Optimal feeding strategies by adaptive mesh selection for fed-batch bioprocesses

The objective of this contribution is the design of optimal feeding strategies for fed-batch bioprocesses, where complex dynamic models with input and state constraints are present. For the solution of this dynamic optimization problem a transformation to a finite dimensional optimization problem is made using piecewise linear control profiles. The optimization of these profiles is performed by a sequential approach, that includes an ODE solver for the solution of the model ODE's. Further an adaptive mesh selection algorithm was investigated for an appropriate discretization of the control profiles. The implementation of the resulting optimal feeding profiles is shown for a process example, namely the production of nikkomycin by Streptomyces tendae. This implementation uses a hierarchical process control framework, that consists of components for process monitoring, state estimation, and trajectory control.     

Optimization of feeding profile for baker's yeast production by dynamic programming

An optimal substrate feeding for an industrial scale fed-batch fermenter is determined through iterative dynamic programming in order to maximize the cell-mass production and to minimize the ethanol formation. An experimentally validated rigorous dynamic model comprises constraints in the optimization problem. A new objective function is proposed to accommodate the competing requirements of maximum yeast production and minimum ethanol formation. The objective function is maximized with iterative dynamic programming with respect to the sugar feed rate. Results prove the effectiveness of dynamic programming for solving such high-dimensional and nonlinear optimization problems, and the resulting optimal policy indicates that considerable increase in yeast production in fed-batch fermenters can be achieved while minimizing the undesired by-product, ethanol.     

Construction of Selection Indexes with Restrictions

In this paper the problem of unistage selection with inequality constraints is formulated. If the predictor and criterion variables are all normally distributed, this problem can be written as a convex programming problem, with a linear objective function and with linear constraints and a quadratic constraint. Using the duality theory, for convex nonlinear programming it is proved, that its dual problem can be transformed into a convex minimization problem with non-negativity conditions. Good computational methods are known for solving this problem. By the help of the dual problem sufficient conditions for a solution of the original primal problem are derived and illustrated by an example of practical interest.     

Predictive algorithms for neuromuscular control of human locomotion.

The problem of quantifying muscular activity of the human body can be formulated as an optimal control problem. The current methods used with large-scale biomechanical systems are non-derivative techniques. These methods are costly, as they require numerous integrations of the equations of motion. Additionally, the convergence is slow, making them impractical for use with large systems. We apply an efficient numerical algorithm to the biomechanical optimal control problem. Using direct collocation with a trapezoidal discretization, the equations of motion are converted into a set of algebraic constraint equations. An augmented Lagrangian formulation is used for the optimization problem to handle both equality and inequality constraints. The resulting min-max problem is solved with a generalized Newton method. In contrast to the prevalent optimal control implementations, we calculate analytical first- and second-derivative information and obtain local quadratic convergence. To demonstrate the efficacy of the method, we solve a steady-state pedaling problem with 7 segments and 18 independent muscle groups. The computed muscle activations compare well with experimental EMG data. The computational effort is significantly reduced and solution times are a fraction of those of the non-derivative techniques.     

Productivity improvement in xanthan gum fermentation using multiple substrate optimization

A novel and more comprehensive formulation of the optimal control problem that reflects the operational requirements of a typical industrial fermentation has been proposed in this work. This formulation has been applied to a fed-batch bioreactor with three control variables, i.e., feed rates of carbon source, nitrogen source, and an oxygen source, to result in a 148.7% increase in product formation. Xanthan gum production using Xanthomonas campestris has been used as the model system for this optimization study, and the liquid-phase oxygen supply strategy has been used to supply oxygen to the fermentation. The formulated optimization problem has several constraints associated with it due to the nature of the system. A robust stochastic technique, differential evolution, has been used to solve this challenging optimization problem. The infinite dimensional optimization problem has been approximated to a finite dimensional one by control vector parametrization. The state constraints that are path constraints have been addressed by using penalty functions and by integrating them over the total duration to ensure a feasible solution. End point constraints on final working volume of the reactor and on the final residual concentrations of carbon and nitrogen sources have been included in the problem formulation. Further, the toxicity of the oxygen source, H(2)O(2), has been addressed by imposing a constraint on its maximum usable concentration. In addition, the initial volume of the bioreactor contents and feed concentrations have been handled as decision variables, which has enabled a well-grounded choice for their values from the optimization procedure; adhoc values are normally used in the industry. All results obtained by simulation have been validated experimentally with good agreements between experimental and simulated values.     

