Static and dynamic behavior of a class of unstructured models of continuous bioreactors with growth associated product

The static and dynamic behavior of a class of unstructured models of continuous bioprocesses, for which the product is growth associated, are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which both the rates of utilization of limiting substrate and product formation are linearly proportional to the specific cell growth rate. The kinetic expressions are allowed to assume general forms of substrate and nonbiomass product. The steady-state analysis allows the derivation of analytical results and the construction of a useful picture in the models' parameter space delineating the different static behavior these models can predict, including unique steady states and bistability. The analysis of the dynamic behavior allows the derivation of general analytical conditions for the occurrence of periodic behavior in the models. It is also shown that the subclass of these models for which the specific cell growth rate expression is monotonic with respect to the nonbiomass product is unable to predict a stable oscillatory behavior regardless of the expression of the growth rate. These results illustrate the fundamental weakness of this class of unstructured models in predicting transient behavior in continuous cultures. The effect of kinetic and operating parameters on the stability characteristics of these models is also investigated.     

Output regulation of a class of unstructured models of continuous bioreactors: steady-state approaches

This article deals with the output regulation of continuous bioreactors in the face of constant disturbances and inverse dynamics. Nonlinear controllers developed on the basis of approximate equilibrium manifolds can almost attenuate measurable or unmeasurable disturbances on the output. This nonlinear feed-forward/feedback control framework without any tuning parameters can be directly implemented to strictly nonlinear systems. Under dynamic actuator constraints and the availability of only output signals for use in the control law, closed-loop simulations demonstrate that the proposed control techniques are superior to a nonlinear PI control scheme based on the identified Hammerstein model.     

Intelligent modelling of bioprocesses: a comparison of structured and unstructured approaches

This contribution moves in the direction of answering some general questions about the most effective and useful ways of modelling bioprocesses. We investigate the characteristics of models that are good at extrapolating. We trained three fully predictive models with different representational structures (differential equations, differential equations with inheritance of rates and a network of reactions) on Saccharopolyspora erythraea shake flask fermentation data using genetic programming. The models were then tested on unseen data outside the range of the training data and the resulting performances were compared. It was found that constrained models with mathematical forms analogous to internal mass balancing and stoichiometric relations were superior to flexible unconstrained models, even though no a priori knowledge of this fermentation was used.Paper presented at the international conference on trends in monitoring and control of life science applications, 7–8 October 2002, Lyngby, Denmark.     

Fundamental bottlenecks in the application of continuous bioprocesses.

Continuous culture is applied mainly as a research tool and much less as a production process. Fundamental bottlenecks in continuous culture are discussed to help shed light on this apparent contradiction. Based on a discussion of technical, process related, and economic/market bottlenecks it is concluded that the often mentioned productivity argument in favor of continuous processing is much too simple. The optimal choice of a process mode is determined by a full understanding of the equipment and production plant factors and of the economic/market factors. Very often the resulting choice will be the fed batch and/or the cell retention process mode which is characterized by low growth rates. Therefore more research towards product formation at low growth rates (less than 0.05 h-1) is needed.     

Control strategies of continuous bioprocesses based on biological activities

Based on the material balance principle applied to microbial reactions in continuous bioprocesses, the concept of reaction rate control has been developed theoretically. This concept provides a more direct way of controlling biological activities than the control of physical or chemical parameters in practice today. From an analysis of dynamic and steady-state experiments, two control systems for carbon dioxide production rate control during the continuous culture of baker's yeast have been designed and evaluated experimentally. In these control methods, intracellular NADH concentration is used as an immediate indication of the onset of glucose repression. A more sophisticated master controller based on the respiratory quotient can be combined with these control methods. The resulting control system provides a means to indirectly optimize biomass production while preventing ethanol formation in the continuous culture of baker's yeast.     

Chemostat cultures of yeasts, continuous culture fundamentals and simple unstructured mathematical models.

