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Macroscopic Modelling of Hybridoma Cell Fed-batch Cultures with Overflow Metabolism: Model-Based
Optimization and State Estimation
A macroscopic model that takes into account phenomena of overflow metabolism within glycolysis and glutaminolysis is proposed to simulate hybridoma HB-58 cell cultures. The model of central carbon metabolism is reduced to a set of macroscopic reactions describing the three metabolism states: respiratory metabolism, overflow metabolism and critical metabolism. The model is identified and validated with experimental data of fed-batch hybridoma cultures and successfully predicts the dynamics of cell growth and death, substrate consumption (glutamine and glucose) and metabolites production (lactate and ammonia). Based on a sensitivity analysis of the model outputs with respect to the parameters, a model reduction is proposed.
This model allows, on the one hand, quantitatively describing overflow metabolism in mammalian cell cultures and, on the other hand, will be valuable for monitoring and control of fed-batch cultures in order to optimize the process. This is illustrated in this work: First, with the determination of substrate feeding policies in fed-batch cultures with single and multiple feeds based on two different approaches (numerical and analytical approaches); and second, with the design and implementation of a state observer (extended Kalman filter) for online estimation of glucose and glutamine in hybridoma cell fed-batch cultures based on parameter identification for state estimation.
Zakaria Amribt
Macroscopic Modelling of Hybridoma Cell Fed-batch
Cultures with Overflow Metabolism: Model-Based
Optimization and State Estimation
Ph.D. Thesis presented in fulfillment of the requirements for the degree of Docteur en Sciences de l’Ingénieur
Thesis Advisor : Prof. Philippe Bogaerts
UNIVERSITE LIBRE DE BRUXELLES, École Polytechnique de Bruxelles
3BIO-BioControl (Modelling and Control of Bioprocesses)
École Polytechnique de Bruxelles
3BIO-BioControl (Modelling and Control of Bioprocesses)
Zakaria Amribt
Ph.D. Thesis
submitted at theÉcole Polytechnique de Bruxelles Université Libre de Bruxelles and presented on 23 June, 2014
Macroscopic Modelling of Hybridoma Cell Fed-batch
Cultures with Overflow Metabolism: Model-Based
Opti-mization and State Estimation
Dissertation committee:
Prof. Benoît Haut Prof. Patrick Fickers
Prof. Alain Vande Wouwer Prof. Jan Van Impe
Prof. Jean-Philippe Steyer
Université Libre de Bruxelles – President Université Libre de Bruxelles – Secretary Université de Mons
Katholieke Universiteit Leuven Laboratoire de Biotechnologie et de
l’Environnement (INRA Narbonne, France)
Prof. Philippe Bogaerts Université Libre de Bruxelles – Advisor
Amribt, Z. (2014). Macroscopic Modelling of Hybridoma Cell Fed-batch Cultures with Overflow Metabolism: Model-Based Optimization and State Estimation. PhD thesis, 3BIO-BioControl Modelling and Control of Bioprocesses, Université Libre de Bruxelles, Belgium.
À ma mère et à mon père,
À ma femme et à mon futur bébé,
Cette thèse de doctorat de plus de quatre années est réalisée dans le cadre du pro-jet OCPAM (Optimisation d’une Chaine de Production d’Anticorps Monoclo-naux), fruit des accords financiers de l’Université Libre de Bruxelles (ULB), de la Faculté Polytechnique de Mons (FPMs) et le Fonds Européen de Développe-ment Régional (FEDER). Pour accomplir cette thèse et le travail qu’il représente, j’ai bénéficié de l’aide de nombreuses personnes. Je leur adresse à tous mes plus sincères remerciements.
Je voudrais tout d’abord adresser mes remerciements les plus sincères à mon promoteur, Monsieur le Professeur Philippe Bogaerts, pour le temps qu’il a con-sacré à diriger mes travaux de recherche tout au long de ces dernières années, pour son professionnalisme, son soutien constant, sa rigueur, sa sympathie ainsi que ses précieux conseils tant sur le plan personnel que professionnel.
Je tiens à remercier Madame le Professeur Marianne Rooman, pour m’avoir ac-cueilli au sein du service 3Bio (BioModélisation, Bioinformatique et BioProcé-dés), et de m’avoir ainsi permis de réaliser mes travaux de thèse dans d’excellentes conditions.
Un merci particulier à Hongxing Niu, Docteur à Imperial College of Science London, de m’avoir fourni les données intéressantes qui ont donné à cette thèse le côté pratique si important pour mon travail, de m’avoir orienté lors de mes premiers pas de chercheur ainsi que pour sa collaboration et sa gentillesse.
nombreuses discussions et les échanges que nous avons eus, et qui m’ont permis de mener à bien mes travaux de recherche.
Je remercie Monsieur Benoît Haut, Directeur du service Transferts, Interfaces et Procédés (TIPs) de l'ULB, de m’avoir fait l’honneur de présider mon jury. Je tiens à rendre hommage à mes rapporteurs : Monsieur Patrick Fickers, Professeur à l’ULB, Monsieur Jan Van Impe, Professeur à KUL (Katholieke Universiteit of Leuven) et Monsieur Jean-Philippe Steyer, Professeur à l’Institut National de la Recherche Agronomique (INRIA-Narbonne France) pour le temps précieux qu’ils ont consacré à la lecture du présent mémoire ainsi que d’avoir accepté de participer au jury de ma soutenance.
Suite aux remerciements à mon promoteur et aux membres du jury, je tiens à re-mercier tous les membres du Service 3Bio (permanents ou temporaires): Natha-lie, Danièle, Anne, Khadija, Alex, Marie, Souad, Jean Marc, Dimitri, Fabian, Ja-roslav, Yves et Emmanuelle…. Merci à tous pour votre soutien, gentillesse, et pour la bonne ambiance qui a régnée dans le service. Je garde d'excellents souve-nirs de tous ces bons moments passés avec vous!
Ces remerciements ne sont pas encore complets. Si ce travail a pu voir le jour, c’est principalement grâce à l’amour et au soutien de mes proches.
J’ai une pensée toute particulière pour mes chers parents, ma mère, Fatima et mon père, Said ainsi que ma sœur, Zahera qui m’ont toujours soutenu et encoura-gé pendant mes années d’études malgré la longue distance qui nous séparait du-rant les 9 dernières années. Les mots ne suffisent pas pour vous exprimer toute ma reconnaissance. Sans leur intervention, leurs conseils et leurs sacrifices je n’aurai jamais réussi cette thèse. Maman, papa, je vous aime infiniment!
Je ne peux finir ces remerciements sans remercier ma chère épouse Laila pour son amour, sa présence et son soutien constant tout au long de cette thèse. Sans toi, je ne serais pas là où je suis maintenant, je te suis infiniment reconnaissant pour tout ce que tu as fait pour moi.
Au cours des dernières décennies, grâce aux progrès réalisés en biotechnologie et en génie génétique, les cultures de cellules animales (ovaires de hamster chi-nois (CHO), cellules d'hybridome, HEK293, BHK, MDCK et VERO) sont deve-nues le système le plus dominant pour la production d'anticorps monoclonaux (MAbs), qui sont des protéines recombinantes intervenant dans la fabrication des médicaments et des vaccins pour le traitement contre certaines maladies. Plus ré-cemment, des recherches tentent d’utiliser des anticorps monoclonaux pour faire reculer certains cancers.
