental conditions, involving water and/or food exposure, different fish weights and exposure concentrations and the presence/absence of depuration periods. Model predicted, for most compartments and experiments, the tendency of Cd dynamics. However, accumulation in liver and kidney was underestimated in approximately a half of the experiments, due mainly to a rapid metallothionein (MT) sequestration phenomena and subsequent saturation on liver and kidney produced under high exposure concentrations. On the other hand, both generic and condition-specific parameter values were in accordance with the values reported in literature when available. Therefore, the results obtained in this work are an initial step indicating that a generic global input parameter set could be applied to physiology-based pharmacokinetic (PBPK) models for estimating Cd accumulation in fish in different types of scenarios. (C) 2010 Published by Elsevier Ltd.

Abstract: An enhanced scatter search method for the global dynamic optimization of nonlinear processes using the control vector parametrization (CVP) approach is presented. Sharing some features of the scatter search metaheuristic, this new method presents a simpler but more effective design which helps to overcome typical difficulties of nonlinear dynamic systems optimization such as noise, flat areas, nonsmoothness, and/or discontinuities. This new algorithm provides a good balance between robustness and efficiency in the global phase, and couples a local search procedure to accelerate the convergence to optimal solutions. Its application to four multimodal dynamic optimization problems, compared with other state-of-the-art global optimization algorithms, including an advanced scatter search design, proves its efficiency and robustness, showing a very good scalability.

Abstract: Background: Mathematical optimization aims to make a system or design as effective or functional as possible, computing the quality of the different alternatives using a mathematical model. Most models in systems biology have a dynamic nature, usually described by sets of differential equations. Dynamic optimization addresses this class of systems, seeking the computation of the optimal time-varying conditions (control variables) to minimize or maximize a certain performance index. Dynamic optimization can solve many important problems in systems biology, including optimal control for obtaining a desired biological performance, the analysis of network designs and computer aided design of biological units. Results: Here, we present a software toolbox, DOTcvpSB, which uses a rich ensemble of state-of-the-art numerical methods for solving continuous and mixed-integer dynamic optimization (MIDO) problems. The toolbox has been written in MATLAB and provides an easy and user friendly environment, including a graphical user interface, while ensuring a good numerical performance. Problems are easily stated thanks to the compact input definition. The toolbox also offers the possibility of importing SBML models, thus enabling it as a powerful optimization companion to modelling packages in systems biology. It serves as a means of handling generic black-box models as well. Conclusion: Here we illustrate the capabilities and performance of DOTcvpSB by solving several challenging optimization problems related with bioreactor optimization, optimal drug infusion to a patient and the minimization of intracellular oscillations. The results illustrate how the suite of solvers available allows the efficient solution of a wide class of dynamic optimization problems, including challenging multimodal ones. The toolbox is freely available for academic use.

Abstract: An enhanced scatter search method for the global dynamic optimization of nonlinear processes using the control vector parametrization (CVP) approach is presented. Sharing some features of the scatter search metaheuristic, this new method presents a simpler but more effective design which helps to overcome typical difficulties of nonlinear dynamic systems optimization such as noise, flat areas, nonsmoothness, and/or discontinuities. This new algorithm provides a good balance between robustness and efficiency in the global phase, and couples a local search procedure to accelerate the convergence to optimal solutions. Its application to four multimodal dynamic optimization problems, compared with other state-of-the-art global optimization algorithms, including an advanced scatter search design, proves its efficiency and robustness, showing a very good scalability.

Abstract: Background: Mathematical optimization aims to make a system or design as effective or functional as possible, computing the quality of the different alternatives using a mathematical model. Most models in systems biology have a dynamic nature, usually described by sets of differential equations. Dynamic optimization addresses this class of systems, seeking the computation of the optimal time-varying conditions (control variables) to minimize or maximize a certain performance index. Dynamic optimization can solve many important problems in systems biology, including optimal control for obtaining a desired biological performance, the analysis of network designs and computer aided design of biological units. Results: Here, we present a software toolbox, DOTcvpSB, which uses a rich ensemble of state-of-the-art numerical methods for solving continuous and mixed-integer dynamic optimization (MIDO) problems. The toolbox has been written in MATLAB and provides an easy and user friendly environment, including a graphical user interface, while ensuring a good numerical performance. Problems are easily stated thanks to the compact input definition. The toolbox also offers the possibility of importing SBML models, thus enabling it as a powerful optimization companion to modelling packages in systems biology. It serves as a means of handling generic black-box models as well. Conclusion: Here we illustrate the capabilities and performance of DOTcvpSB by solving several challenging optimization problems related with bioreactor optimization, optimal drug infusion to a patient and the minimization of intracellular oscillations. The results illustrate how the suite of solvers available allows the efficient solution of a wide class of dynamic optimization problems, including challenging multimodal ones. The toolbox is freely available for academic use.

Abstract: The potential of mathematical models describing the microbial behavior during food processing and storage largely depends on their predictive capabilities and, in this concern, model calibration plays a crucial role. Unfortunately, model calibration may only be performed successfully if the sources of information are sufficiently rich. Therefore, a careful experimental design is required. This contribution formulated the optimal experimental design (OED) problem as a general dynamic optimization problem where the objective was to optimize a certain criterion depending on the Fisher information matrix. This formulation allows for more flexibility in the experimental design, including initial conditions, sampling times, experimental durations, time-dependent manipulable variables and number of experiments as degrees of freedom. Moreover, the use of robust confidence regions for the parameter estimates was suggested as an alternative to evaluate the quality of the proposed experimental schemes. The OED for the calibration of the thermal death time and Ratkowsky-type secondary models was considered for illustrative purposes, showing how the usually disregarded E-optimality criterion results in the experimental schemes offering the best compromise precision/decorrelation among the parameters.

Abstract: Due to the importance of coastal areas, is of the highest interest to implement purification systems that with minimum cost are able to assure water quality standards in agreement with the regional legislations. This work addresses the optimal design (outfall locations) and optimal operation (level of oxygen discharges) of a wastewater treatment system. This problem can be mathematically formulated as a two-objective mixed design and optimal control problem with constraints on the states and the design and control variables. The two-objective problem is formulated as a single-objective problem through the use of the weighted sum method. The existence of the optimal solution is then demonstrated for an arbitrary set of weights and a first order optimality condition is obtained to characterize that solution. The numerical solution for a realistic case study posed in the ria of Vigo is also accomplished by using the combination of the control vector parametrization approach with a global non-linear programming (NLP) solver. Remark that, as the problem under consideration is two-objective, there is not an unique solution but a set of equivalent solutions, the Pareto solutions, requiring the involvement of a decision maker to select one solution from the set. (C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.