Fundamental aspects of chemostat cultures are reviewed. Using yeast cultures as examples, it is shown that steady states in chemostats may be predicted quantitatively by combining the correct number of unstructured kinetic models with expressions for existing stoichiometric constraints. The necessary number of such kinetic models corresponds to the number of limiting substrates and increases with the number of different metabolic pathways available to the strain. This is demonstrated by an experimental comparison of yeast growth limited by glucose alone for which metabolism is oxidative, and growth doubly limited by both glucose and oxygen, which occurs according to an oxido-reductive metabolism. The steady state data for such experiments can in principle be predicted based on a minimal amount of information by a simple stoichiometric model. It represents the overall stoichiometry of growth by a superposition of a fully oxidative and a fully reductive growth reaction and uses the concept of "aerobicity" to characterize the relative importance of the two reactions.     

A new training method for hybrid models of bioprocesses

Modeling of bioprocesses for engineering applications is a very difficult and time consuming task, due to their complex nonlinear dynamic behavior. In the last years several propositions for hybrid models, and especially serial approaches, were published and discussed, in order to combine analytical prior knowledge with the learning capabilities of Artificial Neural Networks (ANN). These approaches often require synchronous and equidistant sampled training data. However, in practice concentrations are mostly off-line measured, rare, and asynchronous. In this paper a new training method especially suited for very few asynchronously sampled data is presented and applied for modeling animal cell cultures. The achieved model is able to predict the concentrations of the reaction components inside a stirred tank bioreactor.     

Long-time behavior of continuous time models in genetic algebras

In [2] the solutions of Andreoli's differential equation in genetic algebras with genetic realization were shown to converge to equilibria. Here we derive an explicit formula for these limits.     

Estimation of multiple biomass growth rates and biomass concentration in a class of bioprocesses

In this paper, an approach to the estimation of multiple biomass growth rates and biomass concentration is proposed for a class of aerobic bioprocesses characterized by on-line measurements of dissolved oxygen and carbon dioxide concentrations, as well as off-line measurements of biomass concentration. The approach is based on adaptive observer theory and includes two steps. In the first step, an adaptive estimator of two out of three biomass growth rates is designed. In the second step, the third biomass growth rate and the biomass concentration are estimated, using two different adaptive estimators. One of them is based on on-line measurements of dissolved oxygen concentration and off-line measurement of biomass concentrations, while the other needs only on-line measurements of the carbon dioxide concentration. Simulations demonstrated good performance of the proposed estimators under continuous and batch-fed conditions.     

Classification of continuous diffraction patterns: a numerical study

The very intense and short pulses of future X-ray free electron lasers may allow the atomic resolution imaging of small, non-periodic objects. Preliminary estimates show that images obtained from single pulses do not contain statistically enough photons to allow successful reconstruction. Therefore multiple exposures of randomly oriented identical replicas have to be taken and the individual images have to be classified according to the object's orientation. The classification has been analytically treated by Huldt et al. [Huldt, G., Szoke, A., Hajdu, J., 2003. J. Struct. Biol. 144, 219.]. In this paper we extend the analytical results with numerical model calculations. This allows us to simulate realistic situations, which we will face in real experiments. We find significant deviations from the analytical expectations, even in the ideal case of spherical particles with random atomic distributions. We introduce a new norm for the individual scattering patterns and describe a criterion to select images belonging to similar orientation, which makes the classification more reliable in practice. We also discuss the effects of particle shape and size, partial orientational ordering, the measurement's resolution and the charge error caused by the Coulomb explosion.     

Discrete and continuous mathematical models of intraspecific behavior in the pharmacoethology of the mouse

An algorithm and software library were compiled in order to interpret the intraspecies agonistic animal behaviour in terms of discrete or continuous mathematical model. Applied aspects of the use of mathematical models in pharmacoethology were shown on concrete examples. The ways of construction of standard prototypes, and the integrative criteria of psychotropic drugs action were developed. The possibility was shown of identification of unknown substances by comparing with standard drugs by calculating the norm of standardized matrices.     

Global stability in a class of prey-predator models

Global stability is established in a class of prey-predator models. This includes a prey-predator model in which the predator has Type 2 functional response and no intraspecific interactions. Two simple examples demonstrate that Kolmogoroff’s theorem does not apply to some members of this class of models.     

