SOLUTIONS OF POPULATION BALANCE MODELS BASED ON A SUCCESSIVE GENERATIONS APPROACH

Microbial and cell cultures are composed of discrete organisms, each o f which goes through a cell cycle that terminates in production of new cells. The internal state of an individual cell changes as the cell p rogresses through the cell cycle, and randomness in various features o f the cell cycle always produces a distribution of cell states in the culture. Rigorous models of this situation lead to the so-called popul ation balance equations, which are integro-partial differential equati ons. These equations are notoriously difficult to solve, and the diffi culties increase as the number of parameters needed to describe cell s tate increases. The cells in a culture are of different generations, a nd cells of the (k+1)th generation originate only from divisions of ce lls of the kth generation. A population balance equation written for t he (k+1)th generation is therefore not an integral equation, although it contains a source term which is an integral over the distribution o f states of the kth generation. If competition of coexisting generatio ns for environmental resources does not affect growth and reproduction rates, the population balance equations for the various generations i n a culture do not have to be solved simultaneously but rather can be solved successively, and thus, some of the major difficulties of popul ation balance equations written for entire populations are circumvente d. In this paper, the successive generations approach to modeling is i llustrated by its application to two problems where cell state is desc ribed 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 ma ss, are used to describe cell state at the same time; Analytical solut ions of the population balance equations for the successive generation s are found for the cases discussed, and the solutions are used to cal culate the evolutions of the distributions of cell states with time fo r the single parameter cases. (C) 1997 Elsevier Science Ltd.