Abstract: Mathematical models are central in systems biology and provide new ways to Understand the function of biological systems, helping in the generation of novel and testable hypotheses, and supporting a rational framework for possible ways of intervention, like in e.g. genetic engineering, drug development or treatment of diseases. Since the amount and quality of experimental 'omics' data continue to increase rapidly, there is great need for methods for proper model building, which can handle this complexity. In the present chapter we review two key steps of the model building process, namely parameter estimation (model calibration) and optimal experimental design. Parameter estimation aims to find the unknown parameters of the model which give the best fit to a set of experimental data. Optimal experimental design aims to devise the dynamic experiments which provide the maximum information content for subsequent non-linear model identification, estimation and/or discrimination. We place emphasis on the need for robust global optimization methods for proper solution of these problems, and we present a motivating example considering a cell signalling model.

Abstract: Mathematical models of complex biological systems, such as metabolic or cell-signalling pathways, usually consist of sets of nonlinear ordinary differential equations which depend on several non-measurable parameters that can be hopefully estimated by fitting the model to experimental data. However, the success of this fitting is largely conditioned by the quantity and quality of data. Optimal experimental design (OED) aims to design the scheme of actuations and measurements which will result in data sets with the maximum amount and/or quality of information for the subsequent model calibration. New methods and computational procedures for OED in the context of biological systems are presented. The OED problem is formulated as a general dynamic optimisation problem where the time-dependent stimuli profiles, the location of sampling times, the duration of the experiments and the initial conditions are regarded as design variables. Its solution is approached using the control vector parameterisation method. Since the resultant nonlinear optimisation problem is in most of the cases non-convex, the use of a robust global nonlinear programming solver is proposed. For the sake of comparing among different experimental schemes, a Monte-Carlo-based identifiability analysis is then suggested. The applicability and advantages of the proposed techniques are illustrated by considering an example related to a cell-signalling pathway.

Abstract: In this work a conceptual theory of neural networks (NNs) from the perspective of functional analysis and variational calculus is presented. Within this formulation, the learning problem for the multilayer perceptron lies in terms of finding a function, which is an extremal for some functional. Therefore, a variational formulation for NNs provides a direct method for the solution of variational problems. This proposed method is then applied to distinct types of engineering problems. In particular a shape design, an optimal control and ail inverse problem are considered. The selected examples can be solved analytically, which enables a fair comparison with the NN results. Copyright (C) 2008 John Wiley & Sons, Ltd.

Abstract: The potential of mathematical models describing the microbial behavior during food processing and storage largely depends on their predictive capabilities and, in this concern, model calibration plays a crucial role. Unfortunately, model calibration may only be performed successfully if the sources of information are sufficiently rich. Therefore, a careful experimental design is required. This contribution formulated the optimal experimental design (OED) problem as a general dynamic optimization problem where the objective was to optimize a certain criterion depending on the Fisher information matrix. This formulation allows for more flexibility in the experimental design, including initial conditions, sampling times, experimental durations, time-dependent manipulable variables and number of experiments as degrees of freedom. Moreover, the use of robust confidence regions for the parameter estimates was suggested as an alternative to evaluate the quality of the proposed experimental schemes. The OED for the calibration of the thermal death time and Ratkowsky-type secondary models was considered for illustrative purposes, showing how the usually disregarded E-optimality criterion results in the experimental schemes offering the best compromise precision/decorrelation among the parameters. © 2008, Blackwell Publishing.

Abstract: This article is part of a collection entitled "Models for Safety, Quality and Competitiveness of the Food Processing Sector," published in Comprehensive Reviews in Food Science and Food Safety. It has been peer-reviewed and was written as a follow-up of a pre-IFT workshop, partially funded by the USDA NRI grant 2005-35503-16208. ABSTRACT Mathematical models are the basis of modern process engineering methods. Mathematical optimization is at the kernel of systematic and efficient tools for (1) experimental design, model development, and identification, (2) development of optimal operating procedures, and (3) implementation of those procedures by means of model-predictive controllers. Here, we review and discuss how these model-based optimization techniques can be used at the core of computer-integrated manufacturing systems for the food industry. These systems will be able to bring the operation of food processing plants closer to the best possible product quality and safety, at a reduced cost and with minimal environmental impact. © 2008 Institute of Food Technologists.

Abstract: Due to the importance of coastal areas, is of the highest interest to implement purification systems that with minimum cost are able to assure water quality standards in agreement with the regional legislations. This work addresses the optimal design (outfall locations) and optimal operation (level of oxygen discharges) of a wastewater treatment system. This problem can be mathematically formulated as a two-objective mixed design and optimal control problem with constraints on the states and the design and control variables. The two-objective problem is formulated as a single-objective problem through the use of the weighted sum method. The existence of the optimal solution is then demonstrated for an arbitrary set of weights and a first order optimality condition is obtained to characterize that solution. The numerical solution for a realistic case study posed in the ria of Vigo is also accomplished by using the combination of the control vector parametrization approach with a global non-linear programming (NLP) solver. Remark that, as the problem under consideration is two-objective, there is not an unique solution but a set of equivalent solutions, the Pareto solutions, requiring the involvement of a decision maker to select one solution from the set. (C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.

Abstract: In this work a conceptual theory of neural networks (NNs) from the perspective of functional analysis and variational calculus is presented. Within this formulation, the learning problem for the multilayer perceptron lies in terms of finding a function, which is an extremal for some functional. Therefore, a variational formulation for NNs provides a direct method for the solution of variational problems. This proposed method is then applied to distinct types of engineering problems. In particular a shape design, an optimal control and ail inverse problem are considered. The selected examples can be solved analytically, which enables a fair comparison with the NN results. Copyright (C) 2008 John Wiley & Sons, Ltd.

Abstract: Mathematical models of complex biological systems, such as metabolic or cell-signalling pathways, usually consist of sets of nonlinear ordinary differential equations which depend on several non-measurable parameters that can be hopefully estimated by fitting the model to experimental data. However, the success of this fitting is largely conditioned by the quantity and quality of data. Optimal experimental design (OED) aims to design the scheme of actuations and measurements which will result in data sets with the maximum amount and/or quality of information for the subsequent model calibration. New methods and computational procedures for OED in the context of biological systems are presented. The OED problem is formulated as a general dynamic optimisation problem where the time-dependent stimuli profiles, the location of sampling times, the duration of the experiments and the initial conditions are regarded as design variables. Its solution is approached using the control vector parameterisation method. Since the resultant nonlinear optimisation problem is in most of the cases non-convex, the use of a robust global nonlinear programming solver is proposed. For the sake of comparing among different experimental schemes, a Monte-Carlo-based identifiability analysis is then suggested. The applicability and advantages of the proposed techniques are illustrated by considering an example related to a cell-signalling pathway.

