FIRST PRINCIPLES, UNSTRUCTURED, DYNAMIC MODEL FOR REGULATING CO2 AND pH IN BICARBONATE BUFFERED PERFUSION BIOREACTOR

Since its inception, the biotechnology industry has faced the problem of pH control and CO2 management. This research paper explores the challenges of developing a first principles, unstructured, dynamic, nonlinear mathematical model to maintain pH and control carbon dioxide levels in an aerobic Chinese hamster ovary (CHO) cell culture in a perfusion bioreactor. Perfusion bioreactor is an extremely complex bioreactor to model because the cells grow in a quasi-steady state system. Cell growth, feed, substrate consumption, by-product formation and product formation are all time dependent, equations for which can only be solved through numerical methods. Added complexity comes from presence of stiffness in solving the non-linear equations due to the different time scale of each set of equations. Equations related to pH that involve acid/base ionization have reaction rates on the order of nanoor pico-seconds. CO2 or O2 hydration and bubble dynamics involve the reaction rates on the order of seconds and cell growth equations are on the order of days. This research paper also presents a framework for software development to solve these equations without conducting experiments, except as a final trial before using the results of the model on the manufacturing floor.

iv companies because besides insulin, the market did not have many biopharmaceutical products with medical promise.
Before the widespread availability of biopharmaceuticals and in the absence of any treatment or cure, patients inflicted with the rare diseases had to simply learn to live with that disease. All that changed when biopharmaceutical companies began seeing some tangible progress and began discovering drugs for diseases that inflicted a small number of people for which no treatment was not available. For example, when Abbey Meyers could not find any drug to treat her son's Tourette's syndrome, she lobbied the United States Congress to change the law to give companies incentives to develop drugs for rare diseases (Anand, 2005). In 1983, her efforts led the United States Congress to enact the Orphan Drug Act, which in addition to tax credits, entitled the orphan drugs to market protection that was in some respects better than a patent. Id. This law provided the necessary fortitude that catapulted the genetic engineers from their "garage laboratories" to executives in increasingly confident biotechnology companies in the marketplace. Thus began a new era of biotechnology companies that produced biopharmaceutical drugs that act similar to naturally occurring proteins in the human bodies. Today, the biotechnology industry is at the forefront of developing treatments for difficult diseases for which the pharmaceutical drugs are not readily available (Hopkins et al., 2007).
v Proteomic scientists discovered that scientific advances enabled geneticists to identify propitious traits in living organisms which could have biopharmaceutical applications. Biotechnologists physically manipulated the genetic structure of living cells to create genetically engineered living cells programmed to manufacture proteins that promise to improve human life. Bioprocess engineers exploited a variety of technologies to manipulate the artificial growth environment of these genetically modified mammalian cells to produce commercial quantities of desirable biopharmaceutical protein (Shuler and Kargi, 1992). However, unlike process engineers in the chemical industry who can reasonably rely on established scientific and mathematical principles to predict a process and its outcome, even small changes in pH, temperature, pressure, ionic strength or even genetic mutations, render a bioprocess unpredictable. The biological environment of the cells determines the rate, extent and type of biological processes that take place in a growth medium. Consequently, bioprocess engineering challenges involve optimizing the growth environment of genetically engineered cells to maximize product formation and eventually harvesting the product by separating protein from cells and cell debris.
This research paper presents a first principles, unstructured, dynamic, nonlinear mathematical model to maintain pH and control carbon dioxide levels in aerobic Chinese hamster ovary (CHO) cell culture in a perfusion bioreactor. CHO cells are preferred hosts for mass production of recombinant proteins because of all vi animal cells, CHO cells grow more rapidly, have higher stability and are efficient in foreign gene expression (Matsunaga, 2009a). CO 2 is preferred for regulating cell culture pH because (i) CO 2 reduces high pH quickly (ii) At approximately neutral pH, CO 2 is self-buffering (iii) Unlike strong acids, CO 2 does not produce local areas of very low pH, which could harm the sensitive mammalian cells (iv) CO 2 is cheaper and safer to handle in a manufacturing plant because it is not corrosive and its handling can be completely automated.
Chapter 1 discusses the need for the current thesis, its scope and limitations. This chapter also discusses the statement of the problem and the hypothesis. Chapter 2 discusses the background theories relevant to the current project. It gives a brief history of cell culture perfusion systems, types of mathematical models, and then evaluates parameters that influence a mammalian cell culture. Chapter 3 provides methodology employed to solve the problem of CO 2 management and pH control including material balance equations. Chapter 4 provides MATLAB® simulation details and chapter 5 concludes this thesis and makes some suggestions for future work.
vii   A Volumetric agitated gas-liquid interfacial area at the top of the liquid as defined in Gray et al. (1996) B Cell broth bleeding rate from the vessel (L/min) as defined in Gray et al. (1996) Gray et al. (1996) xiii 2 Equilibrium constant for CO 2 hydration, 5.2 × 10 −7 as defined in Gray et al. (1996) 2 Surface aeration contribution to k L a (1/hr) as defined in Zupke and Green (1998) 2 Sparger contribution to k L a (1/hr) as defined in Zupke and Green (1998) 2 Surface aeration contribution to k L a (1/hr) as defined in Zupke and Green (1998) 2 Sparger contribution to k L a (1/hr) as defined in Zupke and Green (1998) k L Individual mass transfer coefficient for liquid phase, based on concentration difference (m/s) K L Overall mass transfer coefficient for liquid phase, based on concentration difference (m/s) 2 Volumetric CO 2 transfer coefficient (L/min) for Gray et al. (1996) equations and CO 2 gas-liquid mass transfer coefficient (L/hr) as defined in Zupke and Green (1998) K S Saturation constant (g/L) Aqueous concentration of O 2 in equilibrium with headspace gas (M) Aqueous concentration of O 2 in equilibrium with sparge gas (M) P Medium withdraw rate from the cell separator (L/min) as defined in Gray et al. (1996) P B Pressure at the bottom of the tank (atm) as defined in Zupke and Green (1998)  Since mammalian cells are extremely sensitive to physiological growth conditions, ongoing operation of these bioreactors required precise and careful CO 2 management and pH maintenance strategies. This thesis provides a mathematical model for maintaining steady CO 2 concentration and pH levels in a perfusion bioreactor growing Chinese hamster ovary cells in a suspension culture.
In the biotech industry, a precise mathematical description of drug producing cells is immensely valuable as such a model, in addition to saving significant capital, will significantly aid process development both to start out initial drug manufacturing trials and subsequent troubleshooting during production. Such a model forms the backbone of adaptive control strategies, which require complete knowledge of the process including complete analytic expressions (Van Impe and Bastin, 1995). Researchers have presented many different kinds of modeling strategies, for example, neural networks (Karim et al., 1997;Nagy, 2007), data based modeling (Karim et al., 2003), and stochastic simulation (Li et al., 2008).
However, none of these strategies have been applied to CO 2 balance and pH control.
One of the primary and well-known challenges of a bioreactor operation is CO 2 management and pH control. Optimum growth conditions, including CO2 concentration and pH levels, highly cell line dependent. If two companies use Cho cells but different cell lines, then growth condition requirements are different for each cell line. Nevertheless, the published scientific literature has a dearth of articles on modeling CO 2 concentration and pH control in the bioreactors (Yoon et al., 2005). Such a model would lead to the development of a control strategy to maintain CO 2 concentration and pH level.
Today, biopharmaceutical industry routinely produces therapeutic proteins from aerobic mammalian cells. Obviously then, the industry has addressed the challenge of CO 2 management and pH control but each company has knowledge regarding its own particular cell line. Conceivably, these individualized solutions to the CO 2 management and pH control issues exist in the proprietary knowledgebase of these respective corporations. Consequently, there is a dire need for a publicly available first principles model of CO 2 concentration and pH control in bioreactors that can be applied to multiple cell lines. This thesis fills this void by presenting a way to model CO 2 concentration and control pH in an perfusion bioreactor.
Methodology developed in this thesis can be modified for application to batch and fed batch modes of operation of bioreactors.

