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.