A Quasi Birth-and-Death Model For Tumor Recurrence

Abstract

A major cause of chemoresistance and recurrence in tumors is the presence of dormant tumor foci that survive chemotherapy and can eventually transition to active growth to regenerate the cancer. In this paper, we propose a Quasi Birth-and-Death (QBD) model for the dynamics of tumor growth and recurrence/remission of the cancer. Starting from a discrete-state master equation that describes the timedependent transition probabilities between states with different numbers of dormant and active tumor foci, we develop a framework based on a continuum-limit approach to determine the time-dependent probability that an undetectable residual tumor will become large enough to be detectable. We derive an exact formula for the probability of recurrence at large times and show that it displays a phase transition as a function of the ratio of the death rate µ A of an active tumor focus to its doubling rate λ. We also derive forward and backward Kolmogorov equations for the transition probability density in the continuum limit and, using a first-passage time formalism, we obtain a drift-diffusion equation for the mean recurrence time and solve it analytically to leading order for a large detectable tumor size N . We show that simulations of the discretestate model agree with the analytical results, except for O(1/N ) corrections. Finally, we describe a scheme to fit the model to recurrence-free survival (Kaplan-Meier) curves from clinical cancer data, using ovarian cancer data as an example. Our model has potential applications in predicting how changing chemotherapy schedules may affect disease recurrence rates, especially in cancer types for which no targeted therapy is available.model is formulated in terms of a continuous-time master equation in the discrete state space that represents the numbers of dormant and active tumor foci. In Section 4, it is shown that an expansion of the master equation for a large detectable-tumor size N leads to a simplified approach by mapping the original discretestate model to a stochastic process in a continuous two-dimensional state space. In Section 5, we find the large-time probability of recurrence in closed analytic form and calculate the mean recurrence time (MRT) 70 analytically to leading order in N . In Section 5, we compare these analytical results to simulations and describe a scheme to fit the model to recurrence-free survival (Kaplan-Meier) curves from clinical cancer data, using ovarian cancer data as an example. Finally, in Section (6) we present our concluding remarks. Overview of the discrete-state modelThe precise discrete model for tumor recurrence that will be described in this section was inspired by 75 previous work on the effect of quiescence (i.e., the presence of dormant tumor foci) on treatment success, such as the work of Komarova and Wodarz [8], which inspired the stochastic model described below, or to deterministic versions of their model, proposed in [9], [10], or [11]. However, in contrast to these earlier studies, the focus of our paper is on finding a relatio...

Prevention of Colon Cancer Recurrence From Minimal Residual Disease: Computer Optimized Dose Schedules of Intermittent Apoptotic Adjuvant Therapy

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