israel david and michal moatty-assa a stylistic queueing-like model for the allocation of organs on...
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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