Tout d'abord, un modèle mathématique macroscopique de cultures de cellules d’hybridomes HB-58 en bioréacteur fed-batch qui tient compte du phénomène d’overflow métabolique a été mis au point, identifié et validé. Le métabolisme central du carbone est réduit à un ensemble de réactions macroscopiques décri-vant les différents métabolismes des cellules : métabolisme respiratoire, métabo-lisme d’overflow et métabométabo-lisme critique. Les paramètres du modèle ainsi que leurs intervalles de confiance sont obtenus par l'intermédiaire d’un critère d’identification des moindres carrés non linéaires.
Le modèle est validé sur la base des données expérimentales des cultures d’hybridomes en mode fed-batch et il parvient à reproduire de façon satisfaisant la dynamique de consommation des substrats (glucose et glutamine), la produc-tion des métabolites (lactate et ammonium), et la croissance de la biomasse. Sur la base de l’analyse de sensibilité des sorties du modèle par rapport aux para-mètres une réduction de modèle est proposée et les 15 parapara-mètres du modèle ré-duit sont identifiés avec une bonne précision.
glu-tamine) à des valeurs critiques de manière à maintenir la culture des cellules d'hybridome à la limite de l'overflow métabolique. Une analyse de robustesse des profils d'alimentation optimaux obtenus avec les différentes stratégies d'op-timisation, a été réalisée, d'une part, par rapport aux incertitudes de paramètres et, d'autre part, par rapport aux erreurs de structure de modèle.
Enfin, un filtre de Kalman étendu pour l'estimation en ligne des concentrations en glucose et en glutamine dans des cultures de cellules d’hybridomes en bio-réacteur fed-batch a été développé, sur la base des mesures disponibles (biomasse (en ligne), lactate et ammoniaque (en ligne ou hors ligne)). Les conditions d'ob-servabilité sont examinées, et les performances d’estimation des concentrations en glucose et en glutamine dans des cultures de cellules hybridomes sont analy-sées. La sensibilité de l’observateur par rapport à l’estimation de la glutamine est améliorée par la ré-identification des paramètres du modèle sur la base d’une fonction de coût combinant le critère des moindres carrés avec un critère de
Monoclonal antibodies (MAbs) have an expanding market for use in diagnostic and therapeutic applications. Industrial production of these biopharmaceuticals is usually achieved based on fed-batch cultures of mammalian cells in bioreactors (Chinese hamster ovary (CHO) and Hybridoma cells), which can express differ-ent kinds of recombinant proteins. In order to reach high cell densities in these bioreactors, it is necessary to carry out an optimization of their production pro-cesses. Hence, macroscopic model equations must be developed to describe cell growth, nutrient consumption and product generation. These models will be very useful for designing the bioprocess, for developing robust controllers and for op-timizing its productivity.
This thesis presents a new kinetic model of hybridoma cell metabolism in fed batch culture and typical illustration of a systematic methodology for mathemati-cal modelling, parameter estimation and model-based optimization and state es-timation of bioprocesses.
In the next step, the effort is directed to the maximization of biomass productiv-ity in fed-batch cultures of hybridoma cells based on the overflow metabolism model. Optimal feeding rate, on the one hand, for a single feed stream contain-ing both glucose and glutamine and, on the other hand, for two separate feed streams of glucose and glutamine are determined using a Nelder-Mead simplex optimization algorithm. Two different objective functions (performance criteria) are considered for optimization; the first criterion to be maximized is the biomass productivity obtained at the end of the fed-batch culture, the second criterion to be minimized is the difference between global substrate consumption and the maximum respiratory capacity.
The optimal multi exponential feed rate trajectory improves the biomass produc-tivity by 10% as compared to the optimal single exponential feed rate. Moreover, this result is validated by the one obtained with the analytical approach in which glucose and glutamine are fed to the culture so as to control the hybridoma cells at the critical metabolism state, which allows maximizing the biomass productivi-ty. The robustness analysis of optimal feeding profiles obtained with different optimization strategies is considered, first, with respect to parameter uncertainties and, finally, with respect to model structure errors.
Acknowledgements ... 1
Résumé ... 3
Abstract ... 7
List of Figures ... 13
List of Tables ... 19
Abbreviations and Notations... 21
Introduction ... 25
Context and motivation... 25
Original contributions ... 29 Thesis outline ... 29 List of publications ... 31 Chapter I : ... 33 Bioprocess Modeling 1.1. Bioprocesses ... 33 1.1.1. Cell cultures ... 33
1.2. Characteristics of bioprocess models ... 36
1.2.1. Structured vs. unstructured ... 37
1.2.2. Segregated vs. unsegregated ... 38
1.2.3. Black box vs. white box ... 38
1.3. General dynamic model of bioprocesses ... 38
1.3.1. Macroscopic reaction scheme ... 39
1.3.2. General dynamic model ... 40
1.3.3. Yield coefficient matrix ... 41
1.3.4. Modelling the kinetics ... 42
1.4.1. Structural identifiability ... 45 1.4.2. Practical identifiability ... 47 1.4.3. Sensitivity functions ... 49 1.4.4. Model validation ... 50 1.5. Conclusion ... 51 Chapter II : ... 53 Model Development 2.1. Metabolic pathways in mammalian cells ... 53
2.2. Overflow metabolism in mammalian cells ... 58
2.2.1. Overflow metabolism ... 58
2.2.2. Macroscopic reactions ... 60
2.3. Kinetics of hybridoma cell cultures and system of macroscopic mass balances ... 62
2.3.1. Modeling cell kinetics ... 62
2.3.2. Kinetics formulation... 63
2.3.3. Macroscopic mass balances ... 65
2.4. Materials and methods ... 66
2.4.1. Cell lines and media ... 66
2.4.2. Bioreactor operation ... 67 2.4.3. Analysis methods ... 67 2.5. Conclusion ... 69 : Chapter III ... 71 Model Identification 3.1. Structural identifiability analysis... 71
3.1.1. Introduction ... 71
3.1.2. Generating series approach ... 73
3.1.3. Illustrative example ... 75
3.1.4. Results and discussion ... 77
3.2. Practical identifiability analysis ... 82
3.2.1. Parameter identification ... 82
3.2.2. Simulation results ... 88
3.2.3. Statistical validation ... 91
3.3. Sensitivity function-based model reduction ... 92
3.3.1. Problem Formulation ... 92
3.3.3. Model reduction ... 95
3.3.4. Uncertainty on the model outputs from parametric estimation errors 99 3.4. Conclusion ... 100
3.5. Complementary results for Chapter 3 ... 101
: Chapter IV ... 105
Process Optimization 4.1. Introduction ... 105
4.2. Optimal feeding strategy ... 107
4.2.1. Problem formulation ... 108
4.2.2. Constant-feed ... 109
4.2.3. Staircase feed ... 110
4.2.4. Exponential feed ... 112
4.3. Process optimization based on numerical approach ... 114
4.3.1. Problem formulation ... 114
4.3.2. Single feed optimization ... 115
4.3.3. Extension to multiple feeds ... 118
4.4. Process optimization based on analytical approach ... 120
4.4.1. Single feed optimization ... 121
4.4.2. Extension to multiple feeds ... 125
4.5. Robustness analysis ... 127
4.5.1. Robustness regarding parameter uncertainties ... 127
4.5.2. Robustness regarding model structure errors ... 131
4.6. Conclusions ... 136
4.7. Complementary results for Chapter 4 ... 137
