Nov. 2005 M. Huang Northwestern Univ. 1 Markov Chain Population Models in Medical Decision Making...

Nov. 2005 M. Huang Northwestern Univ. 1

Markov Chain Population Models in

Medical Decision Making

Gordon HazenMin Huang

Northwestern University


Markov models (individual-level)in medical decision making

Intervention that reduces disease mortality rate


Conventional outcome measure—QALYs for an individual (or a cohort)

x 1 = 12.2

x 0.75 = 3.96

16.16 QALY

12.20 yr

5.28 yr

x 1 = 12.2

x 0.75 = 13.21

25.41 QALY

12.20 yr

17.62 yr


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 equilibrium2. no births


Augment model by allowing “births”



Population model and its routing

λ0

#Well #Diseased

0

μ0 μ1

λ0

Well Diseased

0

μ0 μ1

Population model

Routing process


Population no longer dies out—reaches new equilibrium after intervention


Jjjq

kjJkjkjq

j

jk

,...,0 ,)1,(

,,...,0, ,),(

Time-homogeneous individual-level Markov models

}0),({ ttX

{0,1,2,..,J,-1}, where ‘-1’ representing ‘Death’ is an absorbing state

Individual Markov model

State space

Transition rates


Population models

Population Markov model }0),({ ttn

State space:

Transition rates:

},...,1,0interger negativenon a is :),...,,{( 110

1 J, jnRnnnZ jJ

JJ

jj

jjj

jjkjk

nTnq

kjJkjnnTnq

nnTnq

),(

,,...,0, ,),(

),(

,1

1,

— Open Jackson processes Serfozo

Serfozo R. Introduction to Stochastic Networks. Springer 1999.


Routing processes

}0},1,,...,1,0{)({ tJtr

{0,1,2,..,J,-1}, where ‘-1’ is a source/sink node

Individual-level model

State space:

Transition rates

,),1(

,,...,0, ,)1,(

,),(

j

j

jk

jq

kjJkjjq

kjq


PropertiesIf }0),({ ttr is irreducible, then at equilibrium:

Jnnn ,...,, 10• are independent,

Jjj ,...,0,

• Conditional on total population size |n|, n ismultinomial )~,...,~,~|,(| 10 Jn

0

jj J

jj

equilibrium population means

equilibrium population proportions

j )jn ~Poisson(


Equilibrium population means

J

jkk

kjkj

J

jkk

jkjj00

)( Jj ,...,0

.

is the unique collection of positive numbersJjj ,...,0,

that satisfy balance equations of routing process

1 Q

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}.


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

jG


x = 10.4

41%

59%x = 12.3

22.7 QALY

25.41

20.83

Average Lifetime QALY

ALQ

J

jjjG

0

~

Mean QALY of randomly selected individual from equilibrium population


Total Lifetime QALY

TLQ

Mean total QALYs of all individuals in

equilibrium population

x = 310

12.2

17.6x = 367.1

677 person-QALYs

25.41

20.83

J

jjjG

0


Average QALYs per Year

AQ/yr One-year QALY of randomly selected individual from equilibrium population

41% x 1.00 = 0.41

59% x 0.75 = 0.44

0.85 QALY/yr

J

jjjv

0

~


12.2 x 1.00 = 12.2

17.62 x 0.75 = 13.22

25.42 person-QALY/yr

Total QALYs per Year

TQ/yr

One-year QALY of all individuals in equilibrium population

J

jjjv

0


Discounted Total QALYs

DTQ

Mean total discounted QALYs for this and all subsequent generations of population.

406.5041 x 1.00 = 406.5

176.1518 x 0.75 = 132.1

538.6 person-QALYs

406.5 yr

176.2 yr

406.5041 x 1.00 = 406.5

400.3449 x 0.75 = 300.3

706.8 person-QALYs

406.5 yr

400.3 yr

Discount rate = 3%

0 ,

1,00

0

)](),()([ tdnjivedtvtneE ji

J

jiji

rtJ

jjj

rt


DTQr

1

TLQ

J

jj

0

TQ/yr

J

jj

0

TG

TT LG

/

TQ/yr

AQ/yr

Relationships between measures

if the population is in equilibrium from t=0.

ALQ

AQ/yr

TQ/yr


The simple illustrative example— differences among measures



Evaluating interventions using these measures:


• Problem: average measures do not account for population size increase due to better survival.

• Caution in choosing population measures

Insight


Example: tamoxifen use to prevent breast cancerCol

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: 386-393.


Non-homogeneous individual-levelMarkov models

1. Human background survival

}0},,{)({ tdeadalivetB

Background mortality rate )(1

0100 )( tabba ebt

2. The other factor: a homogeneous Markov process

}0},1,,...,1,0{)({ 0 tJtX

(Gompertz)


Population models

Mean density with respect to age a of the population in state j at time t:

),( tam j

Theorem:

),0(

)(),(),(),(

tm

aQtama

tam

t

tam


equilibrium mean density with respect to age a of the population in state j,

equilibrium expected total population count in state j.

Notations:

)(aj

j

),()( aPa

)(* aP

Conclusions:


: QALYs for an individual starting from age a0 in state j

)( 0aG j

Individual QALYs

Measures of health

Equilibrium population measures

ALQ TLQ

AQ/yr TQ/yr

TLQ


Example: tamoxifen use to prevent breast cancerCol


SummarySummary

• Population Markov models for medical decision making.

• Population measures of interventions

• Age-dependency.

Nov. 2005 M. Huang Northwestern Univ. 1 Markov Chain Population Models in Medical Decision Making...

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