Model-Based Dynamic Optimization of Monoclonal ... - ACS Publications

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Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Model-Based Dynamic Optimization of Monoclonal Antibodies Production in Semibatch OperationUse of Reformulation Techniques Chrysoula D. Kappatou,† Adel Mhamdi,† Ana Quiroga Campano,‡ Athanasios Mantalaris,‡ and Alexander Mitsos*,† †

RWTH Aachen University, Aachener Verfahrenstechnik-Process Systems Engineering, Forckenbeckstraße 51, 52074 Aachen, Germany ‡ Department of Chemical Engineering, Centre for Process Systems Engineering (CPSE), Imperial College London SW7 2AZ, London, U.K. ABSTRACT: Monoclonal antibodies (mAbs) constitute one of the leading products of the biopharmaceutical market with significant therapeutic and diagnostic applications. This has drawn increased attention to the intensification of their production processes, where model-based approaches can be utilized for successful optimization and control purposes. In this manuscript, dynamic optimization of mAb production in mammalian cell cultures in semibatch operation is performed. To develop a model suitable for optimization, reformulation steps consisting of function smoothening, reducing model size, and scaling are applied to a predictive energy-based model for mAb production presented in Quiroga et al. (2016). Optimization of the reformulated model leads to the derivation of an optimal feeding strategy, accounting for indirect quality measures, dilution effects, and the energy requirements of the cells. The results highlight the increased production outcome by using the reformulated model, and thus indicate the strong dependence of model-based optimization results on model structure.

1. INTRODUCTION

The modeling of mammalian cell cultures is a quite challenging task, due to the high complexity of these cells.7 In principle, cell culture modeling can be classified in two categories, so-called structured and unstructured, based on the incorporation or not of intracellular compartmentalization.8 However, for computationally intense optimization studies, detailed structured models turn out to be less appropriate.9 Nowadays, the main challenges in modeling of cell cultures arise from the lack of data and mechanistic understanding. Despite the challenges yet to be confronted, noticeable progress has been accomplished over the last decades in modeling of those organisms, which allows for attempting model-based optimization in cell culture processes in different modes of operation. Currently, fed-batch operation is the usual choice for largescale cell culture systems; it can extend culture longevity compared to the batch case and increase the final mAb

Monoclonal antibodies (mAbs) represent one of the dominant products of the biopharmaceutical industry; it is their high specificity, their general tolerability, and the amenability of their production to platform-based approaches that renders them as first candidates for many novel targets, thus often leading to a “first-to-market” advantage.1 Monoclonal antibodies are mainly produced in mammalian host cell lines, due to the ability of mammalian cells to carry out complex post-translational modifications of high fidelity.2,3 The strong demand for mAb products, for both therapeutic and diagnostic purposes, has led to high research activity toward production improvements. In this direction, process optimization can increase productivity, reduce operational costs of the bioprocess, and thus lead to a more cost-effective production of mAbs. Conventional process intensification approaches are mainly experimental. As an alternative to expensive and time-consuming experimentation, the development of mathematical modeling in biochemical engineering has been gaining increasing attention.4 More precisely, mathematical modeling can advance process understanding, facilitate process improvements and pave the way toward robust optimization and control of the bioprocess.5,6 © XXXX American Chemical Society

Special Issue: 2017 European Symposium on Computer-Aided Process Engineering Received: December 27, 2017 Revised: May 2, 2018 Accepted: May 2, 2018