: Chapter V ... 141 State Estimation 5.1. Introduction ... 141 5.2. System observability ... 143
5.3. The extended Kalman filter ... 146
5.3.1 Introduction ... 146
5.3.2 State estimation results ... 147
5.4. Parameter identification for state estimation ... 148
5.4.1. Parameter identification ... 148
5.4.2. State estimation ... 151
... 157 Conclusion and Perspectives
1.1. Schematic representation of a bioreactor (Hulhoven, 2006). ... 35 1.2. Various operating modes of biological processes - modified from (Dochain,
2008) ... 36 1.3. Structured vs.unstructured models (S: substrates, C: intracellular metabolites, P:
products) - modified from (Gelbgras, 2008) ... 37 1.4. General procedure for bioprocess modelling ... 52 2.1. Energy relationships between catabolic and anabolic pathways. Catabolic
pathways deliver chemical energy in the form of ATP, NADH, NADPH, and FADH2. These energy carriers are used in anabolic pathways to convert small
precursor molecules into cell macromolecules. See (Lehninger et al., 2004) .. 55 2.2. Schematic diagram of simplified reaction network for mammalian cell
metabolism - modified from (Provost and Bastin, 2004). ... 56 2.3. Illustration of the overflow metabolism in hybridoma cells. Case 1: Respiratory
metabolism with glucose and glutamine completely consumed for cell growth; Case 2: Critical metabolism (maximum respiratory capacity) with cells maximum specific growth rate; Case 3: Overflow metabolism with glucose and/or glutamine excess, and production of the associated metabolites (lactate and/or ammonium). ... 59 2.4. Simplified metabolic network for hybridoma cells only presenting glucose and
glutamine metabolic fluxes in the central carbon metabolism. ... 60 2.5. Representation of the phenomena which have been taken into account in
kinetics of the overflow metabolism model. ... 63 2.6. Measured concentrations of variables: viable cells (Xv), dead cells (Xd), glucose
(G), lactate (L), glutamine (Gn) and ammonia (N) during four hybridoma HB-58 fed-batch cultures. ... 69 3.1. Complete and reduced identifiability tableaus of (3.3) ... 76 3.2. The next reduced identifiability tableaus of (3.3) ... 77 3.3. Complete and reduced identifiability tableaus of the overflow metabolism
3.4. Complete and reduced identifiability tableaus of the overflow metabolism model for (Scenario B: Overflow metabolism of glucose and respiratory metabolism of glutamine) ... 79 3.5. Complete and reduced identifiability tableaus of the overflow metabolism
model (Scenario C: Respiratory metabolism of glucose and overflow metabolism of glutamine) ... 80 3.6. Complete and reduced identifiability tableaus of the overflow metabolism
model (Scenario D: Respiratory metabolism of glucose and glutamine) ... 80 3.7. Histograms of cost function J(θ) (50 uniformly distributed pseudo-random
initialization) obtained with Qij defined as (3.6) and two different methods of initialization: i) initialization based on the values obtained with Qij defined as (3.5) and ii) initialization based on the values of the literature. ... 84 3.8. Comparison between model simulation and measurements of culture 3 after
parameter identification with experiments 1, 2 and 3 (direct validation). ... 89 3.9. Comparison between model simulation and measurements of culture 4 after
parameter identification with experiments 1, 2 and 3 (cross validation). ... 89 3.10.Comparison between model simulation and measurements of culture 2 after
parameter identification with experiments 2, 3 and 4 (direct validation). ... 90 3.11.Comparison between model simulation and measurements of culture 1 after
parameter identification with experiments 2, 3 and 4 (cross validation). ... 90 3.12. Evolution of the 6×17 semirelative sensitivity functions with respect to culture
time. Columns from left to right: sensitivities of outputs of XV, Xd, G, Gn, L and N to the 17 parameters. ... 94 3.13.Colored representation of the absolute values of the linear correlation matrix
obtained for parameters identified with experiments 1, 2 and 3. ... 95 3.14.Comparison between model simulation of culture 3 with the original 17
parameter model (red line) and the reduced 15 parameter model (black line) after parameter identification with experiments 1, 2 and 3 (direct validation). 96 3.15.Comparison between model simulation of culture 4 with the original 17
parameter model (red line) and the reduced 15 parameter model (black line) after parameter identification with experiments 1, 2 and 3 (cross validation) .. 96 3.16.Comparison between model simulation of culture 2 with the original 17
parameter model (red line) and the reduced 15 parameter model (black line) after parameter identification with experiments 2, 3 and 4 (direct validation). 98 3.17.Comparison between model simulation of culture 1 with the original 17
3.18.Colored representation of the absolute values of the linear correlation matrix obtained for reduced model parameters (dim θ=15) identified with experiments 1, 2 and 3. ... 99 3.19.Representation of the uncertainty in the model predictions: Monte Carlo
simulations (500 samples) of normally distributed pseudo-random parameter values. In black: confidence intervals at 95% of the model predictions. In red: model predictions. Blue circles: discrete-time experimental data. ... 102 4.1. Optimal feeding rate and corresponding optimal concentration profiles of viable
cells, substrates (glutamine and glucose) and metabolites (lactate and ammonia) in the case of constant feed. ... 110 4.2. Optimal feeding rate and corresponding optimal concentration profiles of viable
cells, substrates (glutamine and glucose) and metabolites (lactate and ammonia) in the case of staircase feed with 3 time partitions (n=3). ... 111 4.3. Optimal feeding rate and corresponding optimal concentration profiles of viable
cells, substrates (glutamine and glucose) and metabolites (lactate and ammonia) in the case of staircase feed with 5 time partitions (n=5). ... 112 4.4. Optimal feeding rate and corresponding optimal concentration profiles of viable
cells, substrates (glutamine and glucose) and metabolites (lactate and ammonia) in the case of an exponential feed. ... 113 4.5. Optimal feeding rate for a single feed stream Case1 (Gin and Gnin are not
included in the optimization problem) and corresponding optimal concentration profiles of viable cells, substrates (glutamine and glucose) and metabolites (lactate and ammonia) obtained by numerical approach optimization of criterion 1 (red line), criterion 2 (α=1, blue line) and analytical approach (green line). ... 116 4.6. Optimal feeding rate for a single feed stream Case 2 (Gin and Gnin are included
in the optimization problem) and corresponding optimal concentration profiles of viable cells, substrates (glutamine and glucose) and metabolites (lactate and ammonia) obtained by numerical approach optimization of criterion 1 (red line), criterion 2 (α=1, blue line) and analytical approach (green line). ... 117 4.7. Optimal feeding rate for two separate feed streams of glucose and glutamine
and corresponding optimal concentration profiles of viable cells, substrates (glutamine and glucose) and metabolites (lactate and ammonia) obtained by optimization of criterion 1 (red line), criterion 2 (α=1, blue line) and analytical approach (green line). ... 119 4.8. Effect of α (the weight of the glucose overflow in the second optimization
criterion) on the biomass productivity. ... 120 4.9. Evolution of NL with respect to the medium composition, i.e., Gin and Gnin.
