Nikolaos V. Mantzaris

Numerical solution of a mass structured cell population balance model in an environment of changing substrate concentration

Cell population balance models are deterministic formulations which describe the dynamics of cell growth and take into account the biological fact that cell properties are distributed among the cells of a population, due to the operation of the cell cycle. Such models, typically consist of a partial integro-differential equation, describing cell growth, and an ordinary integro-differential equation, accounting for substrate consumption. A numerical solution of the mass structured cell population balance in an environment of changing substrate concentration has been developed. The presented method is general. It can be applied for any type of single-cell growth rate expression, equal or unequal cell partitioning at cell division, and constant or changing substrate concentration. It consists of a time-explicit, one-step, finite difference scheme which is characterized by limited requirements in memory and computational time. Simulations were made and conclusions were drawn by applying this numerical method to several different single-cell growth rate expressions. A periodic behavior was observed in the case of linear growth rate, equal partitioning and constant substrate concentration. The periodicity was equal to the average doubling time of the population. In all other cases examined, a state of balanced growth was reached. Unequal partitioning resulted in broader balanced growth distributions which are reached faster. For the specific types of growth rate dependence on the substrate concentration considered, the changing substrate concentration did not affect the balanced growth-normalized distributions, except for the case of linear growth rate and equal partitioning, where the depletion of the substrate destroyed the periodic behavior observed for constant substrate concentration, and forced the system to reach a steady state.

Numerical solution of multi-variable cell population balance models. II. Spectral methods

Abstract Several Galerkin, Tau and Collocation (pseudospectral) approximations have been developed for the solution of the multi-variable cell population balance model in its most general formulation, i.e. for any set of single-cell physiological state functions. Time-explicit methods were found to be more efficient than time-implicit methods for the time integration of the system of ordinary differential equations that results after the spectral approximation in space. The Legendre and Tchebysheff polynomials that were used in Tau algorithms were shown to have significantly worse convergence and stability properties than the Galerkin and collocation algorithms that were applied with sinusoidal trial functions. The collocation method that was implemented with discrete fast Fourier transforms was found to be the most efficient from all the Galerkin and Tau algorithms that were developed. However, the method was inferior to the best finite difference algorithm that was presented in our earlier work.

A new set of population balance equations for microbial and cell cultures

Abstract Population balance equations for microbial or cell cultures contain a state-dependent fission intensity function which is such that the product of this function, evaluated at a given state, and a differential increment of time is the fraction of cells of the given state that divide in that increment of time. Ways to determine experimentally how the fission intensity function depends on cell state have been proposed but, so far as is known to the authors, no one has yet proposed a model that would predict what the state dependence of the fission intensity function is for any population. If one wants to take account of the existence of cell cycle phases, one will have to write a population balance equation for each phase, a transition intensity function for each phase will have to be introduced, and the problem of making models for these functions will arise again, and in multiplied fashion. In this paper, we describe a new and different approach which circumvents the necessity of having intensity functions for transitions between cell cycle phases, and for which the fission intensity function is state-independent.

Nonlinear productivity control using a multi-staged cell population balance model

Abstract Cell growth processes are inherently nonlinear and as such are difficult to control. Moreover, due to the operation of the cell cycle, cell properties are distributed among the cells of a population. The only type of mathematical models that account for this heterogeneous nature of cell growth consists of the so-called cell population balance models. In this paper, we address the problem of controlling the productivity of a desired product using such a cell population balance model for the process description. It is assumed that cells grow in a continuous bioreactor, in two cell cycle stages. We further assume that the product is only being produced during the second stage of the cell cycle. We develop a cell population balance model consisting of a system of two partial differential equations, each describing the dynamics of cell growth during each of the two stages of the cell cycle, and two ordinary differential equations, describing the dynamics of the limiting substrate and product concentrations. The feed substrate concentration is considered as the manipulated input for achieving the control objective. Nonlinear and linear feedback laws that use measurements of the mass distributions of the two stages and the substrate and product concentrations, to induce a pre-specified output response are synthesized and are tested and compared through numerical simulations.

Control of Cell Mass Distribution in Continuous Bioreactors Using Population Balance Models

Abstract The problem of controlling different moments of the cell mass distribution by manipulating the dilution rate in a continuous bioreactor is studied. The mathematical model used for dynamic simulation and controller synthesis consists of a partial integro-differential equation describing a cell population balance and an ordinary integro-differential equation accounting for substrate consumption. Nonlinear feedback laws that induce a desired closed-loop response for the biomass concentration (first moment) and the total number of cells (zeroth moment) are developed and tested via simulations.