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DOI: 10.1021/acs.iecr.7b05357 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research concentration.9,10 However, continuous manufacturing has great potential, allowing for improved sustainability and costeffectiveness.11 This has initiated various research activities on the transition from batch to continuous manufacturing, such as the efforts of the Novartis-MIT Center12 (e.g., Sahlodin and Barton,13 Benyahia et al.,14 Wong et al.15) and the Research Center for Structured Organic Particulate Systems (C-SOPS) headquartered at Rutgers University16 (e.g., Singh et al.17). Although feasibility of end-to-end continuous mAb production has been recently demonstrated,18,19 considerable effort is still required before establishment of a successful commercial fully integrated and continuous process.20 In this direction, the development of advanced computational tools together with advanced optimization and control strategies in the traditional fed-batch mode are powerful means for progressing the current state of the art and bringing the passage to continuous operation one step closer.21 There is only a limited number of publications reporting on model-based optimization for mammalian cell cultures in fedbatch mode of operation.22 As glucose (GLC) and glutamine (GLN) are the main substrates, optimal feeding strategies for increased mAb production rely principally only on glucose and/ or glutamine requirements (e.g., De Tremblay et al.,22 Koumpouras and Kontoravdi23). However, they might not to be growth limiting,24,25 and the availability of energy sources might be affected also by depletion of other nutrients. Therefore, incorporation of the energy balance is significant when attempting to optimize the bioprocess. Optimization (mathematical programming) requires repeated calculation of the solution to the model equations, as well as evaluation of their derivatives. Thus, optimization performance can be significantly affected by both system complexity and model structure. Generally, in order to enhance efficiency and robustness of an optimization method, model reformulation techniques can be applied.26 Especially in modeling of cell cultures, where often discrete phenomena and different scales are necessitated to describe a system’s behavior, limitations and deficiencies of existing solvers in handling these features can lead to inefficient results or even failures of the optimization studies. Hence, the importance of performing model reformulation steps aiming at high-fidelity yet computationally efficient models that intend to be used for optimization and control purposes is highlighted. Herein, we apply model reformulation techniques to a predictive unstructured and nonsegregated energy-based model for mAb production in mammalian cell cultures, presented by Quiroga et al.27 As highlighted in our preliminary work,28 optimization of the feeding profile has increased potential for process intensification; nevertheless as model complexity is high, the optimization attempts can lead to suboptimal solutions, and thus an efficient and systematic way for improving optimization performance is required. Hence, in this study we principally aim at developing an appropriate model for optimization studies that can improve numerical performance, and thus yield reduced computational efforts and increased production outcomes. The reformulation techniques consist of approximations for nonsmooth continuous functions, model scaling, and size reduction essentially targeted in maintaining only the model compounds necessary for the optimization. The results of the present study confirm a strong influence of the model formulation on the solution of the optimization procedure. Particularly in bioprocess optimization, where frequently the product is a high value pharmaceutical,

this behavior further highlights the significance of using a numerically proper model for optimization studies. The rest of the paper is organized as follows. Section 2 introduces the main characteristics of the model used in the present work. Section 3 describes the steps undertaken in order to create a proper model for optimization. Section 4 is dedicated to the problem formulation for the optimization, and section 5 presents the obtained results from the optimizations. Section 6 concludes this work.

2. MODELING Utilizing the in vitro/in silico approach, Quiroga et al.27 developed a model that couples growth kinetics with stoichiometric balances to predict ATP content in mammalian cell cultures, based on the metabolism of glucose and amino acids. It is an unstructured and nonsegregated fed-batch model for mAb production in GS-NS0 cell lines. The model consists of 31 differential and 81 algebraic equations, 3 inputs (decision variables), and 41 parameters (subjected to global sensitivity analysis and re-estimated, when necessary). A short description of the model equations can be found in Appendix A.1. This experimentally validated model, based on the energy metabolism of GS-NS0 cells producing the antibody cB72.3 in serum-free cultures, allows the design of an optimized supplemental medium to ensure nutrients availability for cell proliferation and growth, product assembly, and energy production. This model serves as a basis for this work, in which the reformulation techniques are applied. Both the original and the reformulated model are further used for optimization studies with derivation of optimal feeding profiles of the optimized supplemental medium for increased mAb production. The modeling and optimization procedure is performed by using gPROMS ModelBuilder v.4.1.2.29

3. MODEL REFORMULATIONS 3.1. Smoothening of Discontinuous Functions. Biological systems often encounter intrinsic hybrid (discrete/ continuous) phenomena. These can be captured in models using hybrid systems, which are challenging computationally.30 Metabolic shifts, as self-regulatory attempts of the metabolism to adjust to new environmental conditions, can be interpreted as switches in cell culture models. A switch is a hybrid phenomenon that indicates discrete changes in the form of some model equations, something that is more commonly referred as an alteration in the mode of the hybrid system.31 More specifically, when the concentration of an amino acid (AA) is below a threshold, cells might start consuming an alternative one to cover their growth and energy requirements. Assuming for example that AA 1 is below a critical concentration, [AA1,cr], the cells can search for an alternative source AA2, the consumption rate QAA2 of which, will at that point increase by the value of the consumption rate of AA1, QAA1. This behavior can be modeled in a discrete way, using for example an IF−ELSE condition, as shown in eq 1. The jump in QAA2 describes a discontinuous behavior and poses difficulties for the optimization. Such a discontinuous formulation is implemented in the model presented by Quiroga et al.27 B


