Complete remission is a major objective of leukemia chemotherapy, but treatment progress cannot be verified below a cancer cell detection threshold. As such, mathematical models of cancer cell growth are useful in predicting the effects and to compare the expected population dynamics for possible regimens. However, deviations from expected behavior become significant when the cancer cell population nears extinction but deterministic models, which describe average cell behavior, give no information on the variance about the expected behavior. This presents a quandary when these models are used for treatment evaluation because, while lower expected cell numbers likely correlate with better treatment outcomes, these differences cannot quantify the treatment outcome. What is of interest is the likelihood of complete eradication of the cancer or remission durations. To answer this question, we adopt the concept of stochastic analysis (Fredrickson 1966) by using stochastic formulations of population balances (Ramkrishna 2000) to account for cell number probability distributions. To illustrate the importance of the stochastic considerations, we compare the likelihood of remission durations for potential treatment regimens by calculating higher order product densities for growth/death models such as exponential, Gompertzian, and Fister-Panetta (2000). The technique is then applied to a multi-stage population balance model which we have developed specifically for leukemia chemotherapy (Sherer et al. In Press). The potency of many chemotherapeutic drugs is cell cycle dependent with residual effects on the cell cycle phase transition rates. As such, an age-structured population balance model of the cell cycle describes the dynamics of cell response to a drug. The G0/G1, S, and G2/M are modeled explicitly with transition rates between phases altered by the drug and age used to encompasses the biochemistry involved in phase transitions. This model has been verified experimentally using HL60 and Jurkat leukemia cell lines exposed to the S phase specific drug camptothecin. Also, a methodology to extract age-dependent transition rates using BrdU pulse-labeling has been developed.

References:

Fister KR and Panetta JC. “Optimal control applied to cell-cycle-specific cancer chemotherapy.” SIAM J on Appl Math, 60: 1059-1072, 2000.

Fredrickson AG. “Stochastic models for sterilization.” Biotech and Bioeng, 8: 167-182, 1966.

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

Sherer E, Hannemann RE, Rundell A, and Ramkrishna D. “Analysis of resonance chemotherapy in leukemia treatment via multi-staged population balance models.” J Theo Bio, In Press.