Parameter estimation and method of moments for multi dimensional population balance equations with application to vaccine production processes

Particulate systems are found in many process engineering applications and play an important role in a large number of industries such as pharmaceutical and food industry. In general, the particles are not uniform and properties, like size, shape or internal composition appear distributed within the particle population. These property distributions may have significant effects on the overall product quality or processability. Examples are not only found in chemical process engineering, e.g. crystallization, but also in bioprocess engineering, e.g. cell culture based production of biopolymers or vaccines. Here, cell-to-cell variability with respect to size and intracellular composition is a common feature affecting the overall processes. A model based analysis of the observed effects supports an thorough experimental characterization of the process and is an important step in the design of model based process intensification and control strategies. The dynamics of particulate processes can be characterized conveniently with the population balance modeling approach. Here, the resulting population balance equations represent partial differential equations and describe the temporal evolution of the properties distribution of the particle ensemble. The dimension of these equations increase with the number of modeled properties. To formulate a suitable population balance model, two general strategies can be followed. Within the top-down approach the measured distributed data is modeled directly. In result, low dimensional population balance equations characterizing the particles heterogeneity with respect to a moderate number of measured properties are obtained. To account for the temporal change of the particle properties, global mechanistic kinetics are used. These have to be adapted to the distributed measurements by solution of an inverse problem. In contrast, bottom-up modeling starts from a detailed description of the mechanisms on the single particle level. In particular for bioprocesses, these may include a large number of particle properties. As each modeled property directly translates to an independent coordinate of the corresponding population balance model formulation, high dimensional partial differential equations are obtained. These require the application of efficient numerical solution techniques as standard discretization based solution methods lack effective implementation for high dimensional problems. Within this thesis, focus is on application of both modeling strategies to cell-culture based influenza A vaccine production processes. These represent an interesting example for particulate processes as cellular variability affects the overall virus yield. The first part is concerned with parameter estimation for a top-down model of the process which was developed in a previous work [81, 79]. The model describes the interaction of cell population and virus within a bioreactor and is able to account for cell-to-cell variability with respect to intracellular amount of viral nucleo protein (NP). As model adaption to flow cytometric data requires the estimation of both, constant and distributed parameters, an infinite dimensional inverse problem emerges. It is translated to a finite dimension by a suitable approximation of the distributed parameters. The parameters are estimated within a generalized least squares framework and the resulting model is able to qualitatively reproduce most of the dynamic effects, though not every trait of data is covered. It is also shown that the model fit can be improved when using more complex approximations of the distributed parameters. However, this comes along with an increased numerical effort and a biologically reasonable interpretation of the estimated parameters becomes more and more challenging. This lack of interpretability motivates the application of a bottom-up modeling approach which allows to take into account more structural information. The model formulation is based on a detailed description of the viral replication mechanism on the single cell level presented in [41]. The resulting high dimensional population balance model requires the application of an efficient numerical solution technique. Therefor, in the the second part, an efficient approximate moment method is developed which based on a combination of the direct quadrature method of moments [70] and a specific choice of the abscissas applying monomial cubatures [107]. The method is evaluated for several benchmark processes which can be represented by low dimensional population balance models. Furthermore, performance is extensively discussed for a generic five dimensional model of viral replication in a multi cellular system. The technique represents a good trade off between computational effort and accuracy of the solution and thus enables an efficient numerical solution of high dimensional population balance models. Finally, the method is applied to the high dimensional population balance model for influenza A virus replication in cell cultures. It is shown that an increasing cell-tocell variability negatively affects the overall virus yield. In addition, the application of genetically modified cell lines for improvement of the vaccine production is discussed. In practice these can be achieved using e.g. transduction techniques. However, not all cells are modified with the same efficiency. Therefore, a certain degree of variance w.r. t. the cell specific virus production rate is expected. A model based analysis shows that even for a relatively large heterogeneity a significant increase in the overall virus yield is obtained. “But still try, for who knows what is possible. ” Michael Faraday