Multistage population balances provide a more detailed mathematical description of cellular growth than lumped growth models, and can therefore describe better the physics of cell evolution through cycles. These balances can be formulated in terms of cell age, mass, size or cell protein content and they can be univariate or multivariate. A specific three stage population balance model based on cell protein content has been derived and used recently to simulate evolution of cell cultures for several applications. The behavior of the particular mathematical model is studied in detail here. A one equation analog of the multistage model is formulated and it is solved analytically in the self-similarity domain. The effect of the initial condition on the approach to self-similarity is studied numerically. The three equations model is examined then by using asymptotic and numerical techniques. It is shown that in the case of sharp interstage transition the discontinuities of the initial condition are preserved during cell growth leading to oscillating solutions whereas for distributed transition, the cell distribution converges to a self-similar (long time asymptote) shape. The closer is the initial condition to the self similar distribution the faster is the convergence to the self-similarity and the smaller the amplitude of oscillations of the total cell number. The findings of the present work lead to a better understanding of the multistage population balance model and to its more efficient use for description of experimental data by employing the expected solution behavior.
['Margaritis Kostoglou', 'María Fuentes-Garí', 'David García-Münzer', 'Michael C. Georgiadis', 'Nicki Panoskaltsis', 'Efstratios N. Pistikopoulos', 'Athanasios Mantalaris']
A comprehensive mathematical analysis of a novel multistage population balance model for cell proliferation