Abstract: Background: Modeling and simulation of cellular signaling and metabolic pathways as networks of biochemical reactions yields sets of non-linear ordinary differential equations. These models usually depend on several parameters and initial conditions. If these parameters are unknown, results from simulation studies can be misleading. Such a scenario can be avoided by fitting the model to experimental data before analyzing the system. This involves parameter estimation which is usually performed by minimizing a cost function which quantifies the difference between model predictions and measurements. Mathematically, this is formulated as a non-linear optimization problem which often results to be multi-modal (non-convex), rendering local optimization methods detrimental. Results: In this work we propose a new hybrid global method, based on the combination of an evolutionary search strategy with a local multiple-shooting approach, which offers a reliable and efficient alternative for the solution of large scale parameter estimation problems. Conclusion: The presented new hybrid strategy offers two main advantages over previous approaches: First, it is equipped with a switching strategy which allows the systematic determination of the transition from the local to global search. This avoids computationally expensive tests in advance. Second, using multiple-shooting as the local search procedure reduces the multi-modality of the non-linear optimization problem significantly. Because multiple-shooting avoids possible spurious solutions in the vicinity of the global optimum it often outperforms the frequently used initial value approach (single-shooting). Thereby, the use of multiple-shooting yields an enhanced robustness of the hybrid approach.

Abstract: Mathematical models are the basis of modern process engineering methods. Mathematical optimization is at the kernel of systematic and efficient tools for (1) experimental design, model development, and identification, (2) development of optimal operating procedures, and (3) implementation of those procedures by means of model-predictive controllers. Here, we review and discuss how these model-based optimization techniques can be used at the core of computer-integrated manufacturing systems for the food industry. These systems will be able to bring the operation of food processing plants closer to the best possible product quality and safety, at a reduced cost and with minimal environmental impact.

Abstract:

Abstract: Background: Modeling and simulation of cellular signaling and metabolic pathways as networks of biochemical reactions yields sets of non-linear ordinary differential equations. These models usually depend on several parameters and initial conditions. If these parameters are unknown, results from simulation studies can be misleading. Such a scenario can be avoided by fitting the model to experimental data before analyzing the system. This involves parameter estimation which is usually performed by minimizing a cost function which quantifies the difference between model predictions and measurements. Mathematically, this is formulated as a non-linear optimization problem which often results to be multi-modal (non-convex), rendering local optimization methods detrimental. Results: In this work we propose a new hybrid global method, based on the combination of an evolutionary search strategy with a local multiple-shooting approach, which offers a reliable and efficient alternative for the solution of large scale parameter estimation problems. Conclusion: The presented new hybrid strategy offers two main advantages over previous approaches: First, it is equipped with a switching strategy which allows the systematic determination of the transition from the local to global search. This avoids computationally expensive tests in advance. Second, using multiple-shooting as the local search procedure reduces the multi-modality of the non-linear optimization problem significantly. Because multiple-shooting avoids possible spurious solutions in the vicinity of the global optimum it often outperforms the frequently used initial value approach (single-shooting). Thereby, the use of multiple-shooting yields an enhanced robustness of the hybrid approach.

Abstract: The mathematical modelling of signal transduction pathways has become a valuable aid to understanding the complex interactions involved in intracellular signalling mechanisms. An important aspect of the mathematical modelling process is the selection of the model type and structure. Until recently, the convention has been to use a standard kinetic model, often with the Michaelis-Menten steady state assumption. However this model form, although valuable, is only one of a number of choices, and the aim of this article is to consider the mathematical structure and essential features of an alternative model form - the power-law model. Specifically, we analyse how power-law models can be applied to increase our understanding of signal transduction pathways when there may be limited prior information. We distinguish between two kinds of power law models: a) Detailed power-law models, as a tool for investigating pathways when the structure of protein-protein interactions is completely known, and; b) Simplified power-law models, for the analysis of systems with incomplete structural information or insufficient quantitative data for generating detailed models. If sufficient data of high quality are available, the advantage of detailed power-law models is that they are more realistic representations of non-homogenous or crowded cellular environments. The advantages of the simplified power-law model formulation are illustrated using some case studies in cell signalling. In particular, the investigation on the effects of signal inhibition and feedback loops and the validation of structural hypotheses are discussed. © 2007.

Abstract: Model based methods are fundamental in modern food process engineering. The most realistic models combine the physical laws of conservation and constitutive relations associated with kinetic transformations and physical properties, which usually depend on non-measurable parameters. Therefore, a crucial step in model development is model calibration, that is, the computation of those parameters based on experimental data. In this contribution, a two-step approach for proper model calibration is proposed. The first step, usually disregarded, consists of performing a structural identifiability analysis to evaluate the (im-)possibility of giving unique solutions for the model parameters. The second step consists of using robust parameter estimation techniques, based on global optimization methods as the alternative to surmount the convergence to sub-optimal solutions which may lead to wrong conclusions about model predictive capabilities. A typical model for food air-drying is presented as a case study in order to highlight usual difficulties associated with the calibration of food processing models, and how the proposed two-step procedure can help modelers to overcome such difficulties. © 2007 Elsevier Ltd. All rights reserved.

Abstract: The mathematical modelling of signal transduction pathways has become a valuable aid to understanding the complex interactions involved in intracellular signalling mechanisms. An important aspect of the mathematical modelling process is the selection of the model type and structure. Until recently, the convention has been to use a standard kinetic model, often with the Michaelis-Menten steady state assumption. However this model form, although valuable, is only one of a number of choices, and the aim of this article is to consider the mathematical structure and essential features of an alternative model form - the power-law model. Specifically, we analyse how power-law models can be applied to increase our understanding of signal transduction pathways when there may be limited prior information. We distinguish between two kinds of power law models: a) Detailed power-law models, as a tool for investigating pathways when the structure of protein-protein interactions is completely known, and; b) Simplified power-law models, for the analysis of systems with incomplete structural information or insufficient quantitative data for generating detailed models. If sufficient data of high quality are available, the advantage of detailed power-law models is that they are more realistic representations of non-homogenous or crowded cellular environments. The advantages of the simplified power-law model formulation are illustrated using some case studies in cell signalling. In particular, the investigation on the effects of signal inhibition and feedback loops and the validation of structural hypotheses are discussed. (c) 2007 Published by Elsevier Inc.

