Abstract
The first-passage-time (FPT) that a tumor size reaches a particular barrier is important in evaluating the efficacy of anti-cancer therapies and understanding certain oncological time occurrences. For certain verified stochastic models describing the volume of a tumor, a moving barrier for the tumor size in which an explicit solution of an FPT probability density function (PDF) exists for the first time the tumor size reaches the moving barrier is obtained in this work. The stochastic tumor dynamics incorporate anti-cancer therapies/treatments that are administered at varying rates. The first-passage-time density (FPTD) is derived and utilized to determine the time at which the tumor volume first reaches the moving barrier, providing a framework for analyzing various oncological time metrics. These metrics include key time measurements used to characterize tumor progression, evaluate treatment response, and capture recurrence patterns in cancer dynamics. The treatment effort needed to cause reduction in tumor size is also obtained. We obtained, for a tumor growing initially, the FPTD for the random variables describing the first time that the growth of the tumor starts slowing down following the commencement of treatment, the first time that the tumor starts showing signs of shrinkage after the start of treatment, the first time it takes for the reduction in tumor to start slowing down, and the first time for tumor recurrence after partial remission. This work is applied to experimental data including the Murine Lewis Lung Carcinoma cells originally derived from a spontaneous tumor in twenty control mice. The time at which the volume of the tumor of each mouse doubles in size is estimated using the results obtained in this study. Additionally, tumor volume experiments conducted on another eight control mice are used to validate the findings derived in this study.