Application of stochastic equations of population balances to sterilization processes

This work explores concepts for describing the uncertainty in the total number of cells when cell birth and death rates are age-structured. These ideas are particularly relevant for sterilization processes when the number of cells decreases to levels where the variation in the number of cells is comparable to the expected number. Uncertainty in the number of cells is always present due to the random timing of the birth and death events, but the age-dependence of rates implies that all cells cannot be treated equally. Moreover, due to the age-dependence of the rate functions, correlations between the ages of cells in the population develop and divergence from the expected number density (and expected total number of cells) occurs. Accounting for these variations explicitly is computationally cumbersome, but it is shown how higher order product densities—or averages of the actual number density—provide information about the age interactions. The integrated total product densities provide information about the cell number probability distribution in the form of its moments which are used to describe the distribution qualitatively and approximate it quantitatively. Two age-structured models are examined: (1) a continuous birth model where mother cells continuously age while giving birth to new daughter cells of age zero and (2) a mother and daughter cell model where the mother cell's age is reset to zero after giving birth to a daughter cell of age zero, but the two cells may have different birth and death rates. In both cases it is shown that no more than the three lowest order product densities are necessary to approximate the cell age correlations when a closure approximation is used for calculating higher order total product densities. In fact, either the first or the first and second order product densities are sufficient in several cases examined herein. For quantitative validation of the methodology, the first 10 moments of the cell number probability distribution are calculated by applying the closure approximation and the distribution fit to these moments. Approximations of both the probabilities of a population having zero cells or less than 10 cells are shown to be in good agreement with Monte Carlo simulations. This methodology has a wide range of potential applications from quantifying potential cancer chemotherapy treatment models to testing models of food decontamination procedures.