Optimal control of a bioreactor

This paper is concerned with the analysis of a control problem related to the optimal management of a bioreactor. This real-world problem is formulated as a state-control constrained optimal control problem. We analyze the state system (a complex system of partial differential equations modelling the eutrophication processes for non-smooth velocities), and we prove that the control problem admits, at least, a solution. Finally, we present a detailed derivation of a first order optimality condition - involving a suitable adjoint system - in order to characterize these optimal solutions, and some computational results.     

An examination for mathematical modelling

First year mathematics degree students at Leicester Polytechnic attend a course in mathematical modelling. The aim is to introduce the students to mathematical modelling concepts and to model development. Work is set and marked during the course and this forms a vital part of the students' assessment. In addition to this, however, the students are assessed by means of a three hour examination at the end of the year. This examination is significantly different from the normal ‘five out of eight’ type. The philosophy and organization of the examination are discussed in this paper. An example of a particular examination, that for June 1986, is included as an appendix to illustrate the points made in the discussion.     

An examination of three draping questions

Rising costs and a new emphasis on outcomes are stimulating closer scrutiny of traditional OR practices such as draping. What is truly necessary? What is proven? What makes sense? This is the second of a two-part article on draping. In the July issue, we examined draping for clean-contaminated procedures, total body draping, and adherent drapes. In this issue, we tackle three more draping questions.     

Physiological loading of cartilage replacement materials in a bioreactor environment

Trying to replace injured cartilage by implants is a common practice in biomedical engineering. These implants can be non-seeded or seeded with human cartilage cells. To initiate cell multiplication and oriented cell growth in cell seeded implants, the implants are cultivated and usually stimulated electrically or mechanically in a bioreactor before implanting. In the present study, a knee testing bench combined with a bioreactor environment is developed. Doing so, it is possible to stimulate such implants controlled in a physiologically consistent, multi-dimensional way. The implants are placed in a recreated human knee joint and stimulated with several physiological load cycles of reproduced walking. After some days, the implanted material can be removed and mechanically and biologically evaluated in cooperation with the RWTH Aachen University Hospital. The new experimental set-up enables us for the first time to study the remodelling effect, the efficiency of the preconditioning as well as the influence of the body-conformable load on the material. Furthermore, the need of cell colonisation in the implants shall be investigated. To understand the correlation between tissue remodelling and mechanical load history, the experiment is also numerically investigated, based on a geometrically realistic FE model of the recreated human knee and appropriate material models for the involved structures. Doing so, the strains and stresses, as well as the shear forces in the implant can be evaluated. The results will be compared to experimental data. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

CFD simulation of a packed-bed solid-state fermentation bioreactor

In recent years, Solid-State Fermentation (SSF) has shown much promise for the development of bioprocesses and products. SSF involves the growth of microorganisms within a bed of moist solid particles permeated by a continuous gas phase and a minimum of visible water. SSF offers potential advantages over submerged culture: since the concentrations of products are often higher, smaller bioreactors can be used, reducing operational costs. However, there is a major challenge in obtaining adequate heat and mass transfer when this fermentation method is used at large scales. Mathematical models and computer simulations are useful tools for designing strategies to overcome this challenge; use of these tools can reduce costs of experimental development programs at pilot-scale and production scale, by reducing the number of fermentation experiments required. In the current work, we used the commercial CFD software ANSYS FLUENT® 16.0 to develop a mathematical model for heat and mass transfer in a pilot-scale packed-bed bioreactor. The model was used to simulate two different experiments that had been carried out previously in the bioreactor: first, the cooling of a bed of soybeans and, second, the heating of a bed composed of a mixture of wheat bran and sugarcane bagasse. The simulations considered the dynamics of airflow in the porous substrate bed and the non-equilibrium transfer of heat and moisture between the solid and gas phases. The second simulation considered a heterogeneous distribution of porosity within the substrate bed. Even though the two experiments were quite different, the same mathematical model was able to represent the temperature profiles observed experimentally. In the second simulation, the average temperature difference between the experimental and predicted values was 0.07°C. A third simulation was done for the growth of the filamentous fungus Aspergillus niger in SSF, with the predictions being compared to the results of a traditional mathematical model based on differential equations. Our work provides the basis for the development of a suitable and reliable mathematical model for testing operating conditions and control strategies for large-scale cultivation of microorganisms in SSF bioreactors.     