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Industrial & Engineering Chemistry Research

Figure 1. Optimization procedures using (a) full-size model, (b) reduced-size model.

thorough selection of the equations constituting the model is required.33 The model presented in Quiroga et al.27 is able to predict ATP content in mammalian cell cultures. This feature enables the development of an optimized supplemental medium for fed-batch cultures. This medium accounts for ATP availability for cell proliferation, maintenance, and mAb assembling. Once the medium is designed, the ATP balance could be eliminated from the model formulation to reduce its complexity, and cell viability can be used as a quality measure of the culture. The use of a reduced-size model to optimize mAb production could contribute significant computational savings. In other words, the equivalence of the two approaches as described in Figure 1, allows us to reduce the complexity of the model used for the optimization and check a posteriori for sufficient energy requirements using the full-length model. By following this procedure, the size of the model used for the optimization is decreased to 16 differential equations, 29 algebraic equations, 3 inputs, and 13 parameters, a reduction to more than half the size. The eliminated equations and parameters are relevant for supplemental medium design, for which the formulation has been optimized and fixed, and as no uncertainty information is incorporated, they do not influence the optimization outcomes. The optimization procedure depicted in Figure 1b requires the performance of an additional optimization with the full-size model, in case the energy requirements are not satisfied by the optimal solution derived with optimization with the reducedsize model. In this case, the optimal profile from the first optimization is anticipated to serve as a good starting point for the full-size model optimization, and thus lead to computational advantages. However, this step was not undertaken, since in all cases that we examined the energy requirements of the cells were met at all time points by the optimal feeding profile

IF [AA1] < [AA1,cr] Q AA

2 ,final

= Q AA + Q AA

ELSE Q AA

2,final

= Q AA

2

1

2

END

(1)

Recently, efforts have been devoted to developing continuous nonsmooth formulations of systems that are previously handled as hybrid (discrete/continuous) systems.32 Herein, in order to eliminate this type of discontinuities from the model, and avoid potentially undefined functions, we utilize the approximate step function, which is continuous and smooth to approximate the real behavior of the system (eq 2). Q AA

2,final

= Q AA + Q AA 2

1

tanh(α([AA1,cr] − [AA1])) + 1 2 (2)

In eq 2, the parameter α determines the tightness of the approximation. Herein, the value of this parameter is chosen to ensure that the difference between discrete formulation and its smooth approximation throughout the culture lifetime for the base-case simulation with a reporting interval of 1 h less equal to 5%. This deviation is deemed smaller than the model accuracy, and thus the reformulation is acceptable. As shown in the results, the reformulation simultaneously gives the advantages of smooth formulations for optimization. 3.2. Reducing Number of Model Equations. System complexity is closely related to the degree of incorporating detailed knowledge in describing a phenomenon through a mathematical model, and thus taking into account the high complexity of biological phenomena, it can be a limiting factor in model development of bioprocesses. For this reason, a C