Multilayer adaptive control of continuous bioprocesses using optimising control technique. Case study: Bakers' yeast culture

High costs associated with many fermentation processes in an increasingly competitive industry make any prompt application of modern control techniques to industrial bioprocesses very desirable. However, this is often hampered by the lack of adequate mathematical models, on the one hand, and by the absence of continuous, on-line measurement of the most relevant process variables, on the other hand. This paper addresses these problems and offers a new strategy to control continuous bioprocesses using a hierarchical structure such that neither structured process models nor continuous measurement of all relevant variables have to be available. The control system consists of two layers. The lower layer represents a dynamic adaptive follow-up control of a continuously measured output — in our case dissolved oxygen concentration. This variable is supposed to be strongly correlated with the key output variable — in our case cellular concentration which is not continuously available for measurement. The higher layer is then designed to maintain a desired profile of the process key output using a set-point optimising control technique. The Integrated System Optimisation and Parameter Estimation method used operates on an appropriately chosen steady-state performance criterion. A prerequisite for successful application of the proposed approach is an approximate steady-state model, describing the relationship between the measured output and the process key output variable. Furthermore, occasional in situ, off-line or laboratory measurement values of the key output variable are needed. Promising simulation results of the biomass concentration control, by manipulating the air flow-rate in the continuous bakers' yeast culture are presented.     

Kinetic studies and unstructured models of lymphocyte metabolism in fed-batch culture

The growth of two lymphocyte cell lines, a hybridoma cell line and a human cutaneous T cell lymphoma (HuT78), was studied in fed-batch culture, and unstructured models of growth developed. A criteria was established to insure that the growth rate varied by less than a specified tolerance throughout the culture period. Glutamine and serum were growth-limiting nutrients for both cell lines with half-maximal growth rates at 0. 53 mM glutamine and 0. 55%(v/v) serum for the hybridoma cells and 0. 21 mM glutamine and 1. 5% serum for the HuT-78 cells. Over the range of glucose concentrations from 5. 5 mM to 28 mM, the specific growth rate of hybridoma cells was independent of glucose concentration, whereas glucose concentrations above 5. 5 mM inhibited HuT-78 growth. For both cell lines, the growth rate was significantly inhibited by the addition of ammonium, although the hybridoma cell line was more affected by ammonia than was the HuT-78 cell line. Growth of HuT-78 cells increased in the presence of interleukin-2. Unstructured models for the hybridoma cells were similar to other models presented in the literature. Applications of these models to adoptive immunotherapy are discussed.     

Application of a simple unstructured kinetic and cost of goods models to support T-cell therapy manufacture

Manufacturing of cell therapy products requires sufficient understanding of the cell culture variables and associated mechanisms for adequate control and risk analysis. The aim of this study was to apply an unstructured ordinary differential equation-based model for prediction of T-cell bioprocess outcomes as a function of process input parameters. A series of models were developed to represent the growth of T-cells as a function of time, culture volumes, cell densities, and glucose concentration using data from the Ambr®15 stirred bioreactor system. The models were sufficiently representative of the process to predict the glucose and volume provision required to maintain cell growth rate and quantitatively defined the relationship between glucose concentration, cell growth rate, and glucose utilization rate. The models demonstrated that although glucose is a limiting factor in batch supplied medium, a delivery rate of glucose at significantly less than the maximal specific consumption rate (0.05 mg 1 × 10⁶ cell h⁻¹) will adequately sustain cell growth due to a lower glucose Monod constant determining glucose consumption rate relative to the glucose Monod constant determining cell growth rate. The resultant volume and exchange requirements were used as inputs to an operational BioSolve cost model to suggest a cost-effective T-cell manufacturing process with minimum cost of goods per million cells produced and optimal volumetric productivity in a manufacturing settings. These findings highlight the potential of a simple unstructured model of T-cell growth in a stirred tank system to provide a framework for control and optimization of bioprocesses for manufacture.     

