Journal of Advances in Mathematics and Computer Science

Health-seeking delay and rapid disease progression both contribute to prostate cancer mortality. Whether the underlying prostate-specific antigen (PSA) trajectory is associated with mortality independently of health-seeking delay remains unclear. This retrospective cohort study used a two-stage joint modelling approach using secondary data from the National Cancer Registry of Kenya. It examined 420 patients diagnosed with prostate cancer between January 2016 and December 2025. In the first stage, a linear mixed-effects submodel estimated each patient’s underlying trajectory of log(PSA + 1) as a function of stage at diagnosis, Gleason score, and time since diagnosis. The fitted trajectory and its estimated slope were then entered as time-varying covariates into a Cox survival submodel, alongside total delay and baseline covariates. Stage at diagnosis was strongly associated with the baseline level of the PSA trajectory (metastatic versus localised: coefficient = 2.87, ) P <.001. In the survival submodel, the fitted PSA trajectory was independently associated with mortality hazard ( HR = 2.08 per unit increase in log-PSA trajectory; 95% CI 1.43–3.03; P<.001), while total delay remained an independent predictor (HR = 1.045; 95% CI 1.01–1.08; P = .006). The PSA slope term was also statistically significant (coefficient = −29.37; 95% CI −45.84 to −12.90, per 1 log-PSA unit per month). This estimate is not clinically interpretable because of its scale, its counterintuitive sign, and the limitations of the two-stage approximation. PSA trajectory and health-seeking delay appeared to act as distinct, additive risk pathways. Interventions that shorten delay and interventions that intensify monitoring of patients with adverse PSA trajectories may therefore address complementary components of mortality risk. A full shared-parameter joint likelihood, fitted with dedicated joint-modelling software, is recommended before these effect sizes are applied to clinical risk stratification.