Modeling and State Estimation of Bio-processes using Dynamic Flux Balances

Due to the increasing demand for bio-pharmaceuticals, optimization of bio-processes' productivity and reduction of process variability have become critical goals for manufacturers. Mathematical models of the fermentation processes are instrumental in achieving these goals. \n \nDynamic flux balance analysis (DFBA), sometimes also referred to as dynamic flux balance modeling (DFBM), is a type of mechanistic modeling approach that can describe the dynamic evolution of key metabolites based on the structure of metabolic networks. DFBA predicts the dynamic evolution of metabolites based on the assumption that resources are optimally allocated so as to maximize/minimize a biological objective function, e.g. maximization of cell growth. Accordingly, DFBA is formulated by a linear programming (LP) problem to compute the metabolic fluxes at each time interval. Then, the evolution of concentrations of different metabolites over time is obtained from the integration of mass balances that are based on the calculated fluxes. \n \nGenerally, the LP used to solve a DFBM for a particular microorganism may have multiple solutions. Mathematically, the multiplicity of solutions arises due to the under-determinancy of the LP. On the other hand, from the biological point of view, the occurrence of multiple solutions may correctly describe the behavior of different strains of the same microorganism or alternatively the occurrence of metabolism switches under different operating conditions. The choice of one solution in the presence of multiplicity is further complicated by the fact that different commercial solvers may lead to different solutions of identical LPs. However, a good DFBA model should be solver-independent while it should be able to correctly describe available data for a specific microorganism strain. \n \nFollowing the above a good LP solver should choose the specific solution based on the strain instead of choosing the solution "randomly" as most commercial solvers do. Hence, the first contribution of this research is to construct a solver that can select a specific solution among all possible optima that is compatible with experimental data. The weighted primal-dual method (WPDM) presented in Chapter 3, is a modified version of the interior point method (IPM) which uses interior weights to solve the LP. By manipulating these weights, the specific optimal solution can be obtained when multiple optimal solutions occur. The interior weights can be found by fitting experimental data obtained for a specific strain of a microorganism. \n \nAlthough WPDM was able to select optimal solutions to fit the data, it was found to be computationally expensive and thus less suitable for large networks. To address this, an alternative fast and low-code algorithm called the ellipsoidal reflection method (ERM) was developed as described in Chapter 6. This algorithm is able to select particular solutions among all possible solutions based on the combination of quadratic programming (QP) and LP problems. ERM plays the same role in DFBM but it can greatly reduce the computations thus making it suitable for future real-time applications. \n \nAn important application of mechanistic models such as DFBM in bioreactors is for the purpose of estimation of states that cannot be measured directly from available measurements. The ability of estimate variables such as growth rate, productivity or key nutrients are crucial for controlling and optimizing the process. State estimation for biochemical systems is particularly difficult due to the lack of online measurements in industrial bio-processes. While variables such as dissolved oxygen, temperature and pH are regularly measured and controlled, most metabolites' concentrations cannot be measured online. Thus, lack of observability of unmeasured states from measured ones are a known challenge in bio-processes. \n \nTo address the lack of observability, set membership estimation (SME) is proposed whereby the upper and lower bounds of each state are estimated based on limited measurements. This approach is motivated by the fact that the cell culture media recipe is generally fixed and the variations of the initial concentrations with respect to the nominal recipe are within small ranges. The SME treats the variation of initial concentrations as a set and propagates the initial bounds of the set onto the bounds of each metabolite at each time step. In this research, two methods of SME are proposed to estimate the bounds of metabolites. \n \nThe first state estimation method, described in chapter 4, is based on the identification of active constraints and assumes that the solution is always unique in DFBA. Since the concentration is varying with time, the LP problem in DFBA can be formulated as an LP with varying parameters. Then, Multiparametric linear programming (mpLP) can be used to convert the DFBA system into a variable structure system (VSS). VSS describes the system as composed of multiple subsystems where each subsystem describes a different region of the state space. For each subsystem, an extended Kalman filter (EKF) is constructed to estimate the key states, and the remaining states are estimated by SME. Moreover, the states crossing in or out of each region of the state space are monitored by a special algorithm and switches between different EKFs are determined accordingly. In the \\textit{E. coli} model, it was assumed that only biomass and culture volume are measured and are used to estimate the bounds of the other states. \n \nThe second state estimation method presented in chapter 5 is an extension of the first method but it explicitly considers the existence of multiple solutions. In this second method, WPDM is used to replace the LP solver in DFBA and multiparametric nonlinear programming (mpNLP) is employed to solve the WPDM interior point-based algorithm. To propagate the uncertain sets by nonlinear mapping, the sets are split into smaller sets and are propagated separately by a linear mapping approximation. This is followed by an assembly operation of all these mapped sets together into one set for each state. Again, for the E. coli model, only biomass and culture volume are assumed to be measured and are used to estimate bounds on the other states. This method is shown to generate bounds of all states much faster than a Monte Carlo algorithm. \n \nTo test these methods proposed a platform of culturing B. pertussis has been set up. In chapter 7, a batch culture of B. pertussis and modeling by DFBM are presented. The protocols of shake flask, batch culture, and measurements of concentrations of amino acids in the culture by HPLC are set up. To solve the multiplicity issue, ERM is used in the modeling by DFBM. Based on the experimental data, DFBM adapted from the previous model is used to fit. The DFBM model can roughly capture the dynamics of key amino acids but not of all of them.

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