Each decreasing curve starting from NU = 8,75 mM represents from left to right
current operating conditions (i.e. Gin =15 mM and Gnin = 4 mM in Case 1 and
Gin =38 mM and Gnin = 10 mM in Case 2). ... 124
4.10.Variation of productivity obtained for single feed with respect to the one obtained with the nominal optimal feeding profile (feeding profile criterion 1, Case 1) after simulations (blue circles) and optimization (red circles) based on 100 different values of model parameters (which were randomly chosen around 20% of nominal values). ... 128 4.11.Variation of productivity obtained for multiple feeds with respect to the one
obtained with the nominal optimal feeding profile (feeding profile criterion 1) after simulations (blue circles) and optimization (red circles) based on 100 different values of model parameters (which were randomly chosen around 20% of nominal values). ... 129 4.12.Histograms of productivity obtained with 2000 runs for single (red histogram)
and multiple (blue histogram) feeds case (A): all parameters are randomly chosen in their confidence intervals (B): µGmax2 and a are fixed to their nominal
values (the remaining parameters are randomly chosen in their confidence intervals). ... 130 4.13.Representation of the variation of the identification criterion, between the
modified model and the nominal one, in function of the parameter values (KL,
KL1 and KN1). ... 132
4.14.Comparison of performances obtained with: (a) simulation based on the modified model 1 and the optimal single feed profile using criterion 1 applied to the same modified model 1 (green lines); (b) simulation based on the modified model 1 and the optimal single feed profile using criterion 1 applied to the original nominal model (blue lines); (c) simulation based on the original nominal model and the optimal single feed profile using criterion 1 applied to the same original nominal model (red lines). ... 133 4.15.Comparison of performances obtained with: (a) simulation based on the
modified model 2 and the optimal single feed profile using criterion 1 applied to the same modified model 2 (green lines); (b) simulation based on the modified model 2 and the optimal single feed profile using criterion 1 applied to the original nominal model (blue lines); (c) simulation based on the original nominal model and the optimal single feed profile using criterion 1 applied to the same original nominal model (red lines). ... 134 4.16.Comparison of performances obtained with: (a) simulation based on the
4.17.Comparison of performances obtained with: (a) simulation based on the modified model 1 and the optimal multiple feeds using criterion 1 applied to the same modified model 1 (green lines); (b) simulation based on the modified model 1 and the optimal multiple feeds using criterion 1 applied to the original nominal model (blue lines); (c) simulation based on the original nominal model and the optimal multiple feeds using criterion 1 applied to the same original nominal model (red lines). ... 138 4.18.Comparison of performances obtained with: (a) simulation based on the
modified model 2 and the optimal multiple feeds using criterion 1 applied to the same modified model 2 (green lines); (b) simulation based on the modified model 2 and the optimal multiple feeds using criterion 1 applied to the original nominal model (blue lines); (c) simulation based on the original nominal model and the optimal multiple feeds using criterion 1 applied to the same original nominal model (red lines). ... 139 4.19.Comparison of performances obtained with: (a) simulation based on the
modified model 3 and the optimal multiple feeds using criterion 1 applied to the same modified model 3 (green lines); (b) simulation based on the modified model 3 and the optimal multiple feeds using criterion 1 applied to the original nominal model (blue lines); (c) simulation based on the original nominal model and the optimal multiple feeds using criterion 1 applied to the same original nominal model (red lines). ... 140 5.1. The software sensor principle as defined in (Cheruy, 1997; Bogaerts and Vande
Wouwer, 2003; Luttmann et al., 2012). ... 141 5.2. Estimation of glucose and glutamine using the measurements (blue circles) of
biomass, lactate and ammonia. In black: 50 runs of extended Kalman filter when varying initial conditions randomly around 50% of nominal values. In blue: model evolution. In green: confidence intervals at 95%. ... 148 5.3. Evolution of J(θ) and Fobs(θ) as functions of . ... 150
5.4. Comparison between model simulation of culture 4 (cross validation test) with the original parameter model (red line) and the modified parameter model (black line). ... 151 5.5. Estimation of glucose and glutamine based on modified model parameters and
using the measurements (blue circles) of biomass, lactate and ammonia. In black: 50 runs of extended Kalman filter when varying initial conditions randomly around 50% of nominal values. In blue: model evolution. In green: confidence intervals at 95%. ... 152 5.6. Estimation (with initial conditions of culture 3) of glucose and glutamine based
nominal values. In blue: model evolution. In green: confidence intervals at 95%. ... 154 5.7. Estimation (with initial conditions of culture 1) of glucose and glutamine based
2.1. Summary of culture conditions for the fed-batch experiments ... 68 3.1. Measured initial conditions of the 6 state variables of the model. ... 83 3.2. Parameter values adopted from the literature (Tremblay et al., 1992; Dhir et al.,
2000; Xing et al., 2010; Kontoravdi et al., 2010). ... 85 3.3. Parameter values identified with all sets of 3 experiments (leave-one-out cross
validation) for Qij defined as (3.4)... 86
3.4. Parameter values (and their corresponding variation coefficients (%)) identified with all sets of 3 experiments (leave-one-out cross validation) for Qij defined as
(3.5). ... 87 3.5. Correlation coefficients of each simulated variable with all the sets of 3
experiments. ... 91 3.6. Parameter values (and their corresponding variation coefficients (%)) of the
reduced model identified with all sets of 3 experiments (leave-one-out cross validation) ... 97 3.7. Correlation matrix obtained for original model parameters (dim θ=17) identified
with experiments 1, 2 and 3. ... 103 3.8. Correlation matrix obtained for reduced model parameters (dim θ=15) identified
with experiments 1, 2 and 3 ... 104 4.1. The different considered feeding policies and corresponding parameters. ... 109 4.2. The optimal biomass productivity and final culture time for the various cases. ... 112 4.3. Considered feeding rates and corresponding parameters for single and multiple
feeds... 114 4.4. The optimal biomass productivity and final culture time for the various
optimization strategies ... 126 5.1. Modified values of model parameters with all sets of 3 experiments ... 150 5.2. RMSE of glucose and glutamine obtained for different extended Kalman filter
MAbs Monoclonal antibodies CHO Chinese hamster ovary HEK Human embryonic kidney BHK Baby hamster kidney MDCK Madin darby canine kidney PAT Process analytical technology FDA Food and drug administration EMA European medicines agency ATCC American type culture collection TCA Tricarboxylic acid
ATP Adenosine triphosphate ADP Adenosine diphosphate
NADH, NAD+ Nicotinamide adenine dinucleotide
NADPH Nicotinamide adenine dinucleotide phosphate FADH2, FAD Flavin adenine dinucleotide
DNA Desoxyribonucleic acid RNA Ribonucleic acid
α-KG α -Ketoglutarate Ac-CoA Acetyl coenzyme A CO2 Carbon dioxide
MBA Macroscopic balance analysis MFA Metabolic flux analysis
RMSE Root mean square error
t Time (h)
V Culture volume (L)
F Volumetric feed rate (L/h)
D Dilution rate (1/h)
θ Vector of the parameters ̂ Estimate value of θ
J(θ) Identification criterion
Cr Correlation coefficient for parameters
Fobs(θ) Estimation sensitivity measure criterion
Xv Viable cell concentration (109cells/L)
