Age -structured cell models in the treatment of leukemia: Identification, inversion, and stochastic methods for the evaluation and design of chemotherapy protocols
Mathematical models for the evaluation of leukemia chemotherapy treatments have been investigated. These include models for: cell cycle-specific therapy, the identification of patient-specific active drug metabolite levels, and the quantification of treatment effectiveness. ^ Many of these treatments involve cell cycle specific chemotherapeutic agents, and a cell growth model is developed to explicitly account for cell cycle effects. The timing of transition between phases will vary between cells and these variations are accounted by age-structured rates. Because age cannot be directly measured, age-structured parameters can be difficult to identify, a methodology is developed for the identification of age-structured cell cycle phase transition rates for a culture undergoing balanced growth by labeling with bromodeoxyuridine. ^ The cancer cell population size will also depend on the death rates where inter-patient variability leads to different death rates for identical drug treatment protocols. A methodology is proposed that uses the dynamics of the mean corpuscular volume of red blood cells as a surrogate marker for the likely active metabolite level of purine synthesis inhibitory agents. This inversion relies heavily on a model that can accurately describe the size dynamics of the maturation of red blood cells in the bone marrow. A preliminary model has been created using literature values for the inter-mitotic and inter-maturation stage residence time distributions, and the presence of an inhibitor does produce increases in the mean corpuscular volume. ^ The final model calculates the moments of the cell number probability distribution. These moments are used to approximate the distribution where the probability of zero cells is the cure rate. The first and second moments provide an indication of the treatment necessary to likely remove all cancer cells. Additional moments are needed to calculate the full cell number probability distribution, but computational limits restrict the order of product density equations. Both yeast-like and binary division models are analyzed and to show how the product density equations can be averaged so that rate parameters for the higher order equations can be approximated using lower order equations. ^
Doraiswami Ramkrishna, Purdue University, Robert E. Hannemann, Purdue University.