Development of a large-scale biocalorimeter to monitor and control bioprocesses

Calorimetry has shown real potential at bench-scale for chemical and biochemical processes. The aim of this work was therefore to scale-up the system by adaptation of a standard commercially available 300-L pilot-scale bioreactor. To achieve this, all heat flows entering or leaving the bioreactor were identified and the necessary instrumentation implemented to enable on-line monitoring and dynamic heat balance estimation. Providing that the signals are sufficiently precise, such a heat balance would enable calculation of the heat released or taken up during an operational (bio)process. Two electrical Wattmeters were developed, the first for determination of the power consumption by the stirrer motor and the second for determination of the power released by an internal calibration heater. Experiments were designed to optimize the temperature controller of the bioreactor such that it was sufficiently rapid so as to enable the heat accumulation terms to be neglected. Further calibration experiments were designed to correlate the measured stirring power to frictional heat losses of the stirrer into the reaction mass. This allows the quantitative measurement of all background heat flows and the on-line quantitative calculation of the (bio)process power. Three test fermentations were then performed with B. sphaericus 1593M, a spore-forming bacterium pathogenic to mosquitoes. A first batch culture was performed on a complex medium, to enable optimization of the calorimeter system. A second batch culture, on defined medium containing three carbon sources, was used to show the fast, accurate response of the heat signal and the ability to perfectly monitor the different growth phases associated with growth on mixed substrates, in particular when carbon sources became depleted. A maximum heat output of 1100 W was measured at the end of the log-phase. A fed-batch culture on the same defined medium was then carried out with the feed rate controlled as a function of the calorimeter signal. A maximum heat output of 2250 W was measured at the end of the first log-phase. This work demonstrates that real-time quantitative calorimetry is not only possible at pilot-scale, but could be readily applied at even larger scales. The technique requires simple, readily available devices for determination of the few necessary heat flows, making it a robust, cost-effective technique for process development and routine monitoring and control of production processes.     

Congruent epidemic models for unstructured and structured populations: analytical reconstruction of a 2003 SARS outbreak

Both the threat of bioterrorism and the natural emergence of contagious diseases underscore the importance of quantitatively understanding disease transmission in structured human populations. Over the last few years, researchers have advanced the mathematical theory of scale-free networks and used such theoretical advancements in pilot epidemic models. Scale-free contact networks are particularly interesting in the realm of mathematical epidemiology, primarily because these networks may allow meaningfully structured populations to be incorporated in epidemic models at moderate or intermediate levels of complexity. Moreover, a scale-free contact network with node degree correlation is in accord with the well-known preferred mixing concept. The present author describes a semi-empirical and deterministic epidemic modeling approach that (a) focuses on time-varying rates of disease transmission in both unstructured and structured populations and (b) employs probability density functions to characterize disease progression and outbreak controls. Given an epidemic curve for a historical outbreak, this modeling approach calls for Monte Carlo calculations (that define the average new infection rate) and solutions to integro-differential equations (that describe outbreak dynamics in an aggregate population or across all network connectivity classes). Numerical results are obtained for the 2003 SARS outbreak in Taiwan and the dynamical implications of time-varying transmission rates and scale-free contact networks are discussed in some detail.     

On a class of non-linear transformation cure rate models

In this paper, we propose a generalization of the mixture (binary) cure rate model, motivated by the existence of a zero-modified (inflation or deflation) distribution, on the initial number of causes, under a competing cause scenario. This non-linear transformation cure rate model is in the same form of models studied in the past; however, following our approach, we are able to give a realistic interpretation to a specific class of proper transformation functions, for the cure rate modeling. The estimation of the parameters is then carried out using the maximum likelihood method along with a profile approach. A simulation study examines the accuracy of the proposed estimation method and the model discrimination based on the likelihood ratio test. For illustrative purposes, analysis of two real life data-sets, one on recidivism and another on cutaneous melanoma, is also carried out.     

Evaluation of two unstructured mathematical models for the penicillin G fedbatch fermentation

The mathematical model for the penicillin G fed-batch fermentation proposed by Heijnen et al. (1979) is compared with the model of Bajpai & Reuß (1980). Although the general structure of these models is similar, the difference in metabolic assumptions and specific growth and production kinetics results in a completely different behaviour towards product optimization. A detailed analysis of both models reveals some physical and biochemical shortcomings. It is shown that it is impossible to make a reliable estimation of the model parameters, only using experimental data of simple constant glucose feed rate fermentations with low initial substrate amount. However, it is demonstrated that some model parameters might be key factors in concluding whether or not altering the substrate feeding strategy has an important influence on the final amount of product.It is illustrated that feeding strategy optimization studies can be a tool in designing experiments for parameter estimation purposes.