Nov M. Huang Northwestern Univ. 1 Markov Chain Population Models in Medical Decision Making Gordon Hazen Min Huang Northwestern University

Nov M. Huang Northwestern Univ. 2 Markov models (individual-level) in medical decision making Intervention that reduces disease mortality rate

Nov M. Huang Northwestern Univ. 3 Conventional outcome measure— QALYs for an individual (or a cohort)

Nov M. Huang Northwestern Univ. 4 From individual to population Motivation: To study a whole population 1.Equilibrium distribution of a population 2. Equilibrium measure of effectiveness of an intervention Individual-level models — 1. no equilibrium 2. no births

Nov M. Huang Northwestern Univ. 5 Augment model by allowing “births” Intervention that reduces disease mortality rate

Nov M. Huang Northwestern Univ. 6 Population model and its routing Population model Routing process

Nov M. Huang Northwestern Univ. 7 Population no longer dies out— reaches new equilibrium after intervention

Nov M. Huang Northwestern Univ. 8 Time-homogeneous individual-level Markov models {0,1,2,..,J,-1}, where ‘-1’ representing ‘Death’ is an absorbing state Individual Markov model State space Transition rates

Nov M. Huang Northwestern Univ. 9 Population models Population Markov model State space: Transition rates: — Open Jackson processes Serfozo Serfozo R. Introduction to Stochastic Networks. Springer 1999.

Nov M. Huang Northwestern Univ. 10 Routing processes {0,1,2,..,J,-1}, where ‘-1’ is a source/sink node Individual-level model State space: Transition rates

Nov M. Huang Northwestern Univ. 11 Properties If is irreducible, then at equilibrium: are independent, Conditional on total population size |n|, n is multinomial equilibrium population means equilibrium population proportions )~Poisson(

Nov M. Huang Northwestern Univ. 12 Equilibrium population means. is the unique collection of positive numbers that satisfy balance equations of routing process i.e. Here Q is a submatrix of the rate matrix of the routing process, and also a submatrix of the rate matrix of the underlying individual model, corresponding to all nonabsorbing states, i.e., health states {0,1,…,J}.

Nov M. Huang Northwestern Univ. 13 What measures of quality are possible at the population level? Equilibrium population measures Individual QALYs : QALYs for an individual starting in state j Measures of health

Nov M. Huang Northwestern Univ. 14 Average Lifetime QALY ALQ Mean QALY of randomly selected individual from equilibrium population

Nov M. Huang Northwestern Univ. 15 Total Lifetime QALY TLQ Mean total QALYs of all individuals in equilibrium population

Nov M. Huang Northwestern Univ. 16 Average QALYs per Year AQ/yr One-year QALY of randomly selected individual from equilibrium population

Nov M. Huang Northwestern Univ. 17 Total QALYs per Year TQ/yr One-year QALY of all individuals in equilibrium population

Nov M. Huang Northwestern Univ. 18 Discounted Total QALYs DTQ Mean total discounted QALYs for this and all subsequent generations of population. Discount rate = 3%

Nov M. Huang Northwestern Univ. 19 DTQ TLQ TQ/yr AQ/yr Relationships between measures if the population is in equilibrium from t=0. ALQ AQ/yr TQ/yr

Nov M. Huang Northwestern Univ. 20 The simple illustrative example— differences among measures Intervention that reduces disease mortality rate

Nov M. Huang Northwestern Univ. 21 Evaluating interventions using these measures:

Nov M. Huang Northwestern Univ. 22 Problem: average measures do not account for population size increase due to better survival. Caution in choosing population measures Insight

Nov M. Huang Northwestern Univ. 23 Example: tamoxifen use to prevent breast cancer Col Col N.F., Orr R.K., Fortin J.M. Survival impact of tamoxifen use for breast cancer risk reduction: projections from a patient-specific Markov model, Med Decis Making 2002; 22:

Nov M. Huang Northwestern Univ. 24 Non-homogeneous individual-level Markov models 1. Human background survival Background mortality rate 2. The other factor : a homogeneous Markov process (Gompertz)

Nov M. Huang Northwestern Univ. 25 Population models Mean density with respect to age a of the population in state j at time t: Theorem:

Nov M. Huang Northwestern Univ. 26 equilibrium mean density with respect to age a of the population in state j, equilibrium expected total population count in state j. Notations: Conclusions:

Nov M. Huang Northwestern Univ. 27 : QALYs for an individual starting from age a 0 in state j Individual QALYs Measures of health Equilibrium population measures ALQ TLQ AQ/yr TQ/yr TLQ

Nov M. Huang Northwestern Univ. 28 Example: tamoxifen use to prevent breast cancer Col

Nov M. Huang Northwestern Univ. 29 Summary Population Markov models for medical decision making. Population measures of interventions Age-dependency.