In previous work, age-structured population balances were used to capture leukemia cell dynamics in response to cell cycle specific chemotherapeutic agents (Sherer et al. 2006). The experimentally verified model was then used to optimize treatment scheduling. However, it was noted at the time that the objective criterion used in the optimization – minimizing the expected number of leukemia cells – was only an indicator of what truly quantifies chemotherapy outcomes: the probability of cure. That is, what is the likelihood that zero cancerous cells remain after a treatment regimen? In this presentation, we demonstrate a methodology – which utilizes stochastic equations of population balances – by which the likelihood of having an actual number of cells can be approximated for age-structured models. The technique is applied to a mother-daughter cell division model and a cell cycle model to predict cure rates for hypothetical chemotherapy treatments. The simulation results and computation times are also compared with Monte Carlo results to demonstrate the utility of the method.

Determining exactly when cure occurs is impossible because the timing of cell births, deaths, or other transitions are uncertain. So, the actual number density (the discrete list of each cell and its age) is only known probabilistically (Ramkrishna and Borwanker 1974). When large number of cells are considered, these fluctuations tend to average out and the expected number density (whose behavior is governed by the population balance equation) adequately describes the system. However, variations are inherently present and become more significant as the population size declines – the case in chemotherapy treatments. More generally, such information is necessary to quantify the state of any system in which significant variations from the expected behavior are anticipated.

In answering the question of the likelihood of cure, we show how the stochastic equations of the population balance are derived from the master (or Janossy) density where each higher order equation provides additional information – in the form of an additional moment - about the master density (Ramkrishna 2000). We then show how the behavior of the equations converges as the equation order increases. In fact, only the third or fourth order stochastic equations give a sufficient approximation of all higher order equations in the examples considered. The approximated solutions of the stochastic equations are used to approximate the moments of the actual number density which are then used to approximate the actual number density. Excellent agreement in the actual number density is seen between the approximate solution and multiple Monte Carlo simulation where the stochastic equation simulation have a considerable computation time advantage.

References:

Ramkrishna D and Borwanker JD. “A puristic analysis of population balance II”. Chemical Engineering Science 29: 1711-1721, 1974.

Ramkrishna D, Population balances: Theory and applications to particulate systems in engineering. Academic Press, San Diego, CA, 2000.

Sherer E, Hannemann RE, Rundell A, and Ramkrishna D, “Analysis of Resonance Chemotherapy in Leukemia Treatment via Multi-Staged Population Balance Models.” Journal of Theoretical Biology, 240: 648-661, 2006.