Statement of the Problem
Cell growth in a bioreactor requires steady state conditions i.e. steady nutrient levels and a steady physiological environment. However, in a bioreactor, viable and total cell counts fluctuate appreciably during cultivation due to known and unknown causes and as such, a bioreactor system is unlikely to be in a true steady state (Vits and Hu, 1992). Thus, in a bioreactor, cells exist in a quasi-steady state and the equilibrium continuously shifts. For example, with the passage of time, cells divide and number of cells increase. Each of these mammalian cells alters the pH of the fermentation broth by releasing metabolic by-products -ammonia, CO 2 and lactic acid (Wu et al., 1993). The goal of this model is to develop a mathematical strategy to maintain pH and CO 2 within a specified narrow range.
High-density mammalian cell cultures that use CO 2 to regulate pH have three sources of CO 2 /CO 3 2-: (1) CO 2 that the aerobic cells excrete as part of their normal growth process (Krebs cycle); (2) HCO 3 -/CO 3 2that is added to regulate (raise) pH; and (3) CO 2 sparged into the bioreactor to regulate (lower) pH (Zanghi et al., 1999). Excess CO 2 can be defined as the cell line dependent amount of CO 2 concentration above which cell growth is appreciably negatively impacted. Excess CO 2 is not only toxic to the mammalian cells, it may also affect the quality of protein products (Gray et al., 1996;Matsunaga, 2009b). This thesis presents a mathematical strategy to regulate CO 2 levels in a perfusion bioreactor.
In addition to multiple sources of CO 2 , aerobic cell cultures with high VCD pose their own unique challenges because as VCD increases, so does cell stress due to environmental conditions. Cells experience stress if the physiological growth conditions deviate optimum growth conditions. For example, cells can experience stress in if the broth is highly acidic or highly basic or due to excessive agitation in the bioreactor. For these mammalian cells, the nature of the stress is key to product formation because certain stress would induce the cells to produce the proteinproduct while other types of stress may inhibit cell growth and/or protein synthesis.
Cell cultures with high VCD produce large amounts of CO 2 and require large quantities of O 2 for respiration. If high VCD causes anorexic and/or low O 2 conditions, then in response, the CHO cells may begin to excrete lactic acid, which lowers pH and eventually the CHO cells switch to metabolizing the lactic acid instead of glucose (Wu et al., 1993). CHO cell growth on lactic acid is undesirable because such growth adversely affects the quality of the protein-product.
In high density mammalian cell cultures, perfusion systems address the problem of accumulation of highly soluble CO 2 and the high demand for sparingly soluble O 2 in the bioreactor fermentation broth, by exchanging spent media with oxygen-rich fresh media and by recycling the viable cells into the bioreactor.
Appropriate agitation of the fermentation broth can alter local concentrations of CO 2 and O 2 , as higher agitation rates lead to more homogeneous growth conditions (Arjunwadkar, 1998). However, the mammalian cells in bioreactors must be gently agitated because as the agitation rates increase, risk of cell lyses also increases.
As cells grow in the perfusion bioreactor, increasing VCD increases the amount of NH 3 , CO 2 and lactic acid in the cell culture broth; and correspondingly the cells consume increasing amounts of nutrients and O 2 . Depending on the amount dissolved in the fermentation broth, CO 2 can be both -a nutrient and a toxin. CO 2 is toxic in high concentrations but is also essential for survival and for general well-being of the cells. Fermentation stability is of paramount importance for a viable cell culture.
However, time-dependent increases in CHO cell density in a bioreactor under dissolved oxygen undergo spontaneous bifurcations losing stability. (Chung et. al., 2003). This loss in stability means sustained and amplified perturbation in the bioreactor dissolved oxygen concentration and in oxygen gas flow rate to the bioreactor. Id. In this mathematical model, the high-density CHO cell culture is assumed to be inherently stable.
Cell cultures grown at constant hydrostatic pressures of 10 5 -9 x 10 5 Pa, show little variation in cell growth or specific glucose consumption rates (Takagi et al., 1995). However, as the pressure is increased (to ~ 5 x 10 5 Pa), specific lactate production rate slightly decreases and specific glutamine consumption and ammonia production rates increase. Id. Perfusion bioreactors considered in this thesis typically contain 3-4 m 3 of fermentation broth and CHO cells. For this reason, hydrostatic pressure effects because of liquid height in a bioreactor are ignored as negligible (for example, approximately 3,000 L bioreactor filled to a height of 1.5 m with water, exerts a pressure of ~1.5 x 10 -4 atm or ~15 Pa at the bottom of the tank).
Heat balance is not considered. Heat is transported into and out of a perfusion bioreactor broth through convective and molecular transport when new feed is added to the system and through media exchange. Additionally, each cell produces and consumes heat because of various intra-and intercellular reactions (Riet, 1983). Similarly, mechanical energy or power applied to propel the agitator does work on the system, which degrades into the broth as thermal energy: as the agitator rotates, each shell of broth that is adjacent to the agitator blade rubs against that agitator blade. This friction between adjacent layers of fluid and the wall of the agitator blades produces heat. Finally, the electric wires around the bioreactor may produce electric and magnetic fields that may affect the temperature of the broth. All these sources and sinks of heat are ignored. The mathematical model assumes that a heat jacket around the reactor keeps the temperature inside the bioreactor steady i.e. the system is diabatic or nonadiabatic. In solving the modeling equations, the temperature of the broth is assumed to be within the range of 36.5 °C to 37.5 °C.
From inoculation until the time when the downstream processes begin, many generations of CHO cells produce the desired protein product. It is conceivable that genetic mutations are introduced over several generations that can alter the metabolism of the cells. This model assumes that number of cells with such mutations is very small to have any material effect on the behavior of the bulk of the bioreactor. Likewise, this thesis ignores the effects of microgravity (Anderson, 2004) and various mechanical and environmental factors on cell behavior and gene expression.
Bubble dynamics play an important role in gas transport and cell survivability in a bioreactor. However, effects of bubbles that are produced as a result of CO 2 and O 2 sparging and as a result of agitation are not considered (Wang et al., 1994;Meier et al., 1999;Ma et al., 2006). Similarly, position dependence of agitators is neither considered nor investigated, which has been shown to have an impact on oxygen and carbon dioxide transfer rates (Arjunwadkar, 1998).
Eddy dynamics affect material transport and diffusion into and out of gas and liquid phases in a bioreactor (Croughan and Wang, 1989). This model assumes that mixing between randomly dispersed eddies is instantaneous. Dynamics of mixing within eddies is ignored. Finally, a bioreactor has many coupled reactions.
That is, they require co-substrates or co-factors, which are either recycled or regenerated by another companion reaction. This unstructured model also ignores these coupling effects.

