Mathematical analysis of multistage population balances for cell growth and death
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abstract
2015 Elsevier B.V. The cell cycle is a biologically timed process by which cells duplicate. It consists of 4 phases, during which cells undergo different mandatory transformations. Modelling the cell cycle therefore requires capturing the evolution of those processes inside of each phase. A specific three stage biologically supported population balance model employed to simulate evolution of several cell cultures is studied here in detail. The three governing equations of this model are composed by growth and transition terms. A one equation analogue of the multistage model is formulated and it is solved analytically in the self-similarity domain. The effect of initial conditions at the system evolution is studied numerically. The three model equations are then considered by using asymptotic and numerical techniques. It is shown that in the case of sharp interstage transition the discontinuities of the initial conditions are preserved during cell growth leading to eternal oscillations whereas for distributed transition the cell distribution converges to a self-similar (long time asymptote) shape. It is also illustrated that the closer the initial condition to the self similar distribution the faster the convergence to self-similarity and the smaller the oscillations of the total cell number. Exact results are given for the growth parameters of the population balance and lumped models.