Abstract: Thermal processing is widely used for ensuring food safety and extended shelf life. However, standard methods of thermal processing have a significant impact on food quality due to thermal degradation of nutrients and other quality factors. Model-based methods can be successfully used for thermal process design, optimization and control. However, building sound models requires suitable estimation of the unknown kinetic parameters. Further, the accuracy of these estimates will largely depend on the quality and quantity of the available experimental data. Optimal experimental design (OED) of dynamic experiments allows for the calculation of the scheme of controls and measurements which improve the estimation of model parameters. In this contribution, the OED problem is formulated as a general dynamic optimization problem where the objective is to find those experimental conditions which result in maximum information content, as measured by the Fisher information matrix. The numerical solution of this problem is then approached using a combination of the control vector parameterization approach with a non-linear global optimization solver. As an illustrative application, we consider the optimal experimental design for the parameter estimation of the thiamine degradation kinetic parameters during the thermal processing of canned tuna. Results confirm that the use of optimal dynamic experiments not only improves identifiability but also results in reduced confidence regions for the parameters (a maximum error of the 2% in the parameter estimates), substantially decreasing the experimental effort (up to a 50%). Particularly the use of six optimally designed experiments results in a 30% reduction of the confidence regions with respect to previously published results using 10 typical experiments. (C) 2007 Elsevier Ltd. All rights reserved.

Abstract: Model based methods are fundamental in modern food process engineering. The most realistic models combine the physical laws of conservation and constitutive relations associated with kinetic transformations and physical properties, which usually depend on nonmeasurable parameters. Therefore, a crucial step in model development is model calibration, that is, the computation of those parameters based on experimental data. In this contribution, a two-step approach for proper model calibration is proposed. The first step, usually disregarded, consists of performing a structural identitiability analysis to evaluate the (im-)possibility of giving unique solutions for the model parameters. The second step consists of using rohust parameter estimation techniques, based on global optimization methods as the alternative to surmount the convergence to sub-optimal solutions which may lead to wrong conclusions about model predictive capabilities. A typical model for food air-drying is presented as a case study in order to highlight usual difficulties associated with the calibration of food processing models, and how the proposed two-step procedure can help modelers to overcome such difficulties. (c) 2007 Elsevier Ltd. All rights reserved.

Abstract: Thermal processing is widely used for ensuring food safety and extended shelf life. However, standard methods of thermal processing have a significant impact on food quality due to thermal degradation of nutrients and other quality factors. Model-based methods can be successfully used for thermal process design, optimization and control. However, building sound models requires suitable estimation of the unknown kinetic parameters. Further, the accuracy of these estimates will largely depend on the quality and quantity of the available experimental data. Optimal experimental design (OED) of dynamic experiments allows for the calculation of the scheme of controls and measurements which improve the estimation of model parameters. In this contribution, the OED problem is formulated as a general dynamic optimization problem where the objective is to find those experimental conditions which result in maximum information content, as measured by the Fisher information matrix. The numerical solution of this problem is then approached using a combination of the control vector parameterization approach with a non-linear global optimization solver. As an illustrative application, we consider the optimal experimental design for the parameter estimation of the thiamine degradation kinetic parameters during the thermal processing of canned tuna. Results confirm that the use of optimal dynamic experiments not only improves identifiability but also results in reduced confidence regions for the parameters (a maximum error of the 2% in the parameter estimates), substantially decreasing the experimental effort (up to a 50%). Particularly the use of six optimally designed experiments results in a 30% reduction of the confidence regions with respect to previously published results using 10 typical experiments. © 2007 Elsevier Ltd. All rights reserved.

Abstract: Food processing plants are usually operated in batch or semi-continuous mode. Dynamic optimization techniques can be used to compute optimal operating policies in order to ensure maximum profits and product quality while guaranteeing food safety. However, the nonlinear and highly constrained nature of food processing models can make their dynamic optimization a daunting task. Here, we analyze the performance of several state of the art methods considering two selected case studies: a semi-continuous fermentor and a thermal sterilization unit. We also propose novel sequential re-optimization strategies in order to avoid convergence problems and to improve computational efficiency. © 2005 Elsevier Ltd. All rights reserved.

Abstract: Food processing plants are usually operated in batch or semi-continuous mode. Dynamic optimization techniques can be used to compute optimal operating policies in order to ensure maximum profits and product quality while guaranteeing food safety. However, the nonlinear and highly constrained nature of food processing models can make their dynamic optimization a daunting task. Here, we analyze the performance of several state of the art methods considering two selected case studies: a semi-continuous fermentor and a thermal sterilization unit. We also propose novel sequential re-optimization strategies in order to avoid convergence problems and to improve computational efficiency. (c) 2005 Elsevier Ltd. All rights reserved.

Abstract: The recent literature on simulated moving bed (SMB) chromatography suggests the modulation of either the feed concentration or the flow rates in order to improve the process performance with respect to the classical (constant feeding) approach. The �best� profiles are selected from a finite set of combinations of step heights and numbers of steps. Therefore, the results, although promising, can be far from truly optimal. In this contribution, the general dynamic optimization (open loop optimal control) of an SMB chromatographic separation process is considered, allowing the calculation of the optimal feed concentration and/or feed flow rate, over each switching period, with maximum flexibility. To numerically solve the problem, the combination of the control vector parametrization scheme with suitable state-of-the art nonlinear programming (NLP) problem solvers is considered. Further, we also show how the use of global optimization methods is required to surmount the convergence difficulties exhibited by local NLP solvers. The advantages of this approach are illustrated through the solution of three case studies, achieving significant improvements in the process productivity and the raffinate purity when compared to traditional feeding profiles.

Abstract: The recent literature on simulated moving bed (SMB) chromatography suggests the modulation of either the feed concentration or the flow rates in order to improve the process performance with respect to the classical (constant feeding) approach. The "best" profiles are selected from a finite set of combinations of step heights and numbers of steps. Therefore, the results, although promising, can be far from truly optimal. In this contribution, the general dynamic optimization (open loop optimal control) of an SMB Chromatographic separation process is considered, allowing the calculation of the optimal feed concentration and/or feed flow rate, over each switching period, with maximum flexibility. To numerically solve the problem, the combination of the control vector parametrization scheme with suitable state-of-the art nonlinear programming (NLP) problem solvers is considered. Further, we also show how the use of global optimization methods is required to surmount the convergence difficulties exhibited by local NLP solvers. The advantages of this approach are illustrated through the solution of three case studies, achieving significant improvements in the process productivity and the raffinate purity when compared to traditional feeding profiles. © 2006 American Chemical Society.

Abstract: A hybrid stochastic-deterministic method, based on the control vector parameterization (CVP) approach, is presented as a reliable and efficient alternative for the solution of dynamic optimization (or open loop optimal control) problems. The problems under consideration are related to free final time single-stage systems and more general multi-stage procecesses that are described by different sets of differential and algebraic equations (DAEs), one for each stage. The operating conditions and the duration of each stage must be computed in order to achieve an overall optimal result for the process subject to constraints in the state and control variables. The solution of three challenging dynamic optimization problems is presented, including a large-scale case study, showing the capabilities of this new strategy.