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decreased ammonia accumulation,10 and therefore ammonia production is not incorporated into the model. Lactate production on the other hand is considered in the model, and can be growth inhibiting. As a result, low concentration of lactate throughout the culture lifetime can serve as an additional indication of process efficiency. Before attempting optimization of mAb production based on a feeding strategy, it is important to identify the growth limiting nutrients and incorporate them into the feeding. Xie and Wang39,40 showed that by feeding all amino acids (rather than only the essential ones), using stoichiometric feeding to maintain the nutrient environment relative low and constant, led to higher cell densities and therefore higher production to their fed-batch experiments. Similar behavior was also experimentally validated by the work of Spens and Häggström.41 Thus, an optimal feeding strategy incorporating feeding of 17 amino acids plus glucose is pursued in the present study. Glycine, alanine, and glutamine are not included in the optimized medium, because the cells produce them. Including these amino acids would cause amino acids accumulation, which could increase the osmolality levels of the culture, and could eventually inhibit growth. The concentration of the amino acids in the feeding is decided by optimization of the feeding composition on a previous study34 and is not presented here. The feeding flow rate with the optimized composition of amino acids and the feeding rate of glucose together with glucose concentration in the feeding constitute the three degrees of freedom (controls) for our optimization problem. The final time of the culture is also left as a degree of freedom for the optimization, and is allowed to vary between 140 h, which is around the batch time presented in Quiroga et al.27 and 288 h, which is double the batch time. From our current experience the upper bound in the culture time is sufficiently large and we do not reach it. Nevertheless, in the hypothetical case that the optimization would result to culture times close to that bound, a larger value for it should be considered. A piecewise constant discretization of the controls is employed, where the time horizon is discretized to 6 h intervals. The duration of each interval is not fixed, but left as optimization variable that can range from 1 to 6 h (adaptive grid). An additional constraint on the total volume of the culture is imposed, accounting for dilution effects. Maintaining the total culture volume relatively close to the volume of the batch case helps with reassuring the predictive ability of the model, which decreases inversely proportional with volume variation.23 Moreover, it allows for a more direct comparison between unoptimized batch and optimized fed-batch operation.42 Finally, a minimum ATP requirement per viable cell (ATPperXv) is imposed. As aforementioned, this is either placed as constraint for the optimization with the original model or checked a posteriori for the optimization with the reformulated and reduced one (see eq 3). Taking into account all the above-mentioned factors, the dynamic optimization problem for the optimization with the full- and reduced-size model can be formulated as follows:

derived from the optimization using the reduced-size model. This indicates that the optimized supplemental medium for fedbatch cultures that was designed using the original model ensures ATP availability over time,34 and thus confirms our assumption that the reduced-size model can be used without any risk of compromising ATP availability. 3.3. Scaling. Biological systems operate in multiple scales, and therefore the modeling of those systems must frequently confront scaling issues. A well-scaled model is constituted by variables in the same order of magnitude, and as a result its use can outweigh considerable advantages. Therefore, model rescaling is generally advisable, as it can substantially contribute toward more efficient and robust optimization methods.26 At a first step, rescaling can be a useful method for checking the consistency of the model equations, ensuring namely same set of units among the terms of an equation.35 Most importantly, proper scaling makes it easier to control roundoff errors by avoiding big differences in the magnitudes of terms and can in general improve the condition number and the natural convergence rate of the problem,36 thus often leading to a better optimization performance. For these reasons, a simple scaling methodology for both variables and equations is additionally implemented to our examined model. The scaling is performed so as to keep the values of the variableswhen possible throughout the culture timebetween 0.001 and 100. 3.4. Summing Up. By applying the above-mentioned steps to the original model, we end up with a reformulated, smaller, and as indicated in the results section, numerically better behaved one. Additional information about both the original and reformulated models can be found in Appendix A.

4. FORMULATION OF OPTIMIZATION PROBLEM Given a model for production of mAbs in mammalian cell cultures, optimization studies can lead to process intensification. Different objectives or coupling between objectives can be used for this purpose, resulting in different optimization outcomes.23,28 For the scope of this study, where we mainly focus on investigating the computational performance and the optimization robustness using a numerically well-behaved model for optimization, we consider the typical objective of maximizing the mAb final concentration (mg/L) as our objective function. When dealing with biopharmaceutical products, quality is an important issue. Since in practice it is extremely difficult for a model to account for direct quality measures, indirect measures of product quality should be incorporated. As an indirect measure for product quality we consider cell viability. Maintaining viability in high levels is significant, since unprogrammed cellular death (necrosis) leads to release of undesired proteins and also release of proteases that can degrade the product. Furthermore, mAb produced by lysed cells is not usually a well-glycosylated product, and therefore lacks high quality.37 Consequently, a lower bound of 60% viability is imposed as path constraint to our optimization problem. This value corresponds to the viability level, at which the batch culture that serves as our base case scenario was terminated. Another indirect measure for product quality, is the concentration of harmful byproducts in the culture. It has been shown that the accumulation of byproducts, such as lactate (LAC) and ammonia (AMM), is toxic for cell growth.38 One of the main advantages of the GS expression system is its D