An examination of the homentropic Euler equations with averaged characteristics

This paper examines the properties of the homentropic Euler equations when the characteristics of the equations have been spatially averaged. The new equations are referred to as the characteristically averaged homentropic Euler (CAHE) equations. An existence and uniqueness proof for the modified equations is given. The speed of shocks for the CAHE equations are determined. The Riemann problem is examined and a general form of the solutions is presented. Finally, numerically simulations on the homentropic Euler and CAHE equations are conducted and the behaviors of the two sets of equations are compared.     

An inverse hyperbolic heat conduction problem in estimating surface heat flux of a living skin tissue

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent surface heat flux in a living skin tissue from the temperature measurements taken within the tissue. The inverse solutions will be justified based on the numerical experiments in which three different heat flux distributions are to be determined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent surface heat flux can be obtained for the test cases considered in this study.     

Electron-microscopic examination of a splat-cooled silver-germanium alloy

Extremely rapid solidification of a Ag-15 at.% Ge alloy results in the formation of a highly irregular structure. The individual grains consist of a lamellar mixture of close-packed structures with different stacking sequences (f.c.c. and h.c.p. phases as well as long-period stacking structures), superposed with a high density of randomly arranged stacking faults. It is concluded that the structure observed at room temperature may have formed by martensitic transformation of a metastable b.c.c. phase which is formed first upon solidification.     

An examination of HMM‐based investment strategies for asset allocation

We develop and analyse investment strategies relying on hidden Markov model approaches. In particular, we use filtering techniques to aid an investor in his decision to allocate all of his investment fund to either growth or value stocks at a given time. As this allows the investor to switch between growth and value stocks, we call this first strategy a switching investment strategy. This switching strategy is compared with the strategies of purely investing in growth or value stocks by tracking the quarterly terminal wealth of a hypothetical portfolio for each strategy. Using the data sets on Russell 3000 growth index and Russell 3000 value index compiled by Russell Investment Services for the period 1995–2008, we find that the overall risk‐adjusted performance of the switching strategy is better than that of solely investing in either one of the indices. We also consider a second strategy referred to as a mixed investment strategy which enables the investor to allocate an optimal proportion of his investment between growth and value stocks given a level of risk aversion. Numerical demonstrations are provided using the same data sets on Russell 3000 growth and value indices. The switching investment strategy yields the best or second best Sharpe ratio as compared with those obtained from the pure index strategies and mixed strategy in 14 intervals. The performance of the mixed investment strategy under the HMM setting is also compared with that of the classical mean–variance approach. To make the comparison valid, we choose the same level of risk aversion for each set‐up. Our findings show that the mixed investment strategy within the HMM framework gives higher Sharpe ratios in 5 intervals of the time series than that given by the standard mean–variance approach. The calculated weights through time from the strategy incorporating the HMM set‐up are more stable. A simulation analysis further shows a higher performance stability of the HMM strategies compared with the pure strategies and the mean–variance strategy. Copyright © 2009 John Wiley & Sons, Ltd.     

Stability and bifurcation of an unstructured model of a bioreactor with cell recycle

An unstructured model of a bioreactor with cell recycle and substrate inhibition kinetics is used to investigate the bifurcation and stability characteristics of this unit. The singularity theory used for this investigation allows a global analysis of steady-states multiplicity and the different bifurcation mechanisms occurring in the system including hysteresis and pitchfork. Analytical criteria are also derived for the safe operation of the reactor and to prevent wash-out conditions. The investigation of the dynamic bifurcation, on the other hand, shows that the model cannot exhibit periodic attractors with any growth kinetics model. The inability of this widely used model to exhibit periodicity despite the experimental results that support the existence of periodic behavior in many bioreactors suggests that new approaches are to be taken for the modeling.     