Dynamic optimization of bioprocesses: efficient and robust numerical strategies

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.     

Computational optimal control for the time fractional convection-diffusion-reaction system

This paper proposes a numerical approximation method for computational optimal control of a time fractional convection-diffusion-reaction system. The proposed method involves discretizing the spatial domain by finite element method, approximating the admissible controls by control parameterization, and then obtaining an optimal parameter selection problem which can be solved by numerical optimization algorithms such as sequential quadratic programming. Specifically, an implicit finite difference method is employed to solve the time fractional system, and the sensitivity method for gradient computation in integer order optimal control problems is adjusted to the fractional order case. Simulation results demonstrate the validity and accuracy of the proposed numerical approximation method.     

Framework for online optimization of recombinant protein expression in high-cell-density Escherichia coli cultures using GFP-fusion monitoring

A framework for the online optimization of protein induction using green fluorescent protein (GFP)-monitoring technology was developed for high-cell-density cultivation of Escherichia coli. A simple and unstructured mathematical model was developed that described well the dynamics of cloned chloramphenicol acetyltransferase (CAT) production in E. coli JM105 was developed. A sequential quadratic programming (SQP) optimization algorithm was used to estimate model parameter values and to solve optimal open-loop control problems for piecewise control of inducer feed rates that maximize productivity. The optimal inducer feeding profile for an arabinose induction system was different from that of an isopropyl-beta-D-thiogalactopyranoside (IPTG) induction system. Also, model-based online parameter estimation and online optimization algorithms were developed to determine optimal inducer feeding rates for eventual use of a feedback signal from a GFP fluorescence probe (direct product monitoring with 95-minute time delay). Because the numerical algorithms required minimal processing time, the potential for product-based and model-based online optimal control methodology can be realized.     

Computational method for the real-time calculation of the full-body muscle load distribution

A method is described for minimising a quadratic function subject to equality and inequality constraints. This approach is applicable to solving the full-body muscle load distribution problem and calculating joint contact loads. It has been found that this approach can provide the solution on modest computing facilities and in significantly less time than using active set and interior point quadratic programming techniques. Hence the approach is suitable for providing real-time feedback to subjects undergoing biomechanical analysis of muscle, skeletal and joint loadings.     

Simple nonsingular control approach to fed-batch fermentation optimization

Determination of the optimal feed rate for fed-batch fermentation is normally a problem in singular control with a state inequality constraint and as such is, in general, difficult to solve, especially for those described by a large number of dynamic mass balance equations. In this article we use a new set of state variables and the culture volume as the control variable. In this way the problem is converted to one of nonsingular control with the magnitude and rate constraints on the manipulated variable and can be numerically solved by a gradient-based technique, thus avoiding the difficulty associated with singular control problems. Examples are given to illustrate the method.     

Genetic algorithms with filters for optimal control problems in fed-batch bioreactors

When using a genetic algorithm (GA) to solve optimal control problems that can arise in a fed-batch bioreactor, the most obvious direct approach is to rely on a finite dimensional discretization of the optimal control problem into a nonlinear programming problem. Usually only the control function is discretized, and the continuous control function is approximated by a series of piecewise constant functions. Even though the piecewise discretized controls that the GA produces for the optimal control problem may give good performances, the control policies often show very high activity and differ considerably from those obtained using a continuous optimization strategy. The present study introduces a few filters into a real-coded genetic algorithm as additional operators and investigates the smoothing capabilities of the filters employed. It is observed that inclusion of a filter significantly smoothens the optimal control profile and often encourages the convergence of the algorithm. The applicability of the technique is illustrated by solving two previously reported optimal control problems in fed-batch bioreactors that are known to have singular arcs.     