Abstract: A hybrid stochastic-deterministic method, based on the control vector parameterization (CVP) approach, is presented as a reliable and efficient alternative for the solution of dynamic optimization (or open loop optimal control) problems. The problems under consideration are related to free final time single-stage systems and more general multi-stage procecesses that are described by different sets of differential and algebraic equations (DAEs), one for each stage. The operating conditions and the duration of each stage must be computed in order to achieve an overall optimal result for the process subject to constraints in the state and control variables. The solution of three challenging dynamic optimization problems is presented, including a large-scale case study, showing the capabilities of this new strategy.

Abstract: The dynamic optimization (DO) of complex distributed parameter systems (DPSs), like e.g., reaction-diffusion processes, is a challenging task. Most of the existing numerical approaches imply a large computational effort, therefore precluding its application to demanding applications like real time DO or model predictive control. This work, based on the control vector parameterization (CVP) approach, describes two ways to enhance the efficiency of the resulting nonlinear programming (NLP) problem solution. On the one hand, the convergence properties of the NLP solver are enhanced through the use of exact gradients and projected Hessians (H.p). On the other hand, simulation efficiency is improved through the use of reduced order descriptions of the DPSs. The capabilities and possibilities of these two enhancements are illustrated with a number of complex distributed case studies.

Abstract: The dynamic optimization (DO) of complex distributed parameter systems (DPSs), like e.g., reaction-diffusion processes, is a challenging task. Most of the existing numerical approaches imply a large computational effort, therefore precluding its application to demanding applications like real time DO or model predictive control. This work, based on the control vector parameterization (CVP) approach, describes two ways to enhance the efficiency of the resulting nonlinear programming (NLP) problem solution. On the one hand, the convergence properties of the NLP solver are enhanced through the use of exact gradients and projected Hessians (H.p). On the other hand, simulation efficiency is improved through the use of reduced order descriptions of the DPSs. The capabilities and possibilities of these two enhancements are illustrated with a number of complex distributed case studies. © 2005 Institution of Chemical Engineers.

Abstract: The dynamic optimization (open loop optimal control) of non-linear bioprocesses is considered in this contribution. These processes can be described by sets of non-linear differential and algebraic equations (DAEs), usually subject to constraints in the state and control variables. A review of the available solution techniques for this class of problems is presented, highlighting the numerical difficulties arising from the non-linear, constrained and often discontinuous nature of these systems. In order to surmount these difficulties, we present several alternative stochastic and hybrid techniques based on the control vector parameterization (CVP) approach. The CVP approach is a direct method which transforms the original problem into a non-linear programming (NLP) problem, which must be solved by a suitable (efficient and robust) solver. In particular, a hybrid technique uses a first global optimization phase followed by a fast second phase based on a local deterministic method, so it can handle the nonconvexity of many of these NLPs. The efficiency and robustness of these techniques is illustrated by solving several challenging case studies regarding the optimal control of fed-batch bioreactors and other bioprocesses. In order to fairly evaluate their advantages, a careful and critical comparison with several other direct approaches is provided. The results indicate that the two-phase hybrid approach presents the best compromise between robustness and efficiency. (c) 2005 Elsevier B.V. All rights reserved.

Abstract: The dynamic optimization (open loop optimal control) of non-linear bioprocesses is considered in this contribution. These processes can be described by sets of non-linear differential and algebraic equations (DAEs), usually subject to constraints in the state and control variables. A review of the available solution techniques for this class of problems is presented, highlighting the numerical difficulties arising from the non-linear, constrained and often discontinuous nature of these systems. In order to surmount these difficulties, we present several alternative stochastic and hybrid techniques based on the control vector parameterization (CVP) approach. The CVP approach is a direct method which transforms the original problem into a non-linear programming (NLP) problem, which must be solved by a suitable (efficient and robust) solver. In particular, a hybrid technique uses a first global optimization phase followed by a fast second phase based on a local deterministic method, so it can handle the nonconvexity of many of these NLPs. The efficiency and robustness of these techniques is illustrated by solving several challenging case studies regarding the optimal control of fed-batch bioreactors and other bioprocesses. In order to fairly evaluate their advantages, a careful and critical comparison with several other direct approaches is provided. The results indicate that the two-phase hybrid approach presents the best compromise between robustness and efficiency. © 2005 Elsevier B.V. All rights reserved.

Abstract: The dynamic optimization of combined lumped and distributed process systems, governed by nonlinear ordinary and partial differential equations (ODEs and PDEs), is considered in this work. The application of a recently developed method based on the combination of the control vector parametrization approach with the calculation of exact gradients and projected Hessians (Hp�s), is presented as an alternative for the efficient computation of the control policies needed to optimize a specific performance criterion. The exact first- and second-order information is calculated by means of the solution of an extended initial value problem (IVP) whose particular time-varying Jacobian structure is exploited by a sparse implicit solver to increase efficiency. Finally, the applicability of this method is shown through the solution of a number of case studies demonstrating that a significant speed-up can be obtained through the use of exact second-order information.

Abstract: Dynamic optimization of distributed process systems has received considerable attention over the last couple of years. Most approaches proposed for the solution of these types of problems are based on the use of the control vector parametrization method, which transforms the original dynamic optimization problem into an outer nonlinear programming (NLP) problem. The solution of this NLP problem requires the simulation of the process under consideration for each function evaluation. Unfortunately, this task is usually very demanding for this class of dynamic systems, which calls for reduced-order descriptions of the distributed process system. In this work, we exploit the use of low-order models based on the Galerkin projection on a set of proper orthogonal functions as a very efficient alternative to the solution of dynamic optimization problems for nonlinear distributed process systems.

Abstract: The dynamic optimization of combined lumped and distributed process systems, governed by nonlinear ordinary and partial differential equations (ODEs and PDEs), is considered in this work. The application of a recently developed method based on the combination of the control vector parametrization approach with the calculation of exact gradients and projected Hessians (Hp's), is presented as an alternative for the efficient computation of the control policies needed to optimize a specific performance criterion. The exact first- and second-order information is calculated by means of the solution of an extended initial value problem (IVP) whose particular time-varying Jacobian structure is exploited by a sparse implicit solver to increase efficiency. Finally, the applicability of this method is shown through the solution of a number of case studies demonstrating that a significant speed-up can be obtained through the use of exact second-order information.

Abstract: Dynamic optimization of distributed process systems has received considerable attention over the last couple of years. Most approaches proposed for the solution of these types of problems are based on the use of the control vector parametrization method, which transforms the original dynamic optimization problem into an outer nonlinear programming (NLP) problem. The solution of this NLP problem requires the simulation of the process under consideration for each function evaluation. Unfortunately, this task is usually very demanding for this class of dynamic systems, which calls for reduced-order descriptions of the distributed process system. In this work, we exploit the use of low-order models based on the Gale?rkin projection on a set of proper orthogonal functions as a very efficient alternative to the solution of dynamic optimization problems for nonlinear distributed process systems.

