Probability and Mean Time of Eradication in Stochastic Models of Tumor Growth and Treatment

The quantitative assessment of tumour eradication under therapy is a key problem in mathematical oncology, since stochastic fluctuations become particularly relevant when the tumour burden is low and may ultimately determine whether extinction occurs. In this work we propose a method to estimate the probability of tumour eradication and the corresponding mean eradication time in stochastic models of tumour growth under treatment. The modelling framework is based on the Chemical Reaction Network (CRN) formalism, which allows tumour dynamics to be represented as a Continuous-Time Markov Chain. By exploiting a quasi-steady-state approximation (QSSA), the original multi-dimensional stochastic model is reduced to a one-dimensional birth-death process governing the proliferating tumour cell population. Within this reduced framework, explicit expressions for the eradication probability and, when meaningful, for the mean eradication time are derived as functions of the model propensities. Unlike previous approaches, the proposed method does not require linearity assumptions on the reduced propensities and relaxes restrictive steady-state hypotheses on the remaining model variables. The effectiveness of the approach is illustrated through simulations on a tumour growth model from the literature, showing how the quasi-steady-state reduction captures qualitative behaviours that may be missed by earlier steady-state approximations.