Local versus global optimal sports techniques in a group of athletes

Various optimization algorithms have been used to achieve optimal control of sports movements. Nevertheless, no local or global optimization algorithm could be the most effective for solving all optimal control problems. This study aims at comparing local and global optimal solutions in a multistart gradient-based optimization by considering actual repetitive performances of a group of athletes performing a transition move on the uneven bars. Twenty-four trials by eight national-level female gymnasts were recorded using a motion capture system, and then multistart sequential quadratic programming optimizations were performed to obtain global optimal, local optimal and suboptimal solutions. The multistart approach combined with a gradient-based algorithm did not often find the local solution to be the best and proposed several other solutions including global optimal and suboptimal techniques. The qualitative change between actual and optimal techniques provided three directions for training: to increase hip flexion–abduction, to transfer leg and arm angular momentum to the trunk and to straighten hand path to the bar.     

Identification of optimal flow rate for culture media,cell density,and oxygen toward maximization of virus production in a fed-batch baculovirus-insect cell system

In recent times, it has been realized that novel vaccines are required to combat emerging disease outbreaks, and faster optimization is required to respond to global vaccine demands. Although, fed-batch operations offer better productivity, experiment-based optimization of a new fed-batch process remains expensive and time-consuming. In this context, we propose a novel computational framework that can be used for process optimization and control of a fed-batch baculovirus-insect cell system. Since the baculovirus expression vector system (BEVS) is known to be widely used platforms for recombinant protein/vaccine production, we chose this system to demonstrate the identification of optimal profile. Toward this, first, we constructed a mathematical model that captures the time course of cell and virus growth in a baculovirus-insect cell system. Second, the proposed model was used for numerical analysis to determine the optimal operating profiles of control variables such as culture media, cell density, and oxygen based on a multiobjective optimal control formulation. Third, a detailed comparison between batch and fed-batch culture was perfromed along with a comparison between various alternatives of fed-batch operation. Finally, we demonstrate that a model-based quantification of controlled feed addition in fed-batch culture is capable of providing better productivity as compared to a batch culture. The proposed framework can be utilized for the estimation of optimal operating regions of different control variables to achieve maximum infected cell density and virus yield while minimizing the substrate/media, uninfected cell, and oxygen consumption.     

An Efficacious Multi-Objective Fuzzy Linear Programming Approach for Optimal Power Flow Considering Distributed Generation

This paper proposes a new formulation for the multi-objective optimal power flow (MOOPF) problem for meshed power networks considering distributed generation. An efficacious multi-objective fuzzy linear programming optimization (MFLP) algorithm is proposed to solve the aforementioned problem with and without considering the distributed generation (DG) effect. A variant combination of objectives is considered for simultaneous optimization, including power loss, voltage stability, and shunt capacitors MVAR reserve. Fuzzy membership functions for these objectives are designed with extreme targets, whereas the inequality constraints are treated as hard constraints. The multi-objective fuzzy optimal power flow (OPF) formulation was converted into a crisp OPF in a successive linear programming (SLP) framework and solved using an efficient interior point method (IPM). To test the efficacy of the proposed approach, simulations are performed on the IEEE 30-busand IEEE 118-bus test systems. The MFLP optimization is solved for several optimization cases. The obtained results are compared with those presented in the literature. A unique solution with a high satisfaction for the assigned targets is gained. Results demonstrate the effectiveness of the proposed MFLP technique in terms of solution optimality and rapid convergence. Moreover, the results indicate that using the optimal DG location with the MFLP algorithm provides the solution with the highest quality.     

Combining flux and energy balance analysis to model large-scale biochemical networks

Stoichiometric Network Theory is a constraints-based, optimization approach for quantitative analysis of the phenotypes of large-scale biochemical networks that avoids the use of detailed kinetics. This approach uses the reaction stoichiometric matrix in conjunction with constraints provided by flux balance and energy balance to guarantee mass conserved and thermodynamically allowable predictions. However, the flux and energy balance constraints have not been effectively applied simultaneously on the genome scale because optimization under the combined constraints is non-linear. In this paper, a sequential quadratic programming algorithm that solves the non-linear optimization problem is introduced. A simple example and the system of fermentation in Saccharomyces cerevisiae are used to illustrate the new method. The algorithm allows the use of non-linear objective functions. As a result, we suggest a novel optimization with respect to the heat dissipation rate of a system. We also emphasize the importance of incorporating interactions between a model network and its surroundings.     