Hypothesis
Contois equation (Contois, 1959) best describes substrate-limited specific growth rate of CHO cells in a perfusion bioreactor that has a high viable cell density (Shuler and Kargi, 1992).
Maintaining CO 2 concentration and pH at a steady state would require dynamic CO 2 material balance and dynamic control of pH.

Justification for and Significance of the Study
Biopharmaceutical drugs are prohibitively expensive due to high research costs and long development time that is required to understand and then develop the drug manufacturing process. A mathematical model that addresses a primary challenge of a mammalian cell culture would greatly reduce process development related time and costs. The fact that work described in this thesis is highly sought after and timely can be gauged from the fact that even though bicarbonate management is a long standing problem, there is no effective solution available.
In 1957, Dr. Theodore T. Puck initiated the use of Chinese hamster ovary cells for research in mammalian somatic cell genetics (Gamper et al., 2005). Since then, CHO cells have become the most popular industrial cells that are used to produce tissue plasminogen activator, erythropoietin, granulocyte colony stimulating factor, factors VIII and IX, deoxyribonuclease I, glucocerebrosidase, beta-interferon, MAb against GPIIb/IIIa, CD20, tumor necrosis factor alpha, tumor necrosis factor receptor and HER2 (Xie and Zhou, 2006). Today, CHO cells produce nearly 70% of all recombinant protein therapeutics that account for $30 billion in annual sales (Jacob et al., 2009). Manufacture of these blockbuster biopharmaceutical therapies from mammalian cell cultures is capital intensive and requires implementation of robust and reliable bioprocesses that consistently produce uniform product. Such production in turn requires a detailed understanding of dynamics and interplay amongst critical bioprocess parameters such as temperature, dissolved CO 2 and O 2 and concentrations of lactic acid and ammonia.
Increasing the product titer requires extensive manipulation of growth conditions inside a bioreactor (Leist, 1990). And although this manipulation is routinely performed in the industry, it is usually a closely guarded trade secret (Yoon et al., 2005;Frick and Junker, 1999). Similarly, despite their numerous applications for producing therapeutic proteins, publications on CHO cell culture processes are quite limited (Xie and Zhou, 2006). Rarer still are publications and purported solution to the problem of carbon dioxide/bicarbonate management in a bioreactor even though bicarbonate management is an overarching issue that biotechnologists deal with every day. Currently, empirical models exist in the industry that are cell line specific, in which, cells are grown under well-characterized conditions. This research thus fills a major void and advances the knowledge about CHO cells growth processes by focusing on mathematically modeling one of the primary challenges that such cultures face -accumulation of CO 2 in bioreactors (Frick and Junker, 1999).
A perfusion bioreactor requires elaborate i.e. expensive support system as compared to a batch or fed-batch bioreactor. The goal of such a support system is to ensure optimum growth conditions for CHO cells while minimizing conditions that are damaging to cell health and growth. Mathematical and computer models aid in estimation of capital outlay by forecasting bioprocess system needs.
Frequently changing culture conditions can influence the consistency of protein product. However, consistent process performance and product quality is a regulatory requirement (21 C.F.R. part 211; Woodside et al., 1998). Thus, in the highly regulated biopharmaceutical manufacturing industry, compliance and regulatory promises and not just economic considerations often dictate bioprocess choices.
Ideally, an accurate representation of cell growth in a bioreactor accounts for every conceivable variation in the process. Not surprisingly, such a description would be computationally onerous and financially impractical (Haag et al., 2005).
This thesis has a direct impact on such considerations by examining the relationships between the critical parameters by focusing specifically on CO 2 management.

LITERATURE REVIEW
Living cells produce proteins that have become blockbuster biopharmaceutical drugs like Enbrel® and Remicade®. Chinese hamster ovary (CHO) cells are primary host cells that the biopharmaceutical industry uses to produce recombinant therapeutic proteins. Manufacture of these proteins in bioreactors consists of two distinct processes: upstream process during which the viable cell density (VCD) increases to a predetermined level and downstream process during which the biopharmaceutical protein that the living cells have produced is purified and isolated. As with any other manufacturing process, protein production through living cells is rife with opportunities for improvement.
A longstanding problem that the biotechnology industry faces is carbon dioxide/bicarbonate management especially in mammalian cell cultures. This thesis focuses on developing a first principles, unstructured, dynamic mathematical model for regulating carbon dioxide and pH in a bicarbonate buffered perfusion bioreactor for aerobic Chinese hamster ovary (CHO) cell system. A perfusion bioreactor is the focus of this research because it is the established bioreactor system in large-scale recombinant protein commercial production facilities. However, perfusion systems are significantly more complex, hence more expensive, and more prone to contamination than batch or fed-batch systems (Fenge and Lüllau, 2006).

Cell Culture Perfusion Systems
Production of biopharmaceutical protein molecules from mammalian cells begins with a single genetically engineered living cell when it is inoculated into a bioreactor containing favorable growth media and environmental conditions.
According to Graff and McCarty (1958), a cell culture is a highly artificial expedient in which host influences are abolished but can be simulated. Suitably then, bioprocess engineers induce the genetically engineered cells to multiply and produce desired protein drug by maintaining optimum growth conditions. In addition to mechanical characteristics of a bioreactor, the growth conditions and critical parameters include optimum concentrations of ammonia, dissolved carbon dioxide, dissolved oxygen, lactic acid, osmolality, pH, substrate and temperature and ionic strength of the fermentation broth. Controlling all these parameters at the same time to ensure optimum environment for the cells is a challenging task that requires simultaneous calculations and real time decision-making. As one scientist noted, "[e]ven the simplest living cell is a system of such forbidding complexity that any mathematical description of it is an extremely modest approximation" (Bailey, 1998).
As the aerobic mammalian cells grow, they consume nutrients and O 2 and excrete CO 2 , NH 3 and lactic acid. As the cell density increases, the concentration of excreted by-products progressively increases in the medium, which can lead to unnatural extremes. Both, dissolved carbon dioxide and lactic acid are mild acids and tend to lower the pH of the biological medium, which is unfavorable for cell growth. Thus, bioprocess engineers must closely monitor dissolved carbon dioxide and nutrient levels.

Bioreactor Operation Modes: Batch, Fed Batch and Perfusion
A bioreactor can be operated in one of three different modes based on the how often cell culture media is replenished and removed: (a) batch; (b) fed-batch; (c) continuous or perfusion with cell recycle.

Batch and Fed Batch Bioreactors
In a batch bioreactor, where there is neither inflow nor outflow of cells or media, the cells grow undisturbed in the initially supplied media. Frequently, oxygen is sparged. Thus, only pH, temperature and aeration are controlled in batch reactors. However, over time, local cell environment constantly changes as the cells multiply and consume available nutrients and excrete by-products like CO 2 , NH 3 and lactic acid.
In a fed-batch operation, a substrate feed maintains the concentration of the nutrients at a predetermined steady state. However, there is no outflow of cells or media. The disadvantage of fed-batch operation is that accumulation of metabolic by-products such as lactate and ammonia limit cultivation times (Vits and Hu, 1992).