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ment of final mAb concentration was approximately 82% compared again to the batch case, and that with less than onethird of the computational time needed for the optimization with the original model. Since culture duration is a degree of freedom for our optimization problem, the optimization studies terminate at different final culture times. In general, we anticipate that an extended culture longevity yields to an increased product concentration. However, the upper value of the allowed time horizon is never reached, due to the imposed lower bound on viability, which serves as termination criterion for our studies. This actually indicates that at a certain point no further extension of culture lifetime can occur, without decreasing viability below its lower bound. This would violate the system’s constraint and lead to an infeasible solution. In addition, since in our case viability represents the main quality attribute, a constraint violation would have a biological impact in terms of compromising our product quality. The optimized culture durations for the different optimization studies are presented in Table 1, and they all refer to the same end point condition, namely a viability level of 60%. Additional information about the optimizations with the two different models can be obtained from the execution output files of gProms. More precisely, for the case of optimization with the full-size model gProms reports termination following lack of improvement in optimization variables and objective function. It describes that the final point is feasible and could correspond to an optimal solution if the optimality tolerances are too tight. On the contrary, the optimization using the reformulated model terminates with all the optimality tolerances met. Despite the theoretical equivalence of the optimization problem solved with the two models, it can been seen in Figure 2c that we end up with quite different feedings of amino acids for the two cases. In the optimization with the original model the optimized result implies feeding at an early stage of the culture, which leads to sooner depletion of amino acids, and consequently shorter culture duration. On the contrary, the optimized results with the reformulated model suggest feeding at a later culture time, shortly before glutamate exhaustion. This strategy besides increased mAb titer, leads also to extended culture lifetime and decreased lactate production, Figure 2a. This confirms the assumption that an improved model structure can lead the optimization to more efficient pathways. No glucose feeding profiles are shown in Figure 2, as both optimizations resulted in no additional glucose supplementation. Since in both cases the initial glucose concentration in the culture was not depleted throughout the culture time, this can justify the absence of glucose supplementation in the optimized case studies. The fact that no additional glucose feed was introduced is also in accordance with the literature,43 where excessive glucose feeding has been characterized as energy inefficient. It is quite important for validation purposes to test the results derived from the reformulated (reduced-size) model as inputs to the original model and solve again with the last one. By following this procedure, the simulation calculates a final mAb concentration of 336.26 mg/L (instead of 336.28 mg/L predicted by optimization using the reformulated model, see Table 1). Figure 3 illustrates the absolute percentage error between the prediction of the original and the reformulated model, for mAb and cell concentration, throughout the culture lifetime. Note that after the halftime of the culture, we observe

where u(t) (controls) are the feeding rates of AA, GLC [0−0.02] (L/h), and the feeding concentration of GLC [0−1000] (mm), x,y (states) consist of x(t), the state variables related to mAb production and y(t), the states related to the energy balance, f (objective) is the final mAb concentration (mg/L), h(t) (model equations) comprise h1(t), the model equations related to mAb production and h2(t), the model equations related to the energy balance, g(t) (constraints) include g1(t), the constraints on viability ≥ 60 (%) and volume ≤ 1.1·V0 (L) and g2(t), the constraint ATPperXv ≥ 4 × 10−12 (mM/cell). These values derive from our current experience and from what is typically reported in the literature. Nevertheless, the exact consideration of those values remains ad hoc. The effect of some of these decisions on the optimization outcomes is examined in the results section.

5. RESULTS After performing the above-mentioned optimization study, with the original (full-size) and the reformulated (reduced-size) model, we obtain the results shown in Table 1. The final mAb Table 1. Optimization Results model

mAb (mg/L)

base case27 original reformulated not scaled reformulated reformulated reformulated reformulated