Optimization of human serum albumin production in methylotrophic yeast Pichia pastoris by repeated fed-batch fermentation

An optimization method for repeated fed-batch fermentation was established with the aim of improving the recombinant human serum albumin (rHSA) production in Pichia pastoris. A simulation model for fed-batch fermentation was formulated and the optimal methanol-feeding policy calculated by dynamic programming method using five different methanol-feeding periods. The necessary state variables were collected from the calculated results and used for further optimization of repeated fed-batch fermentation. The optimal operation policy was investigated using the pre-collected state variables by estimating the overall profit per total methanol-feeding time. The calculated results indicated that the initial cell mass from the 2nd fed-batch fermentation on should be set at 35 or 40 g and methanol-feeding time at 264 h. In repeated fed-batch fermentation using the optimal operation policy, actual culture volume was in good agreement with the values simulated by model equations, but some discrepancy was observed in rHSA production. Minimum experiments were therefore carried out to re-evaluate rHSA production levels, which were then applied in re-calculations to determine the optimal operation policy. The optimal policy for repeated fed-batch fermentation established in the present study (i.e., 4-times-repeated fed-batch fermentation) achieved a 47% increase in annual rHSA production. Optimization of the culture period also brought about a 28% increase in annual rHSA production even in simple (not repeated) fed-batch fermentation.     

Optimization of enzymatic lysis of yeast

In enzymatic lysis of yeast for the recovery of intracellular proteins, the rupture of whole cells is caused by the action of a lytic system consisting primarily of protease and glucanase. A first-principles mechanism for the lytic reaction based on a two-layer model of the wall structure and a burst model for the disruption of cells is pre sented. The fed-batch model results in a dynamic optimization problem, with the enzymes, activities being the control variables. Orthogonal collocation is applied to discretize the state equations, and the resulting non linear program is solved using successive quadratic pro gramming to determine the enzyme and protease inhibitor add-in rates and pH control profiles that maximize the recovery of intracellular protein. Applying the proposed approach, optimal profiles were determined such that a significant increase of the production of proteins in a fed-batch reactor is realized. Also, the optimal control policies in a series of continuous-flow stirred tank reactors (CFSTRs) are determined.     

A parameter optimization approach for the optimal control of large-scale musculoskeletal systems.

This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).     

A hybrid model framework for the optimization of preparative chromatographic processes

An optimization framework based on the use of hybrid models is presented for preparative chromatographic processes. The first step in the hybrid model strategy involves the experimental determination of the parameters of the physical model, which consists of the full general rate model coupled with the kinetic form of the steric mass action isotherm. These parameters are then used to carry out a set of simulations with the physical model to obtain data on the functional relationship between various objective functions and decision variables. The resulting data is then used to estimate the parameters for neural-network-based empirical models. These empirical models are developed in order to enable the exploration of a wide variety of different design scenarios without any additional computational requirements. The resulting empirical models are then used with a sequential quadratic programming optimization algorithm to maximize the objective function, production rate times yield (in the presence of solubility and purity constraints), for binary and tertiary model protein systems. The use of hybrid empirical models to represent complex preparative chromatographic systems significantly reduces the computational time required for simulation and optimization. In addition, it allows both multivariable optimization and rapid exploration of different scenarios for optimal design.     

A 'cheap' optimal control approach to estimate muscle forces in musculoskeletal systems

This paper shows a new method to estimate the muscle forces in musculoskeletal systems based on the inverse dynamics of a multi-body system associated optimal control. The redundant actuator problem is solved by minimizing a time-integral cost function, augmented with a torque-tracking error function, and muscle dynamics is considered through differential constraints. The method is compared to a previously implemented human posture control problem, solved using a Forward Dynamics Optimal Control approach and to classical static optimization, with two different objective functions. The new method provides very similar muscle force patterns when compared to the forward dynamics solution, but the computational cost is much smaller and the numerical robustness is increased. The results achieved suggest that this method is more accurate for the muscle force predictions when compared to static optimization, and can be used as a numerically 'cheap' alternative to the forward dynamics and optimal control in some applications.