Abstract:

Abstract: In this contribution, computer-aided optimization is presented as the ultimate tool to improve food processing. The state of the art is reviewed, especially focusing in recent developments using modern optimization techniques. Their potential for industrial applications is also discussed in the light of several important examples. Finally, future trends and research needs are outlined. © 2003 Elsevier Science Ltd. All rights reserved. All rights reserved.

Abstract: In this contribution, computer-aided optimization is presented as the ultimate tool to improve food processing. The state of the art is reviewed, especially focusing in recent developments using modern optimization techniques. Their potential for industrial applications is also discussed in the light of several important examples. Finally, future trends and research needs are outlined. (C) 2003 Elsevier Science Ltd. All rights reserved.

Abstract: In Part I of this study, a method for the derivation of reduced-order models of food thermal processing was presented. In this second part, the capabilities and efficiency of this method is illustrated by applying it to the problems of design and optimization of thermal sterilization. The particular case of conduction-heated foods is considered, without loss of generality. The results clearly indicate that this new methodology allows very fast and accurate solutions of these problems, opening a whole new avenue of possibilities, especially for real-time optimization and control applications. Furthermore, the methodology can be applied to other food processes described by distributed models. © 2002 Elsevier Science Ltd. All rights reserved.

Abstract: A new scheme using a Truncated Newton algorithm with and exact Hessian-search direction vector product is presented for the solution of optimal control problems. The derivation of formulae for second order parametric sensitivity analysis of differential-algebraic equations is presented, following earlier published work [V.S. Vassiliadis, E. Balsa-Canto, J.R. Banga, Second order sensitivities of general dynamic systems with application to optimal control problems. Chem. Eng. Sci. 54 (17) (1999) 3851-3860]. An original result in this work is the derivation of Hessian matrix-vector product forms which are shown to have the same computational complexity as the evaluation of first order sensitivities. This result for optimal control Hessian-vector products using control vector parameterization is shown to be a very effective way to solve optimal control problems. It is also noted that this work introduces the use of suitable Truncated Newton solvers which can exploit the exact vector products in using conjugate gradient iterations to converge the Newton equations. Such a solver is the TN algorithm of Nash [(S.G. Nash-Newton type minimization via the Lanczos method. SIAM J. Num. Anal. 21, (1984) 770-778)]. Because no full Hessian update is necessary it is demonstrated that the resulting optimal control solver performs very well for a very large number of degrees of freedom, limited only by the necessity for many right-hand-side calculations in the first and second order sensitivity equations (the Hessian vector product). It is also demonstrated by several case studies that the scheme is capable of starting far from the solution and yet arrive there in almost invariant performance. (C) 2002 Published by Elsevier Science Ltd. All rights reserved.

Abstract: The design and optimization of thermal processing of foods needs accurate dynamic models through which to systematically explore new operation policies. Unfortunately, the governing and constitutive equations of thermal processing models usually lead to complex sets of highly nonlinear partial differential equations (PDEs), which are difficult and costly to solve, especially in terms of computation time. We overcome such limitation by using a powerful model reduction technique based on proper orthogonal decomposition (POD) which yields simple, yet accurate, dynamic models still based on sound first principles. Model reduction is carried out by projecting the original set of PDEs on a low dimensional subspace which retains most of the relevant features of the original system. The resulting model consists of a small set of differential and algebraic equations (DAEs) suitable for real-time industrial applications (optimization and control). Further, this approach can be easily adapted to handle complex nonlinear convection-diffusion processes regardless of how irregular the domain geometry might be. © 2002 Elsevier Science Ltd. All rights reserved.

Abstract: The design and optimization of thermal processing of foods needs accurate dynamic models through which to systematically explore now operation policies. Unfortunately, the governing and constitutive equations of thermal processing models usually lead to complex sets of highly nonlinear partial differential equations (PDEs), which are difficult and costly to solve, especially in terms of computation time. We overcome such limitation by using a powerful model reduction technique based on proper orthogonal decomposition (POD) which yields simple, yet accurate, dynamic models still based on sound first principles. Model reduction is carried out by projecting the original set of PDEs on a low dimensional subspace which retains most of the relevant features of the original system. The resulting model consists of a small set of differential and algebraic equations (DAEs) suitable for real-time industrial applications (optimization and control). Further, this approach can be easily adapted to handle complex nonlinear convection-diffusion processes regardless of how irregular the domain geometry might be. (C) 2002 Elsevier Science Ltd. All rights reserved.

Abstract: In Part I of this study, a method for the derivation of reduced-order models of food thermal processing was presented. In this second part, the capabilities and efficiency of this method is illustrated by applying it to the problems of design and optimization of thermal sterilization. The particular case of conduction-heated foods is considered, without loss of generality. The results clearly indicate that this new methodology allows very fast and accurate solutions of these problems, opening a whole new avenue of possibilities, especially for real-time optimization and control applications, Furthermore, the methodology can be applied to other food processes described by distributed models. (C) 2002 Elsevier Science Ltd. All rights reserved.

Abstract: The objective of the present research was to establish a combined electromagnetic and heat transfer model to predict the temperature distribution in food loads during microwave and forced air heating in a microwave combination oven. The microwave process was modelled using the finite difference time domain (FDTD) method to numerically solve Maxwell's equations in three dimensions, assuming the food properties to be constant. The power dissipated at each cell in the computational domain was subsequently calculated. Heat transfer was modelled using Fourier's equation for heat conduction with convective boundary conditions. The conduction model was spatially discretised using finite elements. The power dissipation field was transferred to the finite element heat transfer code using interpolation modules to couple the models. Validation experiments were made for comparisons with predicted temperatures inside a model food load with brick-shaped geometry. Good qualitative agreement between predicted and measured temperature profiles was obtained.

Abstract: A new scheme using a Truncated Newton algorithm with and exact Hessian-search direction vector product is presented for the solution of optimal control problems. The derivation of formulae for second order parametric sensitivity analysis of differential-algebraic equations is presented, following earlier published work [V.S. Vassiliadis, E. Balsa-Canto, J.R. Banga, Second order sensitivities of general dynamic systems with application to optimal control problems. Chem. Eng. Sci. 54 (17) (1999) 3851-3860]. An original result in this work is the derivation of Hessian matrix-vector product forms which are shown to have the same computational complexity as the evaluation of first order sensitivities. This result for optimal control Hessian-vector products using control vector parameterization is shown to be a very effective way to solve optimal control problems. It is also noted that this work introduces the use of suitable Truncated Newton solvers which can exploit the exact vector products in using conjugate gradient iterations to converge the Newton equations. Such a solver is the TN algorithm of Nash [(S.G. Nash-Newton type minimization via the Lanczos method. SIAM J. Num. Anal. 21, (1984) 770-778)]. Because no full Hessian update is necessary it is demonstrated that the resulting optimal control solver performs very well for a very large number of degrees of freedom, limited only by the necessity for many right-hand-side calculations in the first and second order sensitivity equations (the Hessian vector product). It is also demonstrated by several case studies that the scheme is capable of starting far from the solution and yet arrive there in almost invariant performance. © 2002 Published by Elsevier Science Ltd. All rights reserved.