Xd Dead cell concentration (109cells/L)
G Glucose concentration in bioreactor (mM)
Gin Glucose concentration in feeding medium (mM)
Gn Glutamine concentration in bioreator (mM)
Gnin Glucose concentration in feeding medium (mM)
L Lactate concentration in bioreactor (mM)
N Ammonia concentration in bioreactor (mM)
G(Gn) Glucose (glutamine) consumptions rate (mmol/(L.h))
G1(Gn1) Global glucose (glutamine) consumptions rate (mmol/(L.h))
max G(Gn) Maximum respiratory rate of glucose (glutamine) (mmol/(L.h))
Over- G(Gn) Overflow metabolism rate of glucose (glutamine) (mmol/(L.h))
µdmax Maximum death rate (1/h)
µGmax1 Maximum specific uptake rate of glucose (mmol/(109cells.h))
µGnmax1 Maximum specific uptake rate of glutamine (mmol/(109cells.h))
µGmax2 Maximum specific respiratory rate of glucose (mmol/(109cells.h))
µGnmax2 Maximum specific respiratory rate of glutamine (mmol/(109cells.h))
KG Monod constant of glucose (mM)
KGn Monod constant of glutamine (mM)
KN Ammonia inhibition constant (mM)
KL Lactate inhibition constant (mM)
KdG Glucose inhibition constant for cell death (mM)
KdGn Glutamine inhibition constant for cell death (mM)
mG Maintenance coefficient of glucose (mmol/(109cells.h))
mGn Maintenance coefficient of glutamine (mmol/(109cells.h))
a, c Stoichiometric coefficients (109cells/mmol)
c, d Stoichiometric coefficients (mmol/mmol)
R2 Correlation coefficient for simulated variables
tf Final culture time (h)
tfeed Duration of batch phase (h)
Gcrit Critical concentration of glucose (mM)
Gncrit Critical concentration of glutamine (mM)
FG Feed stream of glucose
Context and motivation
In recent decades, due to advances in genetic engineering, mammalian cells (Chinese hamster ovary (CHO), Hybridoma, HEK293, BHK and MDCK cells) have become the dominant system for the production of monoclonal antibodies (MAbs), which are used in the manufacture of drug and vaccines, such as
recom-binant therapeutic proteins, insulin, enzymes and hormones, etc. (Wurm, 2004;
Rodrigues et al., 2010; Li et al., 2010).
In mammalian cell culture systems, cells use the nutrients of culture medium to produce the energy necessary for their growth and the production of proteins, by means of a complex network of reactions called cell metabolism. Therefore, the cells survive, grow, die and produce monoclonal antibodies according to their growing environment. For example, deprivation of nutrients or the accumulation of toxic compounds resulting from cellular metabolism can cause death or reduce protein production. This demonstrates a link between living cells, monoclonal antibodies production and composition of the culture medium.
With the recent Process Analytical Technology (PAT) (Dirk, 2006; Teixeira et al., 2009) initiative from the regulatory authorities (Food and Drug Administra-tion (FDA) and European Medicines Agency (EMEA)), it now seems valuable to obtain a thorough metabolic characterization of cell lines and of the relationships between the cell environment and cellular behaviour. Such work can lead to the development of mathematical models, which can be valuable tools to reduce bio-process development time and cost.
Mathematical modelling is a very powerful tool in physics, chemistry, and en-gineering, it is of great interest in all situations where achieving real experiments outside the normal operating range of a process is to be avoided for practical, economic and security reasons. In biotechnology, these situations are frequently encountered so that the use of models is essential to the development or im-provement of many bioprocesses. In addition, the goal of mathematical model-ling is to derive simple dynamical models which are the basis for the design of on-line algorithms for process monitoring, control and optimization of these bio-processes.
computationally tractable (Sidoli et al., 2004; Sidoli et al., 2006), which makes the model easier to identify and to use in controllers or software sensors.
The issue of bioprocess modelling from extracellular measurements has been considered for a long time in the literature (Sidoli et al., 2004; Haag et al., 2005a; Haag et al., 2005b; Hulhoven et al., 2005; Mailier and Vande Wouwer, 2010). In classical macroscopic models the cells are just viewed as a catalyst for the con-version of substrates into products which is represented by a set of chemical “macro-reactions” that directly connect extracellular substrates and products without paying much attention to the intracellular behavior. Dynamical mass bal-ance models are then established on the basis of these macro-reactions by identi-fying appropriate kinetic models from the experimental data. Such macroscopic models rely on the category of “unstructured models”. The goal of macroscopic modeling is clearly to derive simple models that have been proved of paramount importance in bioengineering for the design of on-line algorithms for process monitoring, control and optimization. Though understanding of detailed mech-anisms is not a requirement for macroscopic modeling, the more information is quantitatively incorporated, the more prediction strength it has. The methodology developed in this thesis can be classified in the same category as macroscopic approaches.
Overflow metabolism denotes the incomplete oxidation, despite aerobic condi-tions, of an abundantly supplied energy source resulting in the excretion of or-ganic end products. These metabolic byproducts are often inhibitory.
The present work focuses on constructing a simple and identifiable macroscopic model of hybridoma cells that takes into account the phenomenon of overflow metabolism within glycolysis and glutaminolysis. These phenomena have been widely recognized in microbial biotechnology, e.g. ethanol production by Sac-charomyces cerevisie (Sonnleitner and Käppeli, 1986; Fiechter and Gmünder, 1989) and acetate by Escherichia coli (Xu et al., 1999).
In animal cell cultures, it has been widely recognized (Häggström and Ljunggren, 1994; Häggström et al., 1996; Doverskog et al., 1997; Quesney at al., 2003) that cells convert a significant amount of glucose and glutamine to lactate and ammo-nia and very little carbon enters the tricarboxylic acid cycle (TCA cycle) under high glucose and glutamine concentrations, which is referred to as the “aerobic glycolysis and glutaminolysis” or “glucose and glutamine overflow metabolism”. Therefore, mathematically describing this phenomenon is important for the de-sign of optimal feeding profile and the optimal operating conditions (final cul-ture time, duration of batch phase, feeding concentrations of glucose and gluta-mine…) of hybridoma cell fed-batch cultures in order to control cells at the most desirable metabolic state without unnecessary overflow metabolism. This strate-gy should allow to avoid the waste of substrates (glucose and glutamine) via the overflow metabolism and to maintain maximum biomass productivity.
Original contributions
In short, the significant and original contributions of this work are summarized as follows:
In bioprocess modelling:
Macroscopic modelling of overflow metabolism in hybridoma cell
fed-batch cultures.
Typical illustration of a systematic methodology for mathematical
model-ling and parameter estimation of biological systems.
In bioprocess optimization:
Development of simple optimization strategies based on the overflow
metabolism model for the determination of substrate feeding policies in fed-batch cultures with single and multiple feeds.
Robustness analysis of optimal feeding profiles obtained with different
optimization strategies, first, with respect to parameter uncertainties and, finally, with respect to model structure errors.
In state estimation:
Design of a state observer (extended Kalman filter) for online estimation
of glucose and glutamine in mammalian cell cultures with phenomena of overflow metabolism, based on parameter identification for state estima-tion.