Perfusion Bioreactor Systems
Perfusion systems are characterized by media exchange. These perfusion systems can either be "open" in which fresh, enriched media is continuously added and spent media is correspondingly removed; or the perfusion system can be closed in which the spent media is withdrawn and pumped through an "oxygenator" before being re-circulated back into the bioreactor (Griffiths, 1990;Hu and Wang, 1986).
In addition to replenishing dissolved oxygen in the bioreactor, the exchange of media in perfusion bioreactors removes metabolites like NH 3 , CO 2 and lactic acid that can become toxic in high concentrations (Riley, 2006). According to Hu and Wang, assuming oxygen consumption rate of 1.5 mmol/L.h in the cell culture, a media recirculation rate (or perfusion rate) of approximately 8 volumes/hr is needed to avoid oxygen limitation. In this manner, by varying the perfusion rate according to the demands of the cell population, a perfusion bioreactor avoids alternate periods of "famine and feast" in a cell culture, which are characteristic of fed batch reactors (Graff and McCarty, 1958;Nahapetian, 1986).

Figure 2: A Typical Perfusion Bioreactor
Any media exchange in a suspension culture requires a cell retention device and can consist of either submerged spin filters or membrane filtration devices outside the bioreactor (Woodside et al., 1998). Perfusion bioreactor systems offer a relatively constant culture environment over months of operation, a short product residence time and increased product concentration, while operating at a high cell density of 10 6 -10 7 cells/mL (Drouin et al., 2007;Kumar et al., 2007).

Cell Cultivation: A Brief History
Perfusion systems are characterized by a bioreactor with a filtration system that retains cells during media exchange. Today, the biopharmaceutical industry employs perfusion systems to cultivate high-density cell cultures by providing, controlling and maintaining for the cells, a steady state homogenous environment that attempts to closely approximate the cells' natural physiological growth conditions. However, the perfusion systems were initially developed for medical applications to examine animal tissues in vitro under the microscope and to cultivate organs outside of an organism.

Early Development of Perfusion Techniques
As early as 1907, Ross Granville Harrison of Johns Hopkins University grew a nerve cell outside the animal body (Patterson, 1975). Subsequently, a young medical student, Montrose Burrows and Harrison, together developed a perfusion system as part of their studies involving tissues from frog embryos in which they used blood plasma as growth medium (Butler, 1986;Friedman and Friedland, 1998 (Carrel, 1912a). That same year, Carrel reported cultivating cell suspension culture and a heart of an eighteen-day-old chick fetus (Carrel, 1912b). Finally, in 1935, American aviator, Charles A. Lindbergh developed one of the first perfusion systems and Carrel perfected the techniques for cultivation of whole organs in that perfusion system (Lepicard, 2008 (Darnell et al., 2005;Eagle, 1977

Black Box, Gray Box and White Box Models
Black box (or empirical or input-output) models link input factors with output responses. These models are constructed when the mechanism underlying a process is not understood sufficiently well, or is too complicated, to allow an exact model to be postulated from theory (Lübbert and Jørgensen, 2001;Box et al., 2002;van Lith, 2002). These models are straightforward and their results can be readily interpreted.
White box (or first principles or fundamental or mechanistic) models are based on known biological, chemical and physical laws (van Lith, 2002;Lübbert and Jørgensen, 2001). Such models are typically nonlinear and thus formulated in terms of ordinary or partial differential equations with supporting linear algebraic equations. These models are employed when understanding a process or a system is essential to progress or when the state of the art is sufficiently advanced to make a useful white box model easily reliable (Box et al., 2002). Consequently, these models require a high level of effort and can be constructed even when the system itself is not yet constructed (van Lith, 2002). Thus, a first principles model is mathematically rigorous albeit computationally expensive.
As the name suggests, gray box models are hybrid models, which are a combination of first principles and empirical models. Models that are based on heuristic knowledge are termed gray box models to distinguish them from the purely data driven black box models and models based on mechanistically completely understood mathematical models (Lübbert and Jørgensen, 2001).
Today, most models are gray box models to some extent (van Lith, 2002).

Dynamic and Static Models
Static models can generally be described by = ( ) where the value of the independent variable does not change with time. Dynamic models on the other hand, seek to describe the behavior of independent variables with respect to time.

Segregated and Non-segregated Models
A segregated model considers individual cells that are different from one another depending on some distinguishable characteristic (Shuler and Kargi, 1992).
Non-segregated models consider the population as lumped into one biophase that interacts with the external environment so that a single variable can describe the cell concentration (Blanch and Clark, 1996).

Structured and Unstructured Models
The concept of structure arises when considering the detailed reactions occurring within the cell (Blanch and Clark, 1996). For example, a structured model may consider the kinetics of compounds involved in a Tricarboxylic Acid (TCA) or Krebs cycle and describe reactions and ATP, CO 2 and O 2 consumption and production in detail. That is, structured models break the population into distinct subcomponents (Shuler and Kargi, 1992). Thus, structured models require a thorough knowledge of intracellular reactions and their regulation mechanisms (Zeng and Bi, 2006).
Unstructured models on the other hand, consider cell as an entity and models its interactions with the environment assuming fixed cell composition (Shuler and Kargi, 1992). These models are based on fundamental observations of biological processes (Zeng and Bi, 2006). They can be used to qualitatively and quantitatively describe important features of cell culture, for example, effect of pH on cell growth. However, since the unstructured models quantify cell mass as a single component, they cannot describe transient behavior very well (Shuler and Kargi, 1992).

Transient and Steady State Models
A steady state of a dynamic system is defined as one in which the time derivatives of each state variable are zero (Beers, 2007). In other words, over time, the input and output parameters of a system are held constant. A steady state is stable, if following every infinitesimal perturbation away from a steady state, x s , the system returns to x s . If any infinitesimal perturbation causes the system to move away from x s , then a steady state x s is unstable. Finally, a neutrally stable steady state exists when a perturbation neither grows nor decays with time (Beers, 2007).
A transient state exists when a bioreactor is brought into operation or is being shut down so that concentrations, pH, temperature and other parameters continually change with time. Alternatively, an unstable steady state can be nudged into a transient state due to some perturbation caused by changes in environment or some feedback control.

Biomass, Cell Growth Kinetics and pH Control
The primary objective of every step in the production of protein from mammalian cells is to increase titer -the protein concentration. Amount of titer can be increased by increasing (1) the cell mass; (2) the productivity of each cell; and (3) time. Given ample supply of nutrients, growth of cells in a bioreactor is an autocatalytic process. The amount of change in cell population at any given time is proportional to the initial cell concentration. Thus, the following overall reaction describes cell growth in unstructured models: Cells (biomass) + Substrates  Byproducts + more cells (biomass) (2.1) where μ is specific cell growth rate (hr -1 ), X is cell mass concentration (g/L) and t is time (h) (Shuler and Kargi, 1992). Under favorable growth conditions in an ideal system, a cell population increases exponentially.
As VCD increases, aerobic CHO cells consume O 2 and produce CO 2 and lactic acid. Equation (2.2) includes a multitude of cellular reactions that produce and consume various acids and bases, altering pH in a bioreactor. However, ideal growth conditions require steady state operation at desired pH, temperature and control of other parameters. To maintain a bioreactor at a certain pH, an acid or a base is added at commercial production facilities where mammalian cells are cultivated in a series of bioreactors. However, because of the variability associated with each individual cell, it is difficult to predict how much acid or base must be added.

Criteria, Limitations and Considerations for Modeling Equations
The Ammonia is one of the two major metabolic by-products of mammalian cell growth (the other is lactate) that accumulates in high-density mammalian cell cultures as a result of glutamine metabolism and its spontaneous decomposition (Xie and Wang, 1996;Yoon et al., 2005;Gódia and Cairó, 2006). Ammonia negatively influences cell growth, recombinant protein productivity of CHO cells and protein quality (Chen and Harcum, 2005 where pK has a value of 9.3 at 37°C (Schneider et al., 1996).

