Estimating relapse time distribution from longitudinal biomarker trajectories using iterative regression and continuous time Markov processes

Biomarker measurements obtained by blood sampling are often used as a non-invasive means of monitoring tumour progression in cancer patients. Diseases evolve dynamically over time, and studying longitudinal observations of specific biomarkers can help to understand patients response to treatment and predict disease progression. We propose a novel iterative regression-based method to estimate changes in patients status within a cohort that includes censored patients, and illustrate it on clinical data from myeloma cases. We formulate the relapse time estimation problem in the framework of Piecewise Deterministic Markov processes (PDMP), where the Euclidean component is a surrogate biomarker for patient state. This approach enables continuous-time estimation of the status-change dates, which in turn allows for accurate inference of the relapse time distribution. A key challenge lies in the partial observability of the process, a complexity that has been rarely addressed in previous studies. We evaluate the performance of our procedure through a simulation study and compare it with different approaches. This work is a proof of concept on biomarker trajectories with simple behaviour, but our method can easily be extended to more complex dynamics.