Calibration of piecewise Markov models using a Bayesian change-point analysis through an iterative convex optimization algorithm

Abstract

Purpose

Relative survival, as reported by the Surveillance, Epidemiology, and End Results (SEER) Program, represents cancer survival in the absence of other causes of death. Often, cancer Markov models have a distant metastasis state, a state not directly observed in SEER, from which cancer deaths are presumed to occur. The aim of this research is to use a novel approach to calibrate the transition probabilities to and from an unobserved state of a Markov model to fit a relative survival curve.

Methods

We modeled relative survival through a three-piecewise Markov model (i.e., with a specific Markov chain within each specified pieces) for stage 3 colorectal cancer patients. For each piece we used a constant transition matrix with three states: 1) recurrence free, 2) metastatic recurrence and 3) dead from cancer. We estimated the optimal cutoff time points using a Bayesian Markov chain Monte Carlo (MCMC) change-point model. This technique allowed us to estimate the time points at which the slope of the relative survival changes. We calibrated the transition probabilities using a two-step iterative convex optimization algorithm previously published. The dynamics of the disease can be defined as xt+1= xtM, where xt+1 is the state vector that results from the transformation given by the monthly transition matrix M. The matrix M is a piecewise block-diagonal matrix that includes in each piece a block-diagonal matrix for each Markov chain.

Results

We applied our method to calibrate a Markov model to fit a relative survival curve for stage 3 colorectal cancer patients younger than 75 years old. We compared our piecewise calibration method to a single-piece approach (i.e., a Markov chain). While the single-piece converged faster, the piecewise method improved the goodness of fit by 60%. The mean of the change points estimated from the Bayesian change-point model was at months 3 and 24 (see figure). The observed, and the piecewise and single-piece calibrated relative survival curves are shown in the figure.

Conclusions

By estimating the change points in the relative survival curve we were able to find the optimal transition probabilities for a piecewise Markov model that allowed us to impose a particular structure defined by the progression of the disease. We propose a piecewise calibration method that produces more accurate solutions compared to a single-piece approach.