Thesis outline
cul-tures and bioreactors are introduced before presenting an overview related to bio-process modelling and parameter estimation.
The second chapter “Model Development” is dedicated to the development of a macroscopic model that takes into account phenomena of overflow metabolism within glycolysis and glutaminolysis for simulation of hybridoma HB-58 cell cul-tures. The model of central carbon metabolism is reduced to a set of macroscopic reactions and the kinetic model is formulated. The experimental data of hybrido-ma HB-58 cell cultures are also presented in Chapter 2.
Then, in Chapter 3 “Model Identification”, the proposed macroscopic model is validated with experimental data of fed-batch hybridoma cultures and numerical results obtained for parameter estimation are presented. The number of model parameters is reduced by parameter sensitivity analysis and their values are iden-tified with confidence intervals.
Concerning the optimization of hybridoma cell fed-batch cultures, Chapter 4 “Process Optimization”, develops an optimization procedure based on the over-flow metabolism model for the determination of substrate feeding policies in fed-batch cultures with single and multiple feeds. The optimal control problem is formulated and solved by using two different approaches (numerical and analyti-cal approaches). The robustness analysis of optimal feeding profiles obtained with different optimization strategies is also considered, first, with respect to pa-rameter uncertainties and, finally, with respect to model structure errors.
Journal Papers
Z. Amribt, H. Niu, Ph. Bogaerts, Macroscopic Modelling of Overflow Me-tabolism and Model Based Optimization of Hybridoma Cell Fed-Batch Cul-tures, Biochem. Eng. J. 70 (2013) 196-209.
H. Niu, Z. Amribt, P.Fickers, W.Tan, P. Bogaerts, Metabolic Pathway Analy-sis and Reduction for Mammalian Cell Cultures - Towards Macroscopic Modeling, Chem. Eng. sci. 102 (2013) 461–473.
Z. Amribt, L. Dewasme, A. Vande Wouwer, Ph. Bogaerts, Optimization and robustness analysis of hybridoma cell fed-batch cultures using the overflow metabolism model, Bioprocess Biosyst. Eng. (2014).
L. Dewasme, S. Fernandes, Z. Amribt, L.O. Santos, Ph. Bogaerts, A. Vande Wouwer, State estimation and predictive control of fed-batch cultures of hy-bridoma cells, Submitted in Journal of Process Control.
Proceedings
Z. Amribt, H. Niu, Ph. Bogaerts, Macroscopic Modelling of Overflow Me-tabolism in Fed-batch Cultures of Hybridoma Cells, 6th International
Con-ference on Mathematical Modelling (MATHMOD 2012), Vienna, Austria,
February 15-17, 2012.
L. Dewasme, Z. Amribt, L. O. Santos, A.-L. Hantson, Ph. Bogaerts, A. Vande Wouwer, Hybridoma Cell Culture Optimization Using Nonlinear Model Pre-dictive Control, 12th IFAC Symposium on Computer Applications in
Biotech-nology (CAB 2013), Mumbai, India, December 16-18, 2013.
Z. Amribt, L. Dewasme, A. Vande Wouwer, Ph. Bogaerts, Optimal Operation of Hybridoma Cell Fed-batch Cultures Using the Overflow Metabolism Mod-el: Numerical and Analytical Approach, 12th IFAC Symposium on Computer
Applications in Biotechnology (CAB 2013), Mumbai, India, December 16-18,
Z. Amribt, L. Dewasme, A. Vande Wouwer, Ph. Bogaerts, Parameter Identi-fication for State Estimation: Design of an Extended Kalman Filter for Hy-bridoma Cell Fed-Batch Cultures, Accepted in the 19th World Congress of
the International Federation of Automatic Control (IFAC 2014), Cape Town,
South Africa, August 24-29, 2014.
Conferences with Abstracts
Z. Amribt, H. Niu, Ph. Bogaerts, Macroscopic Modelling of Overflow Me-tabolism in Fed-batch Cultures of Hybridoma Cells, 31th Benelux Meeting on
Systems and Control, Heijden, The Netherlands, March 27-29, 2012.
Z. Amribt, H. Niu, Ph. Bogaerts, Optimization of Feeding Rate for Hybrido-ma Cell Fed-batch Cultures Using an Overflow Metabolism Model, 32th
Benelux Meeting on Systems and Control, Houffalize, Belgium, March 26-28,
2013.
Z. Amribt, H. Niu, Ph. Bogaerts, Optimization of Hybridoma Cell Growth in Fed-batch Cultures, 9th European Congress of Chemical Engineering (ECCE/ECAB 2013), The Hague, The Netherlands, April 21-25, 2013.
Z. Amribt, L. Dewasme, A. Vande Wouwer, Ph. Bogaerts, Robustness analy-sis of optimal feeding profiles for hybridoma cell fed-batch cultures using an overflow metabolism model, 33rd Benelux Meeting on Systems and Control, Heijden, The Netherlands, March 25-27, 2014.
Conference and Posters without Abstracts
Z. Amribt, Ph. Bogaerts, Mathematical modelling and optimal control of hy-bridoma cell cultures in perfused bioreactors, Interuniversity Attraction Poles, Dynamic Systems Control and Optimization, IAP-DSCO study days, Leuven, Belgium, 27/09/2013.
Z. Amribt, H. Niu, Ph. Bogaerts, Macroscopic modelling of overflow metabo-lism in fed-batch cultures of hybridoma cells, Environmental sciences, tech-nologies and management, ENVITAM-GEPROC PhD student day, Gem-bloux, Belgium, 08/02/2013.
Z. Amribt, H. Niu, Ph. Bogaerts, Macroscopic modelling, parameter estima-tion and sensitivity analysis of overflow metabolism in hybridoma cell cul-tures, Matinée des Chercheurs, Research Fair 2012, Bruxelles, Belgium, 01/03/2012.
Z. Amribt, L. Dewasme, A. Vande Wouwer, Ph. Bogaerts, Optimization and Robustness Analysis of Hybridoma Cell Fed-Batch Cultures Using the Over-flow Metabolism Model, Ecole InterFacultaire de BioIngénieur,
Bioprocess Modeling
This chapter proposes a general discussion about a classical modelling approach applied in bioprocesses and that is going to be used, in the following chapters to develop a macroscopic model of hybridoma cell fed-batch cultures. First, we re-mind what are bioprocesses, cell cultures and bioreactors in Section 1.1. After-wards, general characteristics of bioprocess modelling are presented in Section 1.2. A general dynamic model of a bioprocess is then derived and its elements are described in more details in Section 1.3 before tackling parameter estimation in section 1.4. Finally, Section 1.5 proposes some conclusions.
1.1. Bioprocesses
“A bioprocess may be defined as any process that uses complete living cells or their components to effect desired physical or chemical changes” Ziad et al., (1999). The bioprocess is a structure which is characterized by the cell strains capabilities and characteristics, the culture medium composition as well as by the bioreactor performance and the main variables evolution (temperature, pH, agita-tion, aeration...). All these concepts are explained in this section.
1.1.1. Cell cultures
in a favorable artificial environment (under precise and controlled conditions of temperature, nutrition, gas…) and in absence from contamination.