Modeling and simulation of phenol removal from wastewater using a membrane contactor as a bioreactor

Diverse methods have been introduced for phenol removal from wastewater. Among them, membrane bioreactors have attracted considerable attention in the last decade. In this study, modeling and simulation of a hollow fiber membrane contactor as a bioreactor was carried out. The effects of the various parameters such as flow rate, ratio of membrane porosity to tortuosity, membrane length, initial phenol concentration, number of fibers, and inner and outer radius of the membrane on phenol removal efficiency were studied. A proposed set of partial differential equations and related boundary conditions in the model were solved by the finite element method via simulation with computational fluid dynamics techniques. The simulation results were compared with existing empirical data from literature and acceptable agreement was observed. Based on the obtained results, increasing the initial concentration had a reverse impact on the phenol removal efficiency. However, increasing the cell phase flow rate slightly enhanced the removal efficiency. Moreover, extending the membrane length had a desirable effect. Also, augmentation of the number of fibers within the contactor initially increased and then decreased the efficiency.     

The examination of a vignette activity sequence in a secondary mathematics methods course

Because of their brief nature, vignettes are a strategic way to highlight or explore complex instructional practices. Using a qualitative approach, we examined how the use of vignettes in a Vignette Activity Sequence contributed to secondary mathematics preservice teachers’ understanding of the Mathematical Practices and the Mathematics Teaching Practices. By examining three vignettes used in two iterations of a secondary mathematics methods course, the researchers found that preservice teachers were able to draw connections between the vignettes and their own teaching experiences. However, some misconceptions or incomplete understandings related to the practices were revealed. Preservice teachers sometimes provided vague evidence when identifying particular practices in the vignettes that did not clearly indicate if they understood the practices. Taken together, the researchers found the Vignette Activity Sequence to be a valuable formative assessment that could be used to inform instruction in a secondary mathematics methods course. These findings have implications for teacher preparation programs and mathematics teacher educators.     

Biodegradation of substitutable substrates in a continuous bioreactor with cell recycle: A study of static bifurcation

The static bifurcation of an unstructured model of a continuous bioreactor with cell recycle involving the biodegradation of mixed wastes is analyzed using the singularity theory. The biodegradation of the dissimilar substrates follows Andrew's inhibitory kinetic models. It is shown that the steady state locus of the model can be described by a polynomial of seven order. The stability analysis shows that the model exhibits a number of singularities, including hysteresis and pitchfork that occurs at clean feed conditions. It is also shown that the model cannot predict isola, mushroom, or winged cusp singularities. The effect of the model kinetic and operating parameters on its stability characteristics are also discussed.     

Singular limit in a model of tissue inflammation

A nonlinear parabolic system derived by Alt and Lauffenburger and modelling the inflammatory response of a one-dimensional tissue to a bacterial infection and including a chemotactic process is considered. For typical tissue widths the derivation of the model involves a small parameter . The behaviour of the model as goes to zero is investigated and convergence of the solutions is obtained under rather weak requirements on the dependence of the initial data with respect to . Received September 19, 1997     

An anisotropic viscoelastic soft tissue model at large strains

Soft biological tissues represent complex inhomogeneous, and as a rule multiphase materials subjected to large strains under in vivo mechanical conditions. Apart from a number of other structural-related features they are characterized by a ratedependent material behavior which is attributed to fluid-solid interactions as well as intrinsic viscoelastic properties of the solid matrix. The authors propose to model rate-dependent phenomena of the solid phase of soft biological tissues within the context of a thermodynamically consistent phenomenological material approach resulting from an overstress concept. Due to the presence of directed fibrous constituents soft tissues should be considered as anisotropic materials. Therefore, the viscous overstress model has been completed by a transversely isotropic approach. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An optimum finite element technique to calculate oxygen concentration in tumoured tissue

The aim of this paper is to find a suitable finite element method for solving a tissue problem in a semi-infinite strip with a singularity at the origin. The problem is modelled using Laplace's equation in an infinite strip, simplified to a semi-infinite strip. An analytic form of the solution is derived and this is used for comparing with the finite element solution obtained in the semi-infinite strip using novel infinite elements. Numerical results for various values of the parameters present in the solution are reported and an optimum solution is presented.