185.04 275.96

culture time (h) 144 140

CPU (s) N/A 9943

N/A adaptive

grid

322.83 336.28 328.03 328.02 330.56

179.32 158.31 158.31 158.31 158.31

3762 2575 166 1197 102554

adaptive adaptive fixed (6 h) fixed (3 h) fixed (1 h)

concentration for the batch culture (V = 0.2 L) presented in Quiroga et al.27 was 185.04 mg/L, and the culture time 144 h (simulation results), serves as our base-case scenario. Note that the difference in the optimization results presented by Quiroga et al.34 is due to the different formulation of the optimization problem (i.e., different initial conditions, stricter bound on viability). The results indicate the strong influence of model structure on the solution of the optimization problem. More precisely, for the 10% volume variation the optimization by using the original model led to about 49% improvement in the final mAb concentration compared to the batch case. With the model that includes reduction, approximations, and scaling, the improveE


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Figure 3. Absolute percentage error from simulations with the original and reformulated model using the optimal feeding from the reformulated model as fixed input for both simulations.

model shown in Figure 2d. Indeed, the results indicate satisfaction of the minimum energy requirements throughout the culture time. This further establishes the use of the reformulated (reduced-size) model for optimization studies without significant risk of compromising ATP availability. Finally, for identifying the effect of scaling on the optimization performance, we also present in the results (Table 1) the optimization outcomes by using the reformulated (reduced and approximated), yet not scaled model. As indicated in Table 1, the use of a well-scaled model improves our objective function and also decreases the computational time needed for the optimization. 5.1. Effect of Discretization. On section 4 we reported the use of an adaptive grid for the solution of the optimization problem. The optimal feeding profile of the reformulated model in Figure 2c shows a fluctuating behavior. Such a behavior can be attributed to discretization effects. To examine the effect of the control vector parametrization on the optimal feeding profile, different grids were tested for the optimization with the reformulated model. In order to be able to compare the results, we fixed the time horizon to the end time of the optimized grid, namely 158.31 h. Then we performed the optimizations with uniform grids of one-, three-, and six-hour intervals. The results of this comparison are presented in Figure 4. This figure illustrates the presence of discretization effects. Nevertheless, as indicated in Table 1 the computational times escalate with increasing grid density. Noticeably, the adaptive grid, with time intervals varying from one to six hours performs better in terms of optimized mAb concentration even from a

Figure 2. Comparison of optimization results for mAb production of secreting mammalian cell (GS-NS0) cultures in semibatch operation with original (black) and reformulated (gray) model. (a) Concentration of () mAb (mg/L) and (-·-·) LAC (mM). (b) Concentration of (- - -) viable, (---) dead, and (− −) total cells (cells/L). (c) Optimal feeding profiles of amino acids [mL/h]. (d) ATP content per viable cell (mM/cell)

an increase in the error, which is most probably related to the activation of metabolic shifts at this time period. Nevertheless, the error throughout the culture time is far below the predicting ability of the model. This verifies the use of the reformulated model without endangering the model’s accuracy. After simulating the optimized result of the reformulated model with the original one, we are able to create the plot for ATP content for the optimized results using the reformulated

Figure 4. Effect of discretization refinement on optimization solution. F


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Industrial & Engineering Chemistry Research fixed grid with one-hour time intervals, and thus with significantly less computational effort (see Table 1). 5.2. Effect of Viability Lower Bound. As aforementioned high viability index accounts for product quality in this work, where a lower bound on its value throughout the culture time is enforced as path constraint to avoid the release of potentially not well-glycosylated product. Cell viability is determinant for our optimization problem, since once its value cannot be maintained above the imposed threshold, it signals the end of the culture lifetime. When viability drops, the percentage of dead cells increases in the culture, and so does the mAb concentration due to mAb released by the dead cells. Evidently, as we impose a looser bound on viability we anticipate to achieve higher product concentrations and extension of the culture longevity. To quantitatively study the effect of the permitted lower bound of viability we additionally perform optimizations with the reformulated model using different lower bounds for viability, namely 50, 55, 65, and 70%. The optimizations outcomes are shown in Table 2. The results indicate a linear correlation between viability bound and both final product concentration and culture longevity.

