An efficient approximate moment method for multi-dimensional population balance models – Application to virus replication in multi-cellular systems

Abstract

Many particulate processes in process and bioprocess engineering can be described with multi-dimensional population balances. Approximate moment methods are frequently used for their solution. In the present paper a new approach is presented, which is particular efficient when the number of internal coordinates is high. It combines the direct quadrature method of moments with monomial cubatures. With the new method the computational effort increases only polynomially, in the simplest case even only linearly with the number of internal coordinates, compared to an exponential increase for the well known Gausssian cubatures. The technique is evaluated for a five dimensional benchmark problem describing virus replication in continuous cell cultures. Furthermore, the algorithm is applied to analyze influenza virus replication in genetically modified cell lines.

A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations

“…Subsequently, estimators are applied to reconstruct means and (co)variances of these Gaussian distributions. Thereby, the approach guarantees high flexibility, as it is not limited to a specific numerical solution method like the above-mentioned discretization-based approach: Depending on the problem, tailored numerical schemes can be implemented to compute the integral quantities of the Gaussians, e.g., moment methods [20]. Moreover, one is not restricted to a specific type of estimator: in principle, any established state estimation or observer design technique is applicable.…”

Hybrid simulation algorithm for efficient numerical solution of population balance equations

“…More favorable is the last class of cubature formulas for which the sample size increases polynomially, in the best case even linearly, thus representing a significant improvement over the other two approaches. For a detailed discussion we refer to Dürr et al (2017) and the references therein. Special monomial non-product rules exploit certain characteristic properties of the initial distribution, e.g.…”

“…The main idea in this step of the proposed technique is to approximate these moments and the integral coupling by weighted sums over the characteristic scaled PBEs by applying an approximate moment method (a detailed discussion is found in Dürr et al (2017)). …”

“…In contrast to a pure moment method, our approach also allows to infer information on the full cell distribution. The approach also represents an extension of the algorithm presented in Dürr et al (2017), which is only applicable for multicellular systems with negligible cell division.…”

“…Such algorithms can be used to approximately compute integral dynamics of the cell number density distribution and the extracellular compounds, simultaneously. In this connection, the application of the direct quadrature method of moments in combination with monomial cubature formulas as presented by Dürr et al (2017) is particularly suited as it enables the efficient and accurate approximation.…”