Toxic and Inhibitory Effects of Ammonia/Ammonium
High concentration of ammonium ion in the fermentation broth is toxic to CHO cells because ammonium ion rapidly diffuses into the cytoplasm of a cell through the cell membrane forcing ammonia out into the extracellular broth. This release of cellular ammonia causes cytoplasmic acidification and extracellular alkalinization (Martinelle, 1996). Such a ammonium ion transport highway causes cell death -both through necrosis and through apoptosis. Id. Moreover, ammonium ion partially inhibits the TCA cycle shifting metabolic pathways (Lao and Toth, 1997). Ammonium ion also affects the quality of the protein product (Xing et al., 2008;Harcum, 2006). For these reasons, one of the major design parameters of CHO cell bioreactor includes reducing ammonium ion formation and/or removing ammonium ion from the bioreactor.
Just like any other physiochemical parameter in a bioreactor, the toxic and inhibitory effects of ammonia/ammonium ion are highly cell line dependent (Schneider et al., 1996). For example, Kurano et al. (1990b) reported a 50% reduction in the growth of CHO cells at 8 mM ammonium ion concentration.
Hansen and Emborg's study in 1994 corroborated this observation when they reported that in continuous cultures like perfusion bioreactors, CHO cells experience inhibition of growth at ammonium ion concentration above 8 mM.
However, according to Xing et al. (2008), ammonium ion levels above 5.1 mM inhibit CHO cell growth and Takagi  Mammalian cells produce ammonia in direct proportion to their glutamine consumption (Faraday et al., 2001). At low glutamine concentrations, adding nonessential amino acids into the CHO cell media minimizes ammonia production (Xie and Wang, 1996;Chen and Harcum, 2005). Consequently, the strategies to reduce ammonium ion accumulation involve reducing glutamine concentration in the fermentation broth and lowering or eliminating the CHO cell demand for glutamine (Xie and Zhou, 2006;Zeng and Bi, 2006).

Carbon Dioxide
Carbon dioxide (CO 2 ) concentration -both in the fermentation broth and in the headspace above the broth, is one of the most important parameters in a mammalian cell bioreactor. For aerobic CHO cells, the process of CO 2 management in a bioreactor is complicated because CO 2 is a metabolic by-product and in small quantities, a nutrient when it is required for synthesis of pyrimidines, purines and fatty acids in animal cells (Ma et al., 2006;Zeng and Bi, 2006). For example, during carboxylation of pyruvate to oxaloacetic acid, CO 2 acts as a nutrient even for aerobic cells (Gódia and Cairó, 2006). However, in large quantities, not only is CO 2 toxic, it also adversely affects the quality of the protein product (e.g. Gray et al., 1996;Xing et al., 2008). For this reason, bioengineers must design bioreactors with active CO 2 management systems.

Sources and Effects of CO 2 in Fermentation Broth
In a bicarbonate buffered medium, there are three sources of CO 2 -the aerobic mammalian cells, bicarbonate buffer used to regulate pH and sparged CO 2 .
At low cell densities, the buffer capacity of the media compensates for the variations in cell culture pH due to minor changes in concentrations of ammonia, CO 2 and lactic acid that the mammalian cells release into the medium. However, as VCD increases, these three metabolic by-products can significantly alter the pH of the medium easily overwhelming the buffer capacity of the CO 2 /HCO 3 -/CO 3 2buffer system. Addition of sodium bicarbonate can restore the buffer capacity and thus the pH of the medium. Unfortunately, accumulation characterizes HCO 3 and CO 3 2buffered media thereby affecting the CO 2 balance, increasing the pH to above the optimum level of 7.0 -7.4 (Zanghi et al., 1999;Neeleman, 2000). For this reason, it is necessary to sparge CO 2 gas, which helps lower the pH. The sparged CO 2 adds to the complexity of the CO 2 /water system because of the specific reactions that CO 2 undergoes with water even though a small fraction (<1%) of dissolved CO 2 converts to carbonic acid (H 2 CO 3 ).
CO 2 , a non-polar molecule that is 25 times more soluble in fermentation broth as compared to O 2 , readily diffuses across cell membranes and lowers the intracellular pH (Frick and Junker, 1999;Pattison et al., 2000). Significantly, methods that serve to enhance O 2 transfer to the broth also enhance CO 2 transfer rate (Frick and Junker, 1999).
Higher intracellular CO 2 concentration affects the activity of intracellular enzymes and alters cellular metabolism (Nyberg, 1999). Mammalian cells produce relatively low quantities of protein (approximately several pg/cell). Protein quantity and quality is related to cell growth, maintenance and metabolism (Nyberg, 1999).
If the concentration of CO 2 is higher in the media, then cells grow under stress. In such conditions, cells produce an undesirable quality of protein. Thus, during the cell growth phase, bioengineers take care to reduce the stress that cells experience.
Increasing the cell density will cause the dilution and perfusion rates to increase requiring active control of CO 2 and O 2 sparging regime. Increasing the CO 2 sparge rate will strip CO 2 from the cell culture broth causing it to drop below the ideal limit of 5% thus raising the pH. However, it is expected that increased VCD will also cause more CO 2 to be released into the broth and lower the pH. In fact, the local CO 2 concentrations could become so high that CO 2 will become toxic to the cells requiring a higher agitation speed, which in turn must be balanced against the shear tolerance of the CHO cells.

CO 2 Reactions
Concentration of different species of CO 2 in the aqueous phase vary as a function of pH and other factors. In the aqueous solution of a cell culture, following reactions involving CO 2 may take place:  Consequently, at the target pH of 7.0, reactions 2.9 and 2.10 predominate (Goudar et al., 2011).
As these reactions demonstrate, sparging CO 2 will lower the pH of the cell culture. Additionally, maximum amount of CO 2 dissolved in water is a function of pH because the fraction of total dissolved CO 2 in aqueous solution changes as the pH of the solution changes. Thus, total CO 2 concentration affects the pH of the cell culture broth according to the following equations (Gray et al., 1996;Ma et al., 2006

Effect of Lactic Acid/Lactate in Fermentation Broth
CHO cells produce lactate as a result of their inefficient metabolism of glucose consumption (Faraday et al., 2001;Hinterkörner et al., 2007). As VCD increases, glucose consumption increases leading to production of large amounts of lactate (Gramer and Ogorzalek, 2007). High concentration of lactic acid in the cell fermentation broth inhibits CHO cell growth by decreasing specific Adenosine Triphosphate (ATP) production rate and ATP yield from glutamine (Takagi et al., 2001;Hinterkorner et al., 2007). Modeling lactic acid concentration in the mammalian cell fermentation broth is complicated by the fact that although lactate is an inhibitory metabolic product, under high stress i.e. in glucose limiting conditions, CHO cells switch over from glucose to using lactate as a carbon and energy source (Takagi et al., 2001;Tsao et al., 2005).
A weak acid, lactic acid partially hydrolyses into lactate and hydronium ions according to the following equation: According to Gramer and Ogorzalek (2007), at the pH of approximately 7.0, 1 mol of lactic acid produced by the cells requires addition of 1 mol of base.
Assuming that lactic acid is the dominant metabolite contributing to pH change and that primarily pH is raised by base addition, then 1 mole of lactic acid that the cells produced will consume 1 mol of base, which must then leave the bioreactor in the form of CO 2 because in this model, bicarbonate is used as a base.

