DOES SIZE MATTER? A METHOD FOR ADAPTIVELY DETERMINING CYCLE LENGTH IN A STATE-TRANSITION MODEL

Purpose: Markov models in decision-analytic studies require a fixed cycle length (CL).  Using a uniform CL may introduce bias when competing events in the model operate at different time scales.  Events occurring at a higher rate may necessitate shorter CL compared to those with lower rates. We introduce a method for simulating continuous-time model with adaptive time steps borrowed from chemical kinetics to evaluate the potential bias due to having fixed CL.

Methods: We compared two simulation methods: fixed vs. adaptive CL, and compared them against the gold standard of continuous time, Gillespie's method (GIL).  The Tau-leap method (TAU) adaptively determines the largest CL based on the assumption that the transition rates would not change significantly during the proposed CL.  The calculation of the next CL depends only on the current states and the potential change in the state distribution.  For fixed CL, we used a Markov cohort model (CM), which progresses in fixed time steps.   To compare the methods, we used a breast cancer toxicity model to simulate a cohort of 10,000 50-year-old women with local recurrence and compared the gain in life expectancy (gLE) from chemotherapy vs. no chemotherapy. Bias is defined as the percentage difference in the gLE of a method compared to that of GIL.  CM used 1-year CL while TAU used adaptive CL. We also compared metastasis-free-survival (MFS) across all methods.

Results: CM is the fastest method (0.002 seconds per cohort run) followed by TAU (5 seconds) and GIL (11 seconds).  The range of CL generated by TAU is from 2.4 to 11 days with a mean of 3.1 days.  CM generated higher MFS and LE compared to TAU and GIL over a 50-year time horizon.  TAU estimated LE more accurately compared to CM (difference of three days vs. three months).  TAU and CM methods produced biases in gLE of 0.3% and 2.5%, respectively.  

Conclusion: Model simulation with adaptive CL is computationally inexpensive and may reduce bias in key clinical outcomes driven by competing events.  If the size of the bias is approximately equal to the gains in clinical outcomes due to an intervention, then TAU method should be adopted.  Additionally, TAU method may inform modelers of the correct CL to be used in the cohort models to minimize bias.