A Stylistic Queueing-Like
Model for the Allocation of
Organs on the Public List
Israel David and Michal Moatty-Assa
Supply-Demand Discrepancy
23,637
24,690 27,289
29,157
32,371
40,000
32,440
32,589
8,539
30,000
8,667
9,357
9,913
20,000
‫השתלות‬
10,000
‫מועמדים חדשים‬
10,659
2002
10,588
2003
10,551
2004
0
2005
2006
2007
2008
• Increasing shortage in kidneys for
transplant
• 4,252 died waiting (2008)
•
•
(Kidney offers are thrown away)
~50% refuse 1st kidney offered!
Whom do I
?best fit
Who’s the
?youngest
Who waits
?the longest
:Objectives
Clinical Efficiency:
.QALY, % survival
Equity: in waiting,
across social
.groups
Matching Criteria: ABO, HLA, PRA, Age, Waiting
The Israeli “Point System” for kidney allocation
PRA
points
Age
points
0% - 25%
0
0 – 18
4
26% - 50%
2
19 – 40
2
51% - 75%
4
41 – 60
1
>75%
6
>60
0
HLA
mismatches
points
Waiting time
(months)
points
No MM
4
<24
0
1 MM
3
25 – 48
1
No MM in DR
2
49 – 96
2
>97
4
FIFOf – First In First Offered
• FIFO sorting for Offering
Decision
rule
Allocation
rule
• simplifying assumptions, “stylistic” moel
The decision of the single candidate
The future
arrival process
How good is
this offer?
population
statistics by
ABO, HLA
my HLA,
ABO
How long do I
wait?
donors arrival rate
)A continuous, time-dependent, full-info “Secretary”)
Model Assumptions
•
•
•
•
Constant lifetime under dialysis (T)
What is
the of
compromising
t? l)
Poisson
arrival
donor kidneys (rate
Poisson arrival of patients
"Aggregate HLA " – only one relevant
genetic quality
kidney offer
a match
a mismatch
frequency in
population
p
1-p
gain
(life years)
R
r
First candidate
Second candidate
n’th candidate
Simulation
n = 1, basics
X
– Offer random value; x = 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.
l, 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
U (t  s)le ls ds
n = 1, depiction of V and U
n = 1, Explicit t*
V (t )  (1  e
 l (T  t *)
)  [ r (1  p )  R  p ]  r
 x 
ln 

x r 

 t*  T 
l
x = E[X] = Rp + r(1-p)
n = 1, Explicit solution of V(t), U(t)
t  t  T
0  t  t
V (t )  x (1  e l (T t ) )
V (t )  R  (1  p) (t ) p  r  R    (t ) p x   (t )
 (t )  e
 l (t  t )
,  (t )  e l (T t )
.
(A solvable Volterra)
V t  


U  t  s  l e  l s ds
S 0
U (t  s)  pR  (1  p) max{r ,V (t  s)}
t*
T
t
t*
V (t )   l e  l ( t )   p  R  (1  p ) V ( )  d   l e  l ( t )x d
(  t  s)
n = 2, (approx.) outlook for the
second candidate
Non-hom.-Poisson stream with 3 stages
l2  l
l2  l  (1  p1 )
effective
l
l2  0
0
a2
t1 *
T
T  a2
n = 2, conditional expected
gains

x 2 I   x 2  R  p 2  r  (1  p 2 )

x 2   
0

R  p 2  r  (1  p 2  p1 )
 x 2 III  
1  p1

I t  [T , a 2  T ]
II
t  [t1* , T ]
III
t  [a 2 , t1* ]
n = 2, Explicit t*
 l a2



 
x
1

p

e


1
2
1
*
 ln 
 p1   x 2 1  e l a2   r
max a2 , t1 
l q1  x 2  r 


 
*
t2  


1  x 2  
 l a2
T

max
0
,
a


ln
x
1

e
r

 2


2

l  x 2  r  


n > 1, general
input
output
optimizatio
n
optimal
decision rule
(tn*) for cand. n
still… n = 3
l2  l
l3  l
l2  l  q1
effective
l
l3  l  q1  q2
l3  0
l2  0
0
a2 a3 t2 * t *
1
T
T  a2
T  a3
n=3
l2  l
effective
l
l3  l
l2  l  q1
l3  l2  q2
l3  0
l2  0
0
a2 a3 t2 * t1 *
T
T  a2
T  a3
The l-recursion per sub-intervals
ln [ I ]  ln1[ I ]  qn1
for all
I  [0 , an  T ]
Except for intersections with
*
*
l
[
t
[t n1 , an1  T ] where n n1 , an1  T ]  0
or
[an1  T , an  T ] where ln [an1  T , an  T ]  l
leftmost Vn(t)’s for sub-intervals
m 1


vn (l )  π n (l )  ξ n (l )    π n (m)   1  π n ( j )    ξ n ( m)
m l 1 
j l

in
•v n (l ) - optimal value for cand. n in rejecting
at the beginning of sub-val l
•π n (l ) - arrival probability of an offer during
sub-val l
• ξ n (l ) - conditional expected gain if during
sub-val l
(explicit expressions for v n (l ) )
π n (l )  1  e
R f
ξ n (l ) 
antn
n
 k i
 l


ln ( l ) d n


 i 0


(l )  r 
f
i  antn
k

i 1
i
n
f (l )
l  1,..., in
i
n
(l )
l  1,..., in
The critical subinterval and
determining tn*
l  min l | v n (l )  r , l  1  1
*
n
t  inf t | Wn (t )  r
*
n
Wn (t )
is taken to be
vn (ln *)
such that t is substituted for the beginning of subinterval
ln *
blocking and releasing of
simultaneous antigen currents
l1  l  p1
l2  l  p2
l3  l  p3
0
a2
a3 t2 * t1 *
T
T  a2
T  a3
Simulation Measures
• Long-run proportion of "good" transplants
1 X i  R
Yi  
0 otherwise
N (t )
I max  limt   Yi N (t )
i 1
• Long-run death-rate
Xi  0
1
Di  
0 otherwise
N (t )
D  limt   Di N (t )
i 1
• Long-run Waiting Time for allocated candidate
W  limt 
Nc ( t )
W
i 1
i
Nc (t )