Table 3. Effect of Final Volume on Optimization Solution

viability (%)

mAb (mg/L)

culture time (h)

CPU (s)

reformulated reformulated reformulated reformulated reformulated

50 55 60 65 70

402.87 369.02 336.28 303.90 271.95

177.73 167.34 158.31 150.47 142.58

11424 9868 2575 8247 11495

final volume (L)

mAb (mg/L)

culture time (h)

CPU (s)

original original original original reformulated reformulated reformulated reformulated

0.22 0.23 0.24 0.25 0.22 0.23 0.24 0.25

275.96 337.97 311.69 304.77 336.28 395.32 445.40 508.22

140 174.42 149.33 145.21 158.31 165.77 206.12 189.85

9943 22163 31232 10367 2575 7802 4945 13008

6. CONCLUSIONS In biological systems, discrete phenomena such as metabolic shifts often occur and introduce discontinuities. The way of handling these discontinuities is still an open issue for biological modeling. The advantage of utilizing approximations for describing discrete phenomena in a continuous and smooth way has to do with the better performance of existing algorithms in solving smooth problems. Additionally, biological systems might frequently confront scaling issues, which raises the issue of uncertainty. In such cases, a proper scaling or nondimentionalization of the model can be a useful approach. Furthermore, applying model reduction techniques to a model can save a great deal of computational time and effort, although attention should be payed to not subtract any valuable information from the system. In this work, reformulation steps consisting of functions smoothening, scaling, and reducing model-size were performed using a biological model for mAb production. Optimization problems were formulated and investigated for both the original and the reformulated model. The optimization with the reformulated model followed a two-step procedure, exploiting the presence of inactive constraints in the model. As process intensification is achieved for a high value biopharmaceutical product, where time-to-market and costeffectiveness of the production process are determinant, the utilization of the optimization outcomes can contribute substantial benefits. The optimization results underscore the great effect of model formulation on efficiently exploring the search space, and thus leading to improved results. In this direction, another significant attribute that can have a profound influence on optimization performance, and which was not examined here, is the uncertainty of model parameters. For further model-based optimization studies of biological processes, a careful model formulation with a sharper focus on scaling, discontinuity and complexity issues, together with a shift on robust optimization and feedback control strategies is necessary.

Table 2. Effect of Viability Bound on Optimization Solution model

model

It should be remarked though, that the increase in product concentration and the extension of culture lifetime once the lower bound for viability drops is accompanied by a growing uncertainty regarding the quality of the product. Furthermore, as usually there is a high parametric uncertainty associated with death and apoptosis, the predictive ability of the model at lower viability levels is decreased. Therefore, it is often preferable to make a rather conservative and yet safer choice when selecting the lower bound for viability in similar model-based optimization studies. 5.3. Effect of Maximum Culture Volume. Up to this point, we presented the optimization results that derived with a constraint on maximum volume variation of 10%. Keeping the volume variation in low levels is a common strategy to avoid possible errors of the model in calculating the dilution steps. Nevertheless, since culture volume is an important parameter of the system, it is interesting to examine the behavior of the two models, when allowing for bigger volume variations. For this purpose, the dynamic optimization problem described in eq 3 was solved additionally for volume variations of 15%, 20%, and 25%, using both the original and reformulated model each time. The optimized mAb production of the two models, with respect to the different volume variations, can be seen in Table 3. From the results presented in this table, we can observe that the performance of the reformulated model scales almost linearly with volume, while in the original this does not seem to be the case. The better performance in terms of mAb production of the reformulated model in all different volume variations, further establishes the conclusion of a numerically proper model for optimization.

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A. MODEL EQUATIONS In section A.1 we provide for completeness a short description of the equations of the original model given in Quiroga et al.27 The model is highly nonlinear and has discontinuities. In section A.2 we indicate which parts are not included in the reduced model and which parts are reformulated using smooth approximations. A.1. Original (List of Equations Based on Quiroga et al., 2016)

The total volume of the culture is calculated as follows G


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Industrial & Engineering Chemistry Research d(V ) = Fin,GLC + Fin,AA dt

where the different Qk,j refer to consumption rates of a nutrient j for a specific purpose k (x, cell growth; tca, energy production; mAb, mAb production; glyc, energy through glycolysis; ala, energy through glycolysis and alanine). The monoclonal antibody balance results from the following equation, in which the first term is associated with the expected release of product by viable cells, while the second one accounts for accumulated mAb released from apoptotic or dead cells:

(A.1)

The mass balances on viable (Xv), apoptotic (Xa) and dead (Xd) cells are d(VX v ) = VX v (μ − μa − μd ) dt

(A.2)

d(VXa) = VX v μa − VXaμa,d dt

(A.3)

d(VXd) = VXaμa,d + VX v μd − VXdKlys dt

(A.4)

d(V [mAb]) = VX v mAbrelease + VX v (μa + μd )mAbacc dt (A.13)

d([mAbacc ]) = qmAb,x − mAbrelease dt

The mass balances of cells are based on specific rates: growth rate (μ), early apoptotic rate (μa), transition rate from early to late apoptosis (μa,d), and necrotic rate (μd), where STV is the starvation term.