Oxygen and Volumetric Mass Transfer Coefficient
Dissolved oxygen (DO) in the fermentation broth is a vital substrate for optimum growth of aerobic CHO cells. Oxygen is sparingly soluble in water and can easily become a growth limiting substrate as the VCD increases (Shuler and Kargi, 1992 where 2 is in mg/L and T is in °C. Jorjani and Ozturk (1999) quantitatively studied the effect of temperature on oxygen consumption rate (now referred to as oxygen uptake rate, OUR) on three different cell lines -baby hamster kidney (BHK), murine hybridoma and CHO. They reported 10% decrease in OUR for each 1% decrease in cell culture temperature.
For this thesis, Specific Oxygen Uptake Rate, qO 2 , for CHO cells is taken to be 1.99 x 10 -13 mol cell -1 hr -1 and the Respiratory Quotient is assumed to be 1.
In summary, the beneficial effects of low culture temperature on specific productivity depend on cell types and target proteins (Tang et al., 2009) and the benefits are especially pronounced for high density perfusion cultures of mammalian cells (Zeng and Bi, 2006).

Culture pH
Cell culture pH is arguably the most important parameter that affects cell growth in the bioreactors. Elaborate process analytical technology (PAT) systems and hardware are dedicated to controlling and maintaining broth pH. Mammalian cells grow over a narrow pH range, typically ±0.1 pH units, significant deviations from which profoundly impact cell growth, cell metabolism and protein biosysnthesis (Ozturk and Palsson, 1991;Tang, 2009). This impact results in altered substrate and product formation rates and if not addressed, may cause cell damage.
Slightly acidic pH in a CHO cell culture reduces glucose consumption and increases lactic acid buildup (Ozturk and Palsson, 1991;Tsao et al., 2005). Lactic acid is one of the contributors to localized pH deviation from steady state.
Additionally, the region where base is added to raise pH also experiences a local spike in pH that causes cell damage (Langheinrich and Nienow, 1999). Another metabolic by-product, CO 2 also causes pH deviations as mentioned in Section 2.5.3.
For example, although the physiological range of partial pressure of CO 2 in the fermentation broth, pCO 2 , is 31 -54 mmHg, typical range of pCO 2 in a high-density bioreactor is 150 -200 mmHg High pCO 2 levels causing cell growth inhibition if the pH is not actively controlled (deZengotita et al., 2002;Goudar et al., 2006).
Similar to the buffer system in human blood, a CO 2 -bicarbonate buffer system maintains pH in the CHO cell culture in a bioreactor. The Henderson- To control pH, two separate control loops are needed -one to raise pH and one to lower pH. Such a control loop must avoid excessive control impulses otherwise the increase in CO 2 flow will increase cell culture osmolality, which impedes cell growth.

Temperature
Temperature is a cell line specific critical variable in cell growth kinetics that affects product yield and product quality (Kumar et al., 2007). Bioreactors cultivating the CHO cells traditionally tend to simulate the normal body temperature of a Chinese hamster of around 37°C (Kurano et al., 1990a;Tang et al., 2009).
CHO cells are sensitive to variations in temperature. If the bioreactor temperature rises above the optimal temperature range of a particular cell line, then the growth rate decreases and thermal death of the cells may occur leading to a net decrease in VCD (Shuler and Kargi, 1992). To guard against possible excursions from optimal range of temperature, bioprocess engineers use growth media fortified with d-glucose, d-galactose or d-mannose, which increase the survival of CHO cells at higher temperatures in a concentration-dependent and timedependent manner (Henle et al., 1984). Lower temperature, on the other hand, results in a more complex physiological response from the CHO cells. Chuppa et al. (1997) reported that in a high-density perfusion cell culture, where oxygen may become limited, reducing the temperature allows the bioreactor to be operated at a lower perfusion rate and simplify pH control regime while improving product quality. Similarly, while investigating the production of erythropoietin (EPO) by CHO cells, Ahn et al. (2008) observed that lowering the perfusion culture temperature to below 37°C increased cell viability for a longer period of time and resulted in higher cumulative EPO production. They cited high shear resistant characteristics of mammalian cells at low temperatures as one of the possible reasons for this behavior. Additionally, they reported that the quality of EPO as measured by glycosylation also improved at a lower temperature. Id. On the other hand, Chen et al. (2004) reported that lowering the temperature only had a marginal effect on glucose and lactate metabolism.
Variations in temperature of a cell culture must be minimized to obtain desirable culture performance. A perfusion bioreactor poses unique challenges in maintaining a constant temperature by virtue of its design and purpose. Perfusion cell cultures achieve high cell density (10 6 -10 7 cells/mL), which may lead to very high local temperatures in the broth (Lara et al., 2006).  (Drouin et al., 2007;Lee et al., 2008).
Varying solubility of various gases at different temperatures adds a new dimension to the analysis of cell growth kinetics. For example, with an increase in temperature from 30°C to 40°C, the solubility of CO 2 decreases by approximately 25% (Pattison et al., 2000). Moreover, although concentrations of both CO 2 and NH 3 in fermentation broth must be regulated, since dissolution of CO 2 in aqueous ammonia is endothermic (Δh > 0), with rising temperature, the solubility of CO 2 in ammonia solution also increases (Pazuki et al., 2006). However, the model presented in this research does not consider the variations in solubility with variations in temperature for carbon dioxide and oxygen.

Cell Growth Kinetics
Scientists have proposed many equations to model cell growth kinetics (e.g. Gomes and Menawat, 2000). The Monod equation is the most commonly used cell growth equation. This thesis uses Contois equation (Contois, 1959)  where μ m is the maximum growth rate, S is substrate concentration and K S is saturation constant (Shuler and Kargi, 1992). The saturation constant, which is proportional to the cell concentration, describes the substrate-limited growth at high cell densities. Thus, the Contois equation predicts that the specific growth rate decreases with decreasing substrate concentrations eventually becoming inversely proportional to the cell concentration in the medium. Substituting equation (2.2) into the cell growth equation (2.25) gives the rate of cell growth expression: In a perfusion bioreactor, viable and total cell counts fluctuate appreciably during cultivation due to unknown causes and as such, the system is unlikely to be in a true steady state (Vits and Hu, 1992).