The cell is a highly complex system which is defined as the smallest independent biological entity, in a favorable environment containing the necessary nutrients, which is able to maintain and reproduce itself alive. This ability to survive and multiply allows to think of growing cells outside their natural environment in a bioreactor. Indeed, such cell cultures in bioreactor favor cell growth by the moni-toring and control of the cell environmental conditions like temperature, pH, agi-tation rate, dissolved oxygen and gas (i.e., air, oxygen, nitrogen, carbon dioxide) flow rates.
1.1.2. Bioreactor
The interest is here focused only on classical stirred tank reactors such as the one described in Figure 1.1, which is by far the most common design used in bio-technology industry.
A typical bioreactor involves the following elements:
A reactor tank (central part of bioreactor) in which cells are growing in their culture medium.
Agitation system, allowing homogeneity of the culture medium.
Feeding pumps and connectors associated with the various sensors per-forming the online measurements of pH, temperature, gas (oxygen, nitro-gen, carbon dioxide) and/or concentrations of the various components. Note that in the bioreactor the pH is monitored with a pH probe and controlled by addition of acid or base into the reactor. Temperature is monitored by a thermo-couple and controlled with a heat exchanger. Dissolved oxygen is monitored through an in situ probe and may be controlled by agitation rate and/or air flow and/or gas composition.
Figure 1.1. Schematic representation of a bioreactor (Hulhoven, 2006).
In batch mode cells grow in a finite volume (approximately constant) of the cul-ture medium. Indeed, in practice, a slight variation of the volume is due to the sampling, regulation of the pH (addition of acid or base) and evaporation
(ex-change between the liquid phase and the gas phase). The inlet feed rate (Fin) and
the outlet flow rate (Fout) are equal to zero (∀t t0, Fin(t) = Fout(t) = 0 where t
represents the culture time). But the drawback of batch cultures is the accumula-tion of inhibitory metabolites, which leads to low cell densities, low productivity and product degradation.
In continuous (or chemostat) mode, fresh culture medium is continuously add-ed to the culture, accompaniadd-ed by a corresponding continuous subtraction of the medium for recovery of cells and products. The inlet and outlet flow rates are
con-stant). With continuous cultures it is possible to achieve high cell density and high productivity without growth inhibition due to accumulation of metabolites (note that genetic changes of the cell can also occur during continuous cultures leading to a loss of productivity). However, in order to be successful, the contin-uous mode must be manipulated with a higher level of technical skills.
A culture mode different from the continuous one is the perfusion mode, in this particular case the cells are maintained within the reactor (i.e. by a filtration de-vice) and only the medium is renewed.
The fed-batch mode is operated when the medium is added during the culture
without outlet flow rate (∀t t0, Fin 0, Fout(t) = 0 and V(t) V(t0)). For better
use of fed-batch mode, the inlet feed rate must be regulated using a controller which can control concentration of substrates in the culture at desired levels. The case study presented in this work will consider fed-batch cultures aiming at max-imizing the cell productivity.
Figure 1.2. Various operating modes of biological processes - modified from (Dochain,
2008)
1.2. Characteristics of bioprocess models
or unsegregated depending on the heterogeneity of the composition of the cell culture.
1.2.1. Structured vs. unstructured
Classifying a model as structured or unstructured (Figure 1.3) depends on the ob-jective of the model. Hence, structured models constitute a more detailed and biologically consistent representation of cellular activities. As a consequence, they usually present a high number of equations which are difficult to handle. For this reason, they are particularly difficult to use for the design of on-line algo-rithms for process monitoring, control and optimization.
Figure 1.3. Structured vs.unstructured models (S: substrates, C: intracellular
metabo-lites, P: products) - modified from (Gelbgras, 2008)
On the other hand, in contrast to the structured models, unstructured models de-scribe the evolution of some extracellular culture variables (e.g. substrates, bio-mass, and metabolites) by a small set of macroscopic reactions, without paying much attention to the intracellular behavior. Hence, only a few states are consid-ered which makes the model easier to identify and to use in process monitoring and control.
1.2.2. Segregated vs. unsegregated
Classifying a model as being segregated or unsegregated pertains to the hetero-geneity of the composition of the cell culture. The segregated models divide the cell population into segments characterized by different metabolic activities (het-erogeneous).
Conversely, unsegregated models consider the entire cell population as homoge-neous and use a lumped variable such as total biomass per unit of volume to de-scribe the entire population; this is a very simplified representation of reality, which can be adopted if the majority of cells have the same biological activity. Furthermore, segregated models are more computationally difficult and require a higher level of complexity than unsegregated ones which are simplest to handle.
1.2.3. Black box vs. white box
There exist different ways of modelling. The consideration of physical interpreta-tion of the system leads to so-called white-box models in contrast to black-box models where a system is summarized based on the mathematical map between the different variables without real physical interpretation. Between these two extremities, it is possible to distinguish grey box models (Dewasme et al, 2009) corresponding to a combination of black and white box models and generally de-scribing macroscopic aspects of the system on the basis of components mass-balances equations. The model developed in this thesis can be classified in the same category as grey box models.
1.3. General dynamic model of bioprocesses
Bastin and Dochain (1990) have proposed a general approach to describe the dy-namics of a bioprocess. The authors introduce the concept of macroscopic reac-tion scheme and propose a generic mathematical model of bioprocesses namely “general dynamic model”.
de-scribed. A general dynamic model of a bioprocess is then derived from this reac-tion scheme and its elements are described in more details.
1.3.1. Macroscopic reaction scheme
From a macroscopic point of view, cells are just viewed as a catalyst for the con-version of substrates into products which is represented by a set of irreversible “macro-reactions” that describe the main phenomena occurring in the culture. These reactions are represented as follows (Bastin and Dochain, 1990):
] [1 k r ,...,n k j P j r , j R i i r , i r r r
(1.1) where ki,r and kj,r are the pseudo-stoichiometric coefficients or yield coefficients.
They are negative when they concern a reactant and positive when they concern a product.
φr is the reaction rate of the reaction r.
Rr (Pr) is the set of reactants (products) in reaction r.
i is the ith macroscopic component (n macroscopic components are
con-sidered).
nφ is the number of reactions.
According to (1.1) a bioprocess can be defined by a set of nφ reactions involving
n macroscopic components (with the hypothesis n ≥ nφ). However, to avoid
confusion and for a correct understanding of the concept of macroscopic reaction scheme, two main comments have been specified by Bastin and Dochain (1990) as follows:
In contrast to the chemical reactions, the macroscopic reaction scheme
(1.1) does not represent a stoichiometric relation between the components
and does not satisfy elementary mass balances, hence ki,r and kj,r are said
The reaction scheme (1.1) does not represent an exhaustive description of phenomena occurring in the culture. For instance, non-limiting substrates as well as certain byproducts are often omitted if they are not involved in any other reactions or do not present any interest at this macroscopic lev-el.
1.3.2. General dynamic model
By considering the system of mass balances for the different state variables, the dynamic model of each component involved in the macroscopic reaction scheme can be obtained as follows:
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ( ) ( ( t Q t Q t t F t t F t V t , K dt t V t d out in out in in (1.2) where T ] [
1, 2,..., n is the vector of concentrations of the macroscopic
com-ponents.
K is the pseudo-stoichiometric matrix.
