1. QUANTITATIVE RISK STRATIFICATION IN MARKOV CHAINS WITH QUASI-STATIONARY DISTRIBUTIONS David C Chan, MD, MSc 1 ; Philip K Pollett, PhD 2 ; Milton C Weinstein, PhD 3 1 Division of General Internal Medicine, Brigham and Women’s Hospital, Boston, MA; 2 University of Queensland, Queensland, Australia; 3 Harvard School of Public Health, Boston, MA. Contact Information: David Chan, MD, MSc Brigham & Women’s Hospital 75 Francis Street, Boston, MA 02115 (617) 732-6660 E-mail: dcchan@partners.org BACKGROUND PURPOSE METHODS RESULTS CONCLUSIONS DISCUSSION  Markov chains are frequently used to model disease processes in decision analysis.  A Markov chain in which all living patients eventually die is called a transient chain with an absorbing state (death).  Quasi-stationary distributions represent the limiting probability conditional on no absorption (the eventual probability of a surviving patient being in a certain state).  We considered a general Markov chain with N living states and an absorbing death state D.  Using Markov chain theory, we solved for the conditions under which a positive quasi- stationary distribution exists.  We considered the specific example of targeting risky heart failure patients, and we explored the use of quasi-stationary distributions to quantify risk.  In clinical practice, treatment decisions often rely on the concept of risk stratification.  The decision in these cases is not whether all patients should be treated as a group, but how many patients should be treated when the group is risk-stratified.  We considered a general Markov chain with living states L 1, …, L N and an absorbing death state D.  A unique positive quasi-stationary distribution exists for the living states, if and only if patients in L 1 are at lowest risk for progression or death.  As an example, heart failure patients with greater previous hospitalizations (H) are more likely to die or be hospitalized. A positive quasi-stationary distribution therefore exists for surviving patients.  This quasi-stationary distribution allows for quantitative risk stratification of living patients.  Outcomes for quantitatively defined populations at risk (e.g. “highest quintile”) can be evaluated.  Risk can be inferred from observed outcomes, and expected outcomes can be extrapolated from theoretical risk. L1L1 L2L2 L3L3 LNLN D · · · Quasi-stationary distributions exist in Markov chains with properties that are natural for many disease models.  Quasi-stationary distributions allow for quantitative definitions of risk within an established framework of Markov chain modeling.  Patient risk in trials can be quantitatively inferred from observed outcomes. Outcomes can then be extrapolated to populations of different risk.  Limitation: Quasi-stationary distributions do not account for population influx. Risk stratification is a necessary part of many clinical decisions. Quantitative risk stratification is made possible by quasi-stationary distributions intrinsic in many Markov chains. This allows for communication about decisions for precisely how many patients, rather than simplistic ones for all or none.