The ATP balance is calculated by ⎛ n ⎞ ⎞ ⎛ μ d(V [ATP]) = VX v ⎜⎜∑ θiQ tca, i⎟⎟ − VX v ⎜⎜ + m x,ATP⎟⎟ dt ⎠ ⎝ Yx,ATP ⎝ i=1 ⎠

⎛ ⎞⎛ [GLC] [GLU] μ = μmax ⎜ ⎟⎜ ⎝ K GLC + [GLC] ⎠⎝ K GLU + [GLU] ⎛ ⎞⎜ [ARG] 1 + ⎟⎜ KARG + [ARG] ⎠⎜⎜ [LAC] 1+ K ⎝ I,LAC

⎞ ⎟ n⎟ ⎟⎟ ⎠

( )

STV =

Klim,GLC

+

(A.15)

where θi stoichiometrical coefficients indicate the number of ATP molecules produced per molecule of nutrient following the TCA cycle pathway, Yx,ATP accounts for the ATP utilization for proliferation, and mx,ATP accounts for maintenance.

(A.5)

A.2. Reformulated Model

Klim,GLU

Klim,GLC + [GLC] Klim,GLU + [GLU] Klim,ARG Klim,ASP + + Klim,ARG + [ARG] Klim,ASP + [ASP] Klim,ILE Klim,LEU + + Klim,ILE + [ILE] Klim,LEU + [LEU]

⎛ ⎜ 1 μa = μa,max STV + μa,LACmax ⎜ ⎜⎜ 1 + [LAC] K a,LAC ⎝

⎞ ⎟ n⎟ ⎟⎟ ⎠

( )

μa,d = μa ka,d

In the reformulated model the equation related to the energy balance is omitted. Mass balances are considered only for the amino acids affecting the growth and death rates and consequently the mAb production. All remaining parameters and variables are scaled. The specific consumption rates of GLC, Qi,GLC or AA, Qi,AA are described with continuous and smooth equations, whenever the original formulations are discontinuous.

(A.6)

■

⎞ ⎟ n⎟ ⎟⎟ ⎠

( )

*E-mail: [email protected]. (A.7)

ORCID

Alexander Mitsos: 0000-0003-0335-6566 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 675251. The authors also gratefully acknowledge the contribution of Ruth Misener for the development of a scaled model and for providing scientific guidance and helpful discussions throughout the planning and development of this work.

(A.9)

Mass balances for glucose, lactate, and amino acids are given by d(V [GLC]) = [GLCin]Fin,GLC − VX v dt

■

(Q x,GLC + Q mAb,GLC + Q tca,GLC + Q ala,GLC + Q glyc,GLC)

d(V [LAC]) = X v V 2Q glyc,GLC dt

(A.10)

(A.11)

d(V [AA]) = [AA in]Fin,AA − VX v (Q x,AA + Q mAb,AA dt + CAAQ tca,AA )

AUTHOR INFORMATION

Corresponding Author

(A.8)

⎛ ⎜ 1 μd = μd,max STV + μd,LACmax ⎜ ⎜⎜ 1 + [LAC] K d,LAC ⎝

(A.14)

(A.12) H

NOMENCLATURE AA = amino acid AMM = ammonia ATPperXv = ATP requirement per viable cell Fin AA = feeding of amino acids GLC = glucose GLN = glutamine GLU = glutamate GS = glutamine synthetase DOI: 10.1021/acs.iecr.7b05357 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article


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LAC = lactate mAbs = monoclonal antibodies QAA = consumption of an amino acid for cell growth V = volume V0 = initial volume Xv = viable cells

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