Cancer stem cells (CSCs) represent a novel therapeutic target that may revolutionize the treatment of cancer. While they comprise only a small proportion of the total tumor cell population, CSCs have been shown to both originate all cell types found in a malignant tissue and drive cancer progression, raising interesting quantitative issues. While the eradication of CSCs could lead to cure, current challenges in designing targeted therapy include CSC quiescence, heterogeneity in genetic expression and phenotype of CSCs, and the need to ensure that the therapy largely spares the normal healthy stem cells (HSCs) required for tissue repair and maintenance. We propose a birth-death process model of stein cell population dynamics to examine the time to eradication of CSCs. Using extreme value theory from mathematical statistics, we derive an accurate asymptotic distribution and corresponding moments for the extinction times of both CSCs and HSCs, and compare these distributions as a function of killing rates. By conditioning on the asymptotic time to extinction of the cancer stem cells, we calculate the asymptotic mean and variance of N_H the number of HSCs remaining at the extinction time of the CSCs.

We obtain the full distribution of N_H analytically using the finite Fourier transform and eigenfunction expansion and compare with stochastic simulation with good agreement. Stochastic simulation is perforated using a novel update to τ-leaping that provides improved accuracy. We account for stem cell quiescence with both analytic methods and stochastic simulation and conclude that successful therapy must target both quiescent and actively dividing CSCs. Heterogeneity of CSCs is modeled using a multi-type branching process model with each type representing a different mutational status. Finally, we examine the joint effects of targeted CSC therapy and niche signaling inhibitors and predict that the combination could lead to cure.