Israel David and Michal Moatty-Assa

A Stylistic Queueing-Like Model for the Allocation of Organs on the Public List

Supply-Demand Discrepancy

• Increasing shortage in kidneys for transplant

• 4,252 died waiting (2008)

• (Kidney offers are thrown away)

• ~50% refuse 1st kidney offered!

Who waits the longest?

Whom do I best fit?

Who’s the youngest?

Objectives:Clinical Efficiency: QALY, % survival.

Equity: in waiting, across social groups.

Matching Criteria: ABO, HLA, PRA, Age, Waiting

PRApointsAgepoints

25% - 0%018 – 04

50% - 26%240 – 192

75% - 51%460 – 411

>75%6>600

HLA mismatches

pointsWaiting time

(months)points

No MM4<240

1 MM348 – 251

No MM in DR296 – 492

<974

The Israeli “Point System” for kidney allocation

First In First Offered FIFOf –

• FIFO sorting for Offering

• simplifying assumptions, “stylistic” moel

Decision rule

Allocation rule

A continuous, time-dependent, full-info “Secretary )”)

The future arrival process

How long do I wait?

How good is this offer?

my HLA, ABO

population statistics by ABO, HLA

donors arrival rate

The decision of the single candidate

Model Assumptions

• Constant lifetime under dialysis (T)• Poisson arrival of donor kidneys (rate )• Poisson arrival of patients• "Aggregate HLA " – only one relevant

genetic quality

What is the compromising t?

kidney offerfrequency inpopulation

gain (life years)

a matchpRa mismatch1-pr

n = 1, basics

• פונקציית המטרה, הרווח האופטימלי – מהצעת כליה ברגע

• מ"מ, רווח (שנות חיים) מהשתלת הכליה -

• תוחלת הרווח הצפויה מדחיית ההצעה - t ברגע

X – Offer random value; = E[X] = Rp + r(1-p)

U(t) – expected optimal value assuming that at t an offer is pending

V(t) – optimal value from t onwards (exclusive of t if an offer is pending); V(T) = 0.

, T, R, r, p, ant1

Dynamic Programming

1. U)t , x( = max{x, V)t(}

2. U(t) = EX[U(t , X)]

3. V(t) =

0() dsestU s

n = 1, depiction of V and U

n = 1, Explicit t*

) *() ( )1 ( [ )1 ( ]

ln

*

T tV t e r p R p r

rt T

= E[X] = Rp + r)1-p(

n = 1, Explicit solution of V(t), U(t)

t t T

) () ( )1 (T tV t e

tt0 ) ( )1 ( ) ( ) ( ) (p pV t R p t r R t t

()() (),() tTtt etet

.

)A solvable Volterra (

*

) ( ) (

*

) ( )1 ( ) (t T

t t

t t

V t e p R p V d e d

) (t s

0

s

S

V t U t s e ds

) ( )1 ( max{ , ) (}U t s pR p r V t s

*1t0 T2a 2aT

(1) 12 p

02

2effective

n = 2, (approx.) outlook for the second candidate

Non-hom.-Poisson stream with 3 stages

],[1

(1)],[0

],[(1)

*12

1

1222

*1

22222

2

tatIIIp

pprpRIII

TttII

TaTtIprpRI

n = 2, conditional expected gains

2

2

2

2 1*2 1 1 2

1 2*2

22 2

2

11max , 1

1max 0 , ln 1

aa

a

p ea t ln p e r

q rt

T a e rr

n = 2, Explicit t*

n > 1, general

specifics of cand. n

specifics of cand.(n - 1 )

and t* n-1

optimization optimal decision rule (tn

*) for cand. n

input output

still… n = 3

*1t0 T2a 2aT

12 q

02

2

3a2 *t 3T a

213 qq

03

3

effective

n = 3

*1t0 T2a 2aT

12 q

02

2

3a2 *t 3T a

223 q

03

3effective

The -recursion per sub-intervals

for all

Except for intersections with

or

11 ][][ nnn qII

],0[ TaI n

0],[ 1*

1 Tat nnn

],[ 1 TaTa nn ],[ 1 TaTa nnn

],[ 1*

1 Tat nn where

where

leftmost Vn(t)’s for sub-intervals

()lnv

()lnπ

()lnξ

1

1

) ( ) ( ) ( ) ( 1 ) ( ) (ni m

n n n n n nm l j l

l l l m j m

π ξ π π ξv

• - optimal value for cand. n in rejecting at the beginning of sub-val l

• - arrival probability of an offer during sub-val l • - conditional expected gain if during sub-val l

(explicit expressions for ) ()lnv

n

dl

n ilelln

k

i

in

,...,11() 0

()

π

ni

n

k

i

anti

in

antn

n illf

lfrlfR

l n

n

,...,1()

()()

()

1

ξ

The critical subinterval and determining tn*

11,()|min* lrlll nn v

rtWtt nn ()|inf*

*() nn lv

*nl

()tWn is taken to be

such that t is substituted for the beginning of subinterval

1 1p

2 2p

3 3p

2a 2aT 0 T 3aT 3a *1t*2t

blocking and releasing of simultaneous antigen currents

Simulation Measures• Long-run proportion of "good" transplants

• Long-run death-rate

• Long-run Waiting Time for allocated candidate

) (

max1

lim ) (N t

t ii

I Y N t

otherwise

RXY i

i 0

1

) (

1

lim ) (N t

t ii

D D N t

otherwise

XD i

i 0

01

) (

1

lim ) (cN t

t i ci

W W N t