Tank Geometry and Hydrodynamics
Current work focuses on the high-density perfusion cultures of CHO cells wherein oxygen transfer is quite often the rate-limiting step. As mentioned in Section 2.5.6, solubility of oxygen is low in aqueous solutions and even more so in the presence of ionic salts. In such situations, increasing the productivity of mammalian cell cultures requires, in addition to prolific cell lines, a vigorous fermentation process that incorporates an efficient bioreactor that can achieve a uniform gas-liquid mass transfer coefficient, k L a.
A number of variables affect bioreactor performance (e.g. Kompala and Ozturk, 2006;Lara et al., 2006). Many researchers have published reviews of bioreactor design for cultivating mammalian cells (e.g. Prokop and Rosenberg, 1989;Lübbert and Jørgensen, 2001). Maximizing k L a values for gas transfer and minimizing any "dead zones" -areas exhibiting poor local mass transfer, is one of the primary performance parameters for the mechanical construction and hydrodynamics inside the bioreactor. Many techniques have been proposed to minimize these "dead zones," which are more pronounced in high-density perfusion cultures (Gogate and Pandit, 1999). These techniques include optimizing k L a by altering aspect ratio, tip speed, type, location and number of impellers, etc. Each of these techniques has its advantages and limitations and must function under its own set of constraints. Although a complete modeling of these parameters is outside the scope of this thesis, this chapter briefly explores the impact of these techniques on cell growth in a bioreactor.

k L a
In a perfusion reactor or continuous stirred tank reactor with cell recycle, the goal of the agitation scheme it to improve gas-liquid mass transfer coefficient, k L a. Oxygen is an important substrate for CHO cells. Since oxygen is sparingly soluble in water, it is often a growth limiting substrate in the bioreactors.
Consequently, bioreactors are designed to ensure that the k L a is as high as possible. Zeng and Bi (2006) describe static gassing out method and dynamic gassing out method of measuring oxygen uptake rate (OUR) and k L a for a particular bioreactor. In the static method, the cell culture is grown to a known cell density and then killed. Oxygen is removed from the headspace and aeration is turned on at the typical gas flow rate and the impeller is operated at a typical level. The increase in oxygen concentration is followed until oxygen saturation (C*) is reached. Change in dissolved oxygen concentration is given by where C is concentration of oxygen in solution C* is equilibrium solubility of oxygen (oxygen saturation); and t is time.
A plot of ln(1-C/C*) versus time gives a slope of -k L a. In Zeng and Bi's dynamic method, aeration to an active culture is briefly turned off and the unsteady state mass balance of oxygen is tracked while taking care that oxygen concentration does not drop so as to negatively impact the cell growth. This is so that rate of oxygen uptake is independent of oxygen concentration. In this method, where C S is the steady state dissolved oxygen concentration. Zeng and Bi (2006) describe that the OUR can be calculated from * = * ( * − ) (2.29) where X is cell density (cells/L) OUR is oxygen uptake rate (g O 2 /10 6 cells-hr).
The kLa is characteristic of each cell line and bioreactor configuration and is a proprietary value and a closely guarded trade secret. In this thesis, only publicly available values are used which may not be typical of the bioreactor modeled.

Agitation
In 2005  where P g is power consumption in presence of gas (W) V g is superficial gas velocity (m/s) V is volume of liquid (m 3 ) Following are the design considerations: 1. Impeller tip speed is limited by amount of shear stress and rate of shear cells can handle.
2. As impeller speed increases, it increases frictional forces, shear stress and rate of shear. Thus, increase in impeller speed increases temperature of the broth.
3. The distance between any two impellers must be greater than their diameter otherwise the individual impellers will generate liquid streams which are inclined towards each other and combine halfway between the impellers acting as a impeller producing a radial outflow (Puthli et al., 2005).

Bioreactor Parameters: Gas Hold-up, Bubbles and Sparging
Two kinds of hydrodynamic forces are constantly at work in a bioreactordirect forces like agitator or bubble bursting causing cell lysis and indirect forces like microgravity and chronic exposure to energy dissipation rate (EDR), which is used to quantify local mixing performance in stirred tanks (Godoy-Silva et al., 2009). CHO cell lines are sensitive to lower values of energy dissipation rate (EDR) (relative to the values needed for cell lysis in one exposure), if the exposure to such levels of EDR is chronic (Godoy-Silva et al., 2009).
An important hydrodynamic design parameter of any bioreactor, fractional gas hold up, ε, is defined as the ratio of the gas phase volume to total volume in a bioreactor (Arjunwadkar et al., 1998b). Fractional gas hold up, together with mean bubble diameter, determines the gas-liquid interfacial area ( = 6 ) (Arjunwadkar et al., 1998a). Consequently, fractional gas hold-up determines the mass transfer coefficient, k L a. Arjunwadkar et al., (1998a) have observed approximately 30% decrease in actual gas hold-up as compared to simple air-water system. They attribute this behavior to larger mean bubble diameter, which has higher rise velocity and correspondingly lower residence times as compared to bubbles with smaller diameters.

Viscosity
As viable cell density increases, the rheology of fermentation broth, which is initially similar to water becomes viscous and non-Newtonian (Moilanen et al., 2006). At a constant agitation rate, changes in viscosity do not significantly affect the metabolic activity of CHO cells (Moreira et al., 1995). Increased viscosity, however, has a detrimental effect on the mass transfer coefficient, k L a (Puthli et al., 2005). For example, in 1987, Schumpe and Deckwer reported that in aerobic fermentations, the oxygen transfer rates into viscous broths are low in all fermentor types. Puthli et al. (2005) note that with increase in the viscosity, the resistance to the mass transfer increases. They reason that only turbulent eddies with sufficiently high energy can overcome the resistance of the viscous layer to cause bubble break-up or gas-solute transfer, resulting in an overall decrease in the k L a, which is more than that expected on the basis of variation in fractional gas phase hold-up alone. According to Dahod (1993), actual CO 2 dissolved in fermentation broth can far exceed its value calculated from the assumption of an equilibrium between the broth and the air leaving the fermentor. The departure from the equilibrium value increases as the broth viscosity increases.

METHODOLOGY
A large number of factors affect cell growth in a high-density perfusion bioreactor. Although each individual cell in a bioreactor experiences a unique physiochemical environment, technology has enabled bioprocess engineers to collect data regarding various growth factors and predict cell behavior under given set of physiochemical conditions. This multivatiate analysis is most suited for modeling cell growth, requires simultaneous solution to partial differential equations and differential algebraic equations (Xing et al., 2008). In this thesis, the mathematical model considers the impact of various correlated variables.
An a priori mathematical description provides insight into the physical interactions and the nonlinearities involved in the variables that impact cell growth.
Ideally, a

Model Objective
In a CHO cell culture, the model seeks to maintain steady state CO 2 and pH profiles as a function of time as described in the picture below: Figure 4: A Basic Modeling Approach

Modeling inputs include initial concentrations of [NH 3 ], [CO 2 ], [lactic acid]
and [O 2 ], CER, OUR, CO 2 and O 2 sparge rates, perfusion and feed rates, feed composition, power input and cell growth. Additionally, a complete understanding of all the specific rates, rate constants etc. is needed. In practice, such complete information about a bioprocess is never available necessitating a need for simplifying assumptions (Van Impe et al., 1995). The figure below depicts the input and output parameters. Inputs and outputs can be categorized into two groups: cell kinetics parameters like cell growth rate and tank geometry/ physical bioprocess parameters like aspect ratio, impeller configuration. Both types of parameters are equally important in fully describing dynamics inside the bioreactor. Since this thesis focuses on cell kinetics parameters, a few simplifying assumptions make the modeling more manageable.

Simplifying Assumptions
A rigorous first principles model must be computationally efficient to be practical. As the parameters in a model are varied, the qualitative nature of an equation and of its numerical solution can change (Beers, 2007). The goal of the assumptions is to simplify the field by partial differential equations of mass balances into simultaneous differential algebraic equations.
Following assumptions were made in deriving the mathematical model: 1. Bioreactor is completely homogenous, ideally well mixed so that the species concentration and temperature are uniform throughout the bioreactor. Thus, tank geometry, like aspect ratio, power input are not considered in the model.
2. All CHO cells are fully-grown cells of the same size. This assumption is made because big cell diameter and the low surface to volume ratio are limiting factors of the internal mass transfer and uptake rate of nutrients thus influencing cell growth and metabolic rates (Leist et al., 1990).