Abstract: The extension of a recently developed method for the dynamic optimization of chemical and biochemical processes is presented. This method is based on the control vector parameterization approach and makes use of the calculation of first- and second-order sensitivities to obtain exact gradient and projected Hessian information. In order to achieve high discretization levels of the control variables with a moderate computational cost, a mesh refining technique is also presented here. The robustness and efficiency of this strategy is illustrated with the solution of several challenging case studies. (C) 2001 Elsevier Science Ltd. AII rights reserved.

Abstract: The extension of a recently developed method for the dynamic optimization of chemical and biochemical processes is presented. This method is based on the control vector parameterization approach and makes use of the calculation of first- and second-order sensitivities to obtain exact gradient and projected Hessian information. In order to achieve high discretization levels of the control variables with a moderate computational cost, a mesh refining technique is also presented here. The robustness and efficiency of this strategy is illustrated with the solution of several challenging case studies. © 2001 Elsevier Science Ltd.

Abstract: The dynamic optimization (open-loop optimal control) of bioprocesses is considered. It is shown how these problems can be solved using a recently developed method, based on the control vector parametrization concept, which makes use of second-order sensitivities to obtain exact gradients and Hessians for the objective function of the underlying dynamic process model. A further extension of this scheme,which makes use of restricted second-order information, is also presented. This extension results in an efficient way to solve general dynamic optimization problems, even for high levels of control discretization. This new approach allows the efficient and robust solution of two challenging case studies regarding the optimal control of fed-batch bioreactors taken from the open literature.

Abstract: The dynamic optimization (open-loop optimal control) of bioprocesses is considered. It is shown how these problems can be solved using a recently developed method, based on the control vector parametrization concept, which makes use of second-order sensitivities to obtain exact gradients and Hessians for the objective function of the underlying dynamic process model. A further extension of this scheme, which makes use of restricted second-order information, is also presented. This extension results in an efficient way to solve general dynamic optimization problems, even for high levels of control discretization. This new approach allows the efficient and robust solution of two challenging case studies regarding the optimal control of fed-batch bioreactors taken from the open literature. The dynamic optimization (open-loop optimal control) of bioprocesses is considered. It is shown how these problems can be solved using a recently developed method, based on the control vector parameterization concept, which makes use of second-order sensitivitives to obtain exact gradients and Hessians for the objective function of the underlying dynamic process model. A further extension of this scheme, which makes use of restricted second-order information, is also presented. This extension results in an efficient way to solve general dynamic optimization problems, even for high levels of control discretization. This new approach allows the efficient and robust solution of two challenging case studies regarding the optimal control of fed-batch bioreactors taken from the open literature.

Abstract: The derivation of formulae for second-order parametric sensitivity analysis of differential-algebraic equations is presented in this paper, using tensorial analysis. The proposed formulae derive this information in conjunction with the state and first-order sensitivity evaluation. An original result in this work is the derivation of Hessian matrix-vector product forms which are shown to have the same computational complexity as the evaluation of first-order sensitivities. The theoretical result for second-order sensitivities is shown to be a very effective way to solve optimal control problems. The algorithm constructed is demonstrated to have a fine performance on three standard optimal control problems taken from the chemical engineering literature. (C) 1999 Elsevier Science Ltd. All rights reserved.

Abstract: The derivation of formulae for second-order parametric sensitivity analysis of differential-algebraic equations is presented in this paper, using tensorial analysis. The proposed formulae derive this information in conjunction with the state and first-order sensitivity evaluation. An original result in this work is the derivation of Hessian matrix-vector product forms which are shown to have the same computational complexity as the evaluation of first-order sensitivities. The theoritical result for second-order sensitivities is shown to be a very effective way to solve optimal control problems. The algorithm constructed is demonstrated to have a fine performance on three standard optimal control problems taken from the chemical engineering literature. The derivation of formulae for second-order parametric sensitivity analysis of differential-algebraic equations is presented in this paper, using tensorial analysis. The proposed formulae derive this information in conjunction with the state and first-order sensitivity evaluation. An original result in this work is the derivation of Hessian matrix-vector product forms which are shown to have the same computational complexity as the evaluation of first-order sensitivities. The theoretical result for second-order sensitivities is shown to be a very effective way to solve optimal control problems. The algorithm constructed is demonstrated to have a fine performance on three standard optimal control problems taken from the chemical engineering literature.

Abstract: Mathematical optimization is at the core of many problems in systems biology: (1) as the underlying hypothesis for model development, (2) in model identification, or (3) in the computation of optimal stimulation procedures to synthetically achieve a desired biological behavior. These problems are usually formulated as nonlinear programing problems (NLPs) with dynamic and algebraic constraints. However the nonlinear and highly constrained nature of systems biology models, together with the usually large number of decision variables, can make their solution a daunting task, therefore calling for efficient and robust optimization techniques. Here, we present novel global optimization methods and software tools such as cooperative enhanced scatter search (eSS), AMIGO, or DOTcvpSB, and illustrate their possibilities in the context of modeling including model identification and stimulation design in systems biology.

Abstract: Mathematical models are central in systems biology and provide new ways to Understand the function of biological systems, helping in the generation of novel and testable hypotheses, and supporting a rational framework for possible ways of intervention, like in e.g. genetic engineering, drug development or treatment of diseases. Since the amount and quality of experimental �omics� data continue to increase rapidly, there is great need for methods for proper model building, which can handle this complexity. In the present chapter we review two key steps of the model building process, namely parameter estimation (model calibration) and optimal experimental design. Parameter estimation aims to find the unknown parameters of the model which give the best fit to a set of experimental data. Optimal experimental design aims to devise the dynamic experiments which provide the maximum information content for subsequent non-linear model identification, estimation and/or discrimination. We place emphasis on the need for robust global optimization methods for proper solution of these problems, and we present a motivating example considering a cell signalling model.

Abstract: In this contribution we study a set of dynamic optimization problems of fed-batch bioreactors. On account of the highly nonlinear dynamic models of the bioprocesses involved, the solution of the corresponding optimal control problems becomes a challenging task. Basically the dynamic optimization consists of calculating the optimal operating policies, usually the time-varying feed rates, to optimize a certain index. In this regard we apply the novel MATLAB toolbox NDOT, which combines the control vector parameterization approach with local and global nonlinear programming solvers and suitable dynamic simulation methods in order to solve in a robust and efficient way complex problems.

