Solutions of population balance models based on a successive generations approach

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Microbial and cell cultures are composed of discrete organisms, each of which goes through a cell cycle that terminates in production of new cells. The internal state of an individual cell changes as the cell progresses through the cell cycle, and randomness in various features of the cell cycle always produces a distribution of cell states in the culture. Rigorous models of this situation lead to the so-called population balance equations, which are integropartial differential equations. These equations are notoriously difficult to solve, and the difficulties increase as the number of parameters needed to describe cell state increases. The cells in a culture are of different generations, and cells of the ( k + 1)th generation originate only from divisions of cells of the kth generation. A population balance equation written for the ( k + 1)th generation is therefore not an integral equation, although it contains a source term which is an integral over the distribution of states of the kth generation. If competition of coexisting generations for environmental resources does not affect growth and reproduction rates, the population balance equations for the various generations in a culture do not have to be solved simultaneously but rather can be solved successively, and thus, some of the major difficulties of population balance equations written for entire populations are circumvented. In this paper, the successive generations approach to modeling is illustrated by its application to two problems where cell state is described by a single parameter, either cell age or cell mass. It is then applied to a problem where two parameters, namely cell age and cell mass, are used to describe cell state at the same time. Analytical solutions of the population balance equations for the successive generations are found for the cases discussed, and the solutions are used to calculate the evolutions of the distributions of cell states with time for the single parameter cases.