Multi-state Continuous-Time Markov Chain Modeling for Chronic Kidney Disease Progression

Abstract

This paper presents a unified six-state Continuous-Time Markov Chain (CTMC) framework for Chronic Kidney Disease (CKD) progression, with CKD stages 1-5 modeled as transient states and death as an absorbing state. Under a non-homogeneous CTMC formulation, we derive integral representations for transition probabilities, state distributions, sojourn times, and survival-related quantities. We then study the homogeneous case as a tractable baseline and provide explicit formulas for key quantities. Although the methodology is rooted in standard multi-state theory, these expressions are often left implicit in applied analyses; here they are written out explicitly within a unified CKD framework. We construct covariate-dependent transition rates through a proportional hazards structure, using the standard identification of cause-specific hazards with CTMC transition rates. We fit the time-homogeneous baseline model to 335,283 longitudinal observations from 21,100 synthetic electronic health record patients by maximum likelihood. In this synthetic cohort, the covariate model improves held-out log-likelihood per transition over the null model, with stable performance across 10-times-repeated 5-fold cross-validation, and reproduces the main population-level prevalence patterns. The transition-specific estimates can also be translated into sojourn-time and survival summaries. The model suggests that male sex is associated with faster progression across nearly all CKD transitions, and that hypertension shows a stage-dependent association, with lower estimated transition rates in early stages but a substantial acceleration of the Stage 4 to Stage 5 transition. Overall, the proposed framework provides a mathematically explicit approach for studying CKD trajectories from longitudinal health records.