Relationship between prostate-specific antigen, alkaline phosphatase levels, and time-to-tumor shrinkage: understanding the progression of prostate cancer in a longitudinal study

This study delves into the complex interplay among prostate-specific antigen, alkaline phosphatase, and the temporal dynamics of tumor shrinkage in prostate cancer. By investigating the longitudinal trajectories and time-to-prostate cancer tumor shrinkage, we aim to untangle the intricate patterns of these biomarkers. This understanding is pivotal for gaining profound insights into the multifaceted aspects of prostate cancer progression. The joint model approach serves as a comprehensive framework, facilitating the elucidation of intricate interactions among these pivotal elements within the context of prostate cancer . A new joint model under a shared parameters strategy is proposed for mixed bivariate longitudinal biomarkers and event time data, for obtaining accurate estimates in the presence of missing covariate data. The primary innovation of our model resides in its effective management of covariates with missing observations. Built upon established frameworks, our joint model extends its capabilities by integrating mixed longitudinal responses and accounting for missingness in covariates, thus confronting this particular challenge. We posit that these enhancements bolster the model's utility and dependability in real-world contexts characterized by prevalent missing data. The main objective of this research is to provide a model-based approach to get full information from prostate cancer data collected with patients' baseline characteristics ( , body mass index ( ), , , and ) and two longitudinal endogenous covariates ( and ). The results reveal a clear association between prostate-specific antigen and alkaline phosphatase biomarkers in the context of time-to-prostate cancer tumor shrinkage. This underscores the interconnected dynamics of these key indicators in gauging disease progression. The analysis of the prostate cancer dataset, incorporating a joint evaluation of mixed longitudinal prostate-specific antigen and alkaline phosphatase biomarkers alongside tumor status, has provided valuable insights into disease progression. The results demonstrate the effectiveness of the proposed joint model, as evidenced by accurate estimates. The shared variables associated with both longitudinal biomarkers and event times consistently deviate from zero, highlighting the robustness and reliability of the model in capturing the complex dynamics of prostate cancer progression. This approach holds promise for enhancing our understanding and predictive capabilit