T
] [
1, 2,..., n is the vector of reaction rates.
Fin is the inlet flow rate.
Fout is the outlet flow rate.
T
] [ in1 in2 inn
in Q ,Q ,...,Q
Q and Qout [Qout1,Qout2,...,Qoutn]Tare vectors of
input and output gaseus outflow rates.
[ ]T
in in1, in2,..., inn is the vector of concentrations of components in the feeding medium.
V is the culture volume.
) ( ) ( )) ( ( t F t F dt t V d out in (1.3)
the system of mass balances becomes:
) ( ) ( ) ( ) ( ) ( ) ( t Q t F t t D t , K dt t d (1.4) where ) ( ) ( ) ( t V t F t
D in is the dilution rate. ) ( ) ( ) ( ) ( ) ( t V t Q t t D t
F in in is the vector of external (liquid and gaseous) feed
rates. ) ( ) ( ) ( t V t Q t
Q out is the vector of outflow rates of gaseous components.
The main challenge in macroscopic modelling lies in the reaction term Kφ(t,ξ), in the selection of an appropriate pseudo-stoichiometry and good kinetic model structure selection and parameter estimation.
1.3.3. Yield coefficient matrix
The selection of an appropriate pseudo-stoichiometry related to the reaction scheme (1.1) is an important prerequisite, which usually relies on physical knowledge of the bioprocess. In contrast to stoichiometric coefficients commonly used in chemistry, yield coefficients do not necessarily depend on the elemental chemical composition of the components considered in the reaction scheme. They can be identified based on databases containing measurements of macro-scopic entities considered in the reaction scheme.
(1996), which implements a two-step procedure for the identification of the pseudo-stoichiometric coefficients independently of the kinetics under the as-sumption that all states of the model are measured and a condition commonly called C-identifiability is satisfied.
However, Bernard and Bastin (2005) have also proposed a methodology to de-termine the structure of the pseudo-stoichiometric in two steps. The first step consists in estimating the minimal number of reactions that must be taken into account to represent the main mass transfer within the bioreactor, while the sec-ond step is to select among a set of possible macroscopic reaction networks those which are in agreement with the available measurements.
1.3.4. Modelling the kinetics
Modelling of reaction kinetics is usually a very difficult task. On the one hand, it is not easy to define the main biological and physicochemical factors (tempera-ture, pH, biomass, substrates, metabolites, dissolved oxygen) that influence the kinetics and, on the other hand, one has to choose the analytical kinetic structures which are able to describe the different behaviors (like limitations, saturations, inhibitions) of the various components as well as the influences of physicochemi-cal factors in the reaction rates.
The most common kinetic models used in bioengineering show the following structure ) ( ) ( ) (,t ,t X t (1.5)
where µ is the specific growth rate.
Different laws describing µ have been proposed by biologists. We quote here the most classic Monod, Ming, Haldane, Contois and Herbet laws.
2 s 2 S K S max (1.7) where
µmax is the maximum specific growth rate.
Ks is the saturation constant.
To describe a potential growth inhibitory effect for large substrate concentrations the Haldane law is frequently used:
i 2 s hal K S S K S (1.8) ) K K 2 1 ( s i max hal
where Ki is the inhibition constant.
Note that, in the Haldane law, the maximum growth rate μmax is achieved for
√ and if the inhibitory effect of S is negligible (Ki>>Ks) (1.8)
be-comes (1.6).
To reflect the slower growth generally observed for large biomass concentrations, the Contois law (1.9) can also be used:
S X K S x max (1.9)
where Kx is the kinetic constant.
Finally, the Herbert law (1.10) can be used to model maintenance of cells, for example in the case of animal cell cultures.
m S X K S x max (1.10)
where m is the maintenance coefficient.
specific properties in order to facilitate the identification of parameters that char-acterize it (see, Bogaerts (1999)).
As shown in Bogaerts (1999) the reaction rate (1.5) may be described by:
j l . lj j hj P l t R h h j ,t t t e ) ( γ j( ) ( ) ) ( (1.11) where αj(t) > 0 is a kinetic constant describing any reaction rate dependence,
ex-cept the one corresponding to the concentrations of the components.
hj > 0 the activation coefficient of component h in reaction j.
lj ≥ 0 the inhibition coefficient of component l in reaction j.
Pj is the set of indices of all the components which are inhibitors in
reac-tion j.
Rj is the set of indices of the components which activate the reaction j
(reactants, catalysts and auto-catalysts).
This structure has the advantage to be very general in the sense that the activation and/or the inhibition of the reaction by any component can be taken into account. Note that Grosfils et al. (2007) generalize this structure in order to improve the way saturation effects are taken into account, and in turn, improve the biological interpretation of the model parameters. Moreover, this structure is linearizable with respect to its parameters thanks to a logarithmic transformation.
Now that the general dynamic model of bioprocesses is described with its pseu-do-stoichiometric and kinetic parts, we consider the parameter estimation in the following subsection.
1.4. Parameter estimation
There-fore, it will be necessary to attribute a numerical value for each parameter, based on a prior knowledge of the system and on the basis of experimental data.
In addition, the estimation of the parameters will depend on difficulties which arise mainly from the lack of reproducibility of experiments with biological pro-cesses, the lack of quality and information content in the experimental data, the nonlinearity of dynamic models of the bioprocesses and the correlation between parameters. Hence, the identifiability analysis of model parameters before their estimation turns out to be an important task.
According to Dochain (2008), the central question for analyzing the identifiabil-ity can be formulated as follows “let us assume that a certain number of varia-bles of the state of the model are available to the required extent; on the basis of the structure of the model (structural identifiability) or on the basis of the type and quality of the data available (practical identifiability), can we expect to be in a position to attribute a unique value to each parameter of the model by means of parameter estimation?”
In this section we will deal with the questions of structural identifiability and practical identifiability. Afterwards, we propose a short overview of sensitivity functions, before presenting the final modeling stage which consists in model
validation.
1.4.1. Structural identifiability
The structural identifiability is a necessary condition for parameter estimation. It regards the possibility of identifying the parameters (uniquely) from the available measurements (Walter and Pronzato, 1997), assuming that:
model structure is error-free;
experimental data are noise-free;
there are no constraints on system excitation and on the measurement
times.
a parameter is structurally globally identifiable if for al-most any
̂ ̂ (1.12)
a parameter is structurally locally identifiable if for almost any ,
there exists a neighborhood V( ) such that
̂ and ̂ ̂ (1.13)
a parameter is structurally non-identifiable if for almost any , there
exists no neighborhood V( ) such that
̂ and ̂ ̂ (1.14)
where
θ = (θ1,…,θnθ) is the vector of model parameters.
̂ is the estimate value of θ.
is the unknown true value of the parameter vector.
There are several different methods to test the structural identifiability of nonlin-ear models: the transformation of the nonlinnonlin-ear model into a linnonlin-ear one (Dochain et al., 1995), the development in series approach (Taylor series (Walter and Pronzato, 1997), generating series (Walter and Lecourtier, 1982)), the similarity transformation approach (Vajda et al., 1989) and differential algebra methods (Ljung and Glad, 1994; Bellu et al., 2007). The goal in this work is not to review all methods used to test structural identifiability, for more details we refer the reader to the book of Walter and Pronzato (1997).