Abstract: Abstract: Mathematical models of complex biological systems, such as cell signalling cascades, usually consist of sets of nonlinear ordinary differential equations which depend on several non measurable parameters that must be estimated by fitting the model to experimental data. This model calibration is performed by minimizing the differences between model predictions and measurements. Optimal experimental design (OED) aims to design an scheme of actuations and measurements which will result in data sets with the maximum amount and/or quality of information, as measured by the Fisher Information Matrix, for the subsequent model calibration. This work presents new computational procedures for OED in the context of systems biology, with a focus on cell signalling. The OED problem is formulated as a general dynamic optimization problem and its solution is approached using a combination of the control vector parameterization approach and a robust global non-linear programming solver. The applicability and advantages of using optimal experimental design are illustrated by considering a mitogen-activated protein (MAP) kinase cascade, which is frequently involved in larger cell signalling pathways, and it is known to regulate several cellular processes of major importance. Notes: Request copies by e-mail to ebalsa@iim.csic.es

Abstract: Model-based mathematical optimization is a powerful tool for building decision support systems. Such systems can be used to design and/or operate (control) food processes in an optimal way, leading to e.g. maximum profit while satisfying safety and quality constraints. However, most of these problems are very challenging to solve due to their multimodal nature (existence of multiple optima). Global optimization is a challenging research area which has started to receive significant attention during the last decade. Here, we review the state of the art and we also present our experiences regarding the use of a number of global optimization methods to solve problems arising in the area of food process engineering. We also outline advanced computational approaches (e.g. cluster computing) in order to successfully handle realistic problems. Results for selected case studies are presented and discussed.

Abstract: In this article account is given of some work done in the area of model reduction of distributed process systems based on mass, energy and momentum first principles. Reduced order models are derived by projecting the original model on a low dimensional subspace, which retains the most relevant dynamic features of the system. This subspace can be obtained from the structure of the parabolic operators, real data or direct numerical simulation by use of the so-called spectral decomposition techniques or proper orthogonal decomposition (POD�s). Despite non-linearity or spatial domain irregularity, the resulting reduced-model consists of a very small set of ordinary differential and algebraic equations which can be solved very efficiently. As a consequence, such a reduced description becomes appropriate for on-line simulation (and thus prediction), real time optimization, identification and control applications. Different examples related with thermal processing and bioreactors will illustrate the technique as well as its impact on fast development of optimal control strategies and advanced robust control

Abstract: Genetic regulation of metabolic networks may be driven by optimality principles. Dynamic optimization can be used to either predict (a priori) or explain (a posteriori) the activation profile of genes and/or enzymes in metabolic pathways. In the proposed approach, the use of a suitable dynamic optimization direct method is combined with adequate global optimization solvers, implemented in the DOTcvpSB toolbox. The main advantage of this approach is that arbitrarily complex pathways can be considered, including branched ones, while avoiding convergence to local solutions. In order to illustrate the capabilities of the proposed methodology, several examples are presented and solved.

Abstract: The thermal processing of packaged foods in batch retorts is a standard operation in the food industry where it is of vital importance to reduce the activity of the harmful micro-organisms to a given level and to stabilize the bioproduct for subsequent storage. However, as part of the thermal processing, organoleptic properties of the food product are also deteriorated. Thus being pertinent to find the operation conditions that may satisfy both safety and quality criteria. This work presents the theoretical development and experimental validation of the real time dynamic optimization of the process with the objective of guaranteeing the maximization of the surface retention of nutrients while satisfying the microbiological requirements under typical perturbed process operation.

Abstract: In this work, a model for metal bioaccumulation in fish is presented. The model captures the basic principles driving the dynamics of metal bioaccumulation in fish, taking into account different exposure routes and the distribution among representative organs. The model is demonstrated to be robust and structurally identifiable. The complete set of parameters can be estimated univoquely, for a specific pair speciemetal, by using concentration measures in a reduced number of organs.

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Abstract: Indoor location systems using 802.11 standard, based on the comparison of the received and predicted levels of the received signals from the Access Points, are a very interesting research area. The location information is computed by searching the nearest neighbour of the measured signal strength within the radio map. In this chapter, we apply a global optimization algorithm to obtain the Access Points location distribution that yields the best performance of these location systems.

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Abstract: This contribution describes the development and Implementation of a novel software toolbox, NDOT, for the dynamic optimization (open loop optimal control) of nonlinear processes. This modular and flexible toolbox combines the control vector parameterization approach with a number of local and global nonlinear programming solvers and Suitable dynamic simulation methods. NDOT is able to solve dynamic optimization problems for both lumped and distributed nonlinear processes. Its performance (robustness and efficiency) is illustrated considering a representative set of nonlinear (lumped and distributed) benchmark problems.

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Abstract: In this work we present the development of an efficient model-based real time dynamic optimization (DO) architecture for the control of distributed parameter systems (DPS). The approach takes advantage of the dissipative nature of this class of systems to obtain reduced order models (ROM) which are then used by the optimization modules to Compute in real time the optimal operation policy. The DO module is based oil the combination of the control vector parameterization (CVP) approach and a Suitable NLP solver selected among several local and global possibilities.

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Abstract: In this contribution we present an overview of some recent works carried out in our group to develop systematic and efficient methods for model reduction and its application to simulation, dynamic optimization and robust control of complex distributed process systems. The numerical projection methods we developed exploit the underlying finite element structure of the numerical PDE system to efficiently evaluate and to integrate the spatial differential terms, and thus to systematically project the original PDE set into a low dimensional subspace. This results into a reduced order description which is able to capture the relevant dynamics of the original system. Details on computational aspects of the methodology as well as applications in the context of dynamic optimization and robust control will be discussed on a number of representative case studies involving nonlinear diffusion-reaction and fluid dynamic systems.

Abstract: In this contribution, we will present an overview of the state of the art regarding the model-based optimization of industrial food processing. The potential of modern optimization techniques for improving industrial processes will be discussed considering several important problem classes. Finally, we will also outline a number of research needs and probable future trends.

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Abstract: The extension of a recently developed method for the dynamic optimization (DO) of chemical and biochemical processes is presented. This method is based on the control vector parameterization approach and makes use of the calculation of first and second order sensitivities to obtain exact gradient and projected Hessian information. In order to achieve high discretization levels of the control variables with a moderate computational cost, a mesh refining technique is also presented here. The robustness and efficiency of this strategy is illustrated with the solution of several challenging case studies.

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Abstract: The general problem of dynamic optimization of bioprocesses with unspecified final time is considered. Several solution strategies, both deterministic and stochastic, are compared based on their results for three bioprocess case studies. A hybrid (stochastic-deterministic) method is also presented and evaluated, showing significant advantages in terms of robustness and computational effort.