CHO cell division does not result in random genetic mutations that have
any material impact on their predictable behavior in the cell culture.
4. CHO cells do not exhibit shifting metabolic pathways in the middle of the growth cycle, which results in diauxic growth (Shulder and Kargi, 1992 (Pazuki et al., 2006).

Material Balance in a Perfusion Bioreactor
For a perfusion cell culture (or a continuous culture with cell recycle or a chemostat with cell recycle), material balance is performed on three systems: the bioreactor, the recycle system and both the bioreactor and the recycling system combined.

Biomass and Substrate Balance Using Contois Equation
Material balance on biomass in a bioreactor with cell recycle gives:  (Shuler and Kargi, 1992). According to this equation, the specific growth rate decreases with decreasing substrate concentrations and eventually becomes inversely proportional to the cell concentration in the cell culture. Id. Fenge and Lüllau (2006) have proposed the following equation for substrate consumption: And the following equation provides product formation: where, D is medium exchange rate (L/hr) F is feed flow rate into the bioreactor (L/hr) P is product concentration (mol/L) S 0 substrate concentration in the feed (mol/L) S is substrate concentration in the bioreactor q S is specific substrate consumption rate (mmol/cell/hr) q P is specific product production rate (mmol/cell/hr)

Mass Balance
Material balance of carbon involves balancing inorganic carbon that exists in the bioreactor in the form of hydrated and gaseous carbon dioxide. In 1996, Gray et al. modeled the CO 2 concentration in a perfusion bioreactor. Their equations are presented in Chapter 2. However, Zupke and Green (1998) presented a more intuitive model that takes into account the headspace gas composition, sparging, surface and bubble mass transfer and generation by mammalian cells.
Zupke and Green suggested the following equation for headspace gas composition in which the first parenthesis represents gas flow into the headspace, the second parenthesis represents gas flow out of the headspace and the third term represents surface mass transfer. where the following equation gives the value of 〈 〉 in which Zupke and Green (1998) further described the gas sparging requirements: According to Zupke and Green (1998)

Solution Method
In developing an unstructured mathematical model for this thesis, following steps were taken: 1. Describe cell growth in a perfusion bioreactor using Contois equationdescribed in Section 3.2.1.
3. Write unstructured material balance equations for CO 2 -described in Section 3.2.3.
4. Model change in CO 2 and pH profiles as a function of cell growth and time.

Key Parameter Limits
All the constraints, constants, linear and nonlinear inequalities and linear and nonlinear equations must be identified before simulation can begin. The following equations (from Xing et al., 2008) provide some of the constraints and limits that apply to a CHO cell culture: According to Goudar et al. (2011), following are the specific carbon dioxide production and specific oxygen uptake rates: = 6.25 .
(3.47) = 5.53 . (3.48) These rates will be multiplied by instantaneous cell count to get CPR and OUR for that instant.
For IMDM medium, which is considered in this thesis the following equations provide the limits of substrate present (Burgener and Butler, 2006):

Solving the Equations in MATLAB®
The model equations presented in this thesis are stiff. Stiffness arises when the control functions vary unevenly with time. In these systems, small and large time constants occur in the same system -small time constant controls the earlier response, whereas the large constant controls the tailing response (Rice and Do, 1995 (Zeng and Bi, 2006).
In line with the first hypothesis stated in Section 1.4, the Contois equation adequately describes the cell growth in the perfusion bioreactor. As the second hypothesis predicted, maintaining CO 2 concentration and pH at a steady state does require dynamic CO 2 material balance and dynamic control of pH. As discussed in Chapter 3, determining instantaneous CO 2 concentration and pH requires experimentation that is not part of current research even though the equations can be adequately solved using MATLAB®.

Simulation in MATLAB®
To solve the mathematical equations, the following graphical user interface (GUI) was developed using MATLAB® GUIDE®:  of how the mathematical model that is presented in this thesis can be solved. Figure 9: Graphs showing simulated prediction of response of cell growth, glucose consumption, growth rate, and product formation curve in a perfusion bioreactor.
The model generating the graphs in Figure 9, demonstrate that as expected cell growth and product formation increases with time, substrate consumption increases resulting in decrease in substrate concentration. The cell growth rate is stabilized around day 11. Similar to Deschênes et al. (2006), for simplicity, cell death parameter is ignored, this multivariable approach demonstrates that the cell death parameter cannot be completely disregarded. Equations from Fenge and Lüllau (2006) were used to generate the graphs in Figure 9. The code used to generate these graphs is presented in Appendix 1, which uses parameters from both Deschênes et al. (2006) and Fenge and Lüllau (2006).

CONCLUSION
The goal of a mathematical model is to understand cell growth in a bioreactor so that the protein-drug production from the cells could be maximized.
Of course, the resulting efficiency in drug production has a direct and positive impact on it bottom-line and competitiveness -especially if competing against potential biosimilar manufacturer. A mathematical and computer model that is applicable across cell lines and to different types of bioreactors would be highly sought after in the biopharmaceutical industry because it will simulate drugmanufacturing process with limited need for expensive experiments because such a model must still be validated against experimental data. However, the need and cost of those experiments would be greatly reduced due to this model.

Future direction
Further additions to mathematical and computer model of cell growth in a bioreactor might include using equations of state (e.g. NRTL, SAFT, PC-SAFT, UNIFAC or UNIQUAC) to model solubility of inorganic salts and amino acids in a background medium that more closely mimics the actual fermentation medium in the bioreactor. Those models might also use Kirkwood equation �ln S p S 0 � = K i I − K s I (Harrison et al., 2003) to model solubility of protein in the broth and also incorporate broth characteristics like viscosity (Lapasin et al., 1996) and tank geometry, hydrodynamics and bubble dynamics.
As can be seen from the foregoing thesis, these are just a few factors that can be considered in this type of a mathematical and computer model. A mathematical model that considers more of these parameters is correspondingly more accurate in predicting actual cell behavior under actual process conditions.
From a computer-programming standpoint, such a model would be highly complex especially since that model must be applicable across cell lines so that it appeals to a majority biotechnology companies. Additionally, such mathematical models can employ neural network and artificial intelligence programming that "learn" from past runs (Karim et al., 1997;Acuña et al., 1998;Nagy, 2007). Neural networks can be "trained" to anticipate parameter changes needed for scaling the model -further reducing the costs associated with funding pilot scale laboratories.
Such programming would greatly reduce the need for expensive experiments.

Continuation of the Current Work
One of the major challenges of this research has been limiting the scope of the model to make it more manageable. Some equations are available in published literature and some can be derived from laboratory scale experiments. However, incorporating those equations in a computer program requires intimate programming knowledge, which was not the focus of this research project.
Focus of this thesis was to demonstrate how to develop a mathematical model that biopharmaceutical companies can use for maintaining CO 2 levels and a steady state pH in a mammalian cell culture perfusion bioreactor. This thesis analyzes many of the parameters that can have a significant impact on such cell cultures. Mathematical and computer model developed in this thesis is based on the published equations and published laboratory research. Although this model has generated expected results, they must be verified against data from actual laboratory assays. As mentioned in earlier, programming charge and mass balance equations for all the ions present in the cell culture medium is outside the scope of this research. Similarly, the impact of ionic strength must be taken into account as discussed in Nagy (2007) and Goudar et al. (2011). For a more complete model that predicts pH and models how perturbations in one of the parameters affects values of other parameters as a function of time, such programming is essential.