Random Graph Models for Kidney Exchange

Report
Kidney exchange - current challenges
Itai Ashlagi
What are the design issues?
• Initial design efforts were for startup kidney exchange
• Now, hospitals have become players
• Pools presently consist of many to hard to match pairs. In this
environment, non-simultaneous chains become important
• Dynamic matching
• Computational issues
• Reduce “congestion”
Simple two-pair kidney exchange
Donor 1
Blood type
A
Donor 2
Blood type
B
Recipient1
Blood type
B
Recipient2
Blood type
A
Factors determining transplant opportunity
O
• Blood compatibility
A
B
AB
• Tissue
type compatibility
Panel Reactive Body –percentage of donors that will be tissue type
incompatible to the patient
4
Theorem (Roth, Sonmez, Unver 2007, Ashlagi and Roth, 2013): In
almost every large pool (directed edges are created with
probability p) there is an efficient allocation with exchanges of
size at most 3.
O-O
AB-B
A-A
B-B
AB-A
AB-O A-O
B-A
ABAB
B-O
VA-B
B-AB
A-AB
“Under-demanded” pairs
O-A
O-B
O-AB
A-B
Dynamic large pools (Unver, ReStud 2009)
Optimal dynamic mechanism: similar to the offline construction but sets
a threshold of the number of A-B pairs in the pool which determines
whether to save them for a 2-way or use them in 3-ways.
O-O
AB-B
A-A
B-B
AB-A
AB-O A-O
B-A
ABAB
B-O
VA-B
B-AB
A-AB
“Under-demanded” pairs
O-A
O-B
O-AB
A-B
Hospitals became players
• Often hospitals withhold internal matches, and contribute only
hard-to-match pairs to a centralized clearinghouse.
a3
a1
e
a2
c
b
d
National Kidney Registry (NKR) Easy to Match Pairs
Transplanted
60%
9/1/13 – 3/25/14
57%
All In Centers
Not All In Centers
50%
40%
31%
30%
22%
21%
20%
9%
10%
9%
0%
PMPa
PMPb
PMPc
Transplanted internally and through NKR
% O donors
NKR
Internal
40
55
% O to O
(from all O
donor
transplants)
92
73
% O to low PRA
recipients A,B,AB
(from such
transplants)
33
88
Random Compatibility Graphs
n hospitals, each of a size bounded by c>0 .
1. pairs/nodes are randomized –compatible pairs are
disregarded
2. Edges (tissue type compatibility) are randomized
Question: Does there exist an (almost) efficient
individually rational allocation?
Current mechanisms aren’t Individually rational
for hospitals
Ashlagi and Roth (2011):
1. Centers are better off withholding their easy to match
pairs
A-O
O-A
2. “Theorem”: design of an “almost” efficient mechanism
that makes it safe for centers to participate in a large random
pools.
Incentive hard to match pairs!
A-O can be easy to match. Make sure to match at least one O-A
pair (and maybe even more…)
A-O
O-A
(Sometimes A-O can be hard to match if A is very highly
sensitized)
Loss is Small - Simulations
No. of
Hospitals
IR,k=3
2
4
6
8
10
12
14
16
18
20
22
6.8 18.37 35.42 49.3 63.68 81.43 97.82 109.01 121.81 144.09 160.74
Efficient, k=3 6.89 18.67 35.97 49.75 64.34 81.83 98.07 109.41 122.1 144.35 161.07
Possible solution:
• “Frequent flier” program for transplant centers that
enroll easy to match pairs.
• Provide points to centers that enroll O donors
• National Kidney Registry:
– Currently provides incentives for altruistic donors
– A few months ago: all in memo… (but not going forward)
– Proposal for points system for different pairs (to be up for
a vote)
Why? many very highly sensitized patients
Previous simulations: sample a patient and donor from the
general population, discard if compatible (simple live
transplant), keep if incompatible. This yields 13% High
PRA.
The much higher observed percentage of high PRA
patients means compatibility graphs will be sparse
PRA distribution in historical data
16%
14%
40%
35%
Percentage
30%
25%
20%
15%
10%
Percentage
12%
10%
8%
NKR
6%
APD
4%
2%
0%
95-96
96-97
97-98
98-99
PRA Range
99-100
NKR
APD
5%
0%
PRA Range
PRA – “probability” for a patient to pass a “tissue-type” test
with a random donor
Dynamic matching
Question:
Suppose only -way or smaller exchanges are possible.
• Greedy policy: Complete an exchange as soon as possible
• Batch policy: Wait for many nodes to arrive and then
‘pack’ exchanges optimally in compatibility graph
Which policy works better?
Policies implemented by kidney exchanges
All clearinghouses are use batching policies
• APD: monthly → daily
• NKR: various longer batches → daily (even more than
once a day)
• UNOS Kidney exchange program: monthly → weekly →
bi-weekly
Are short batches/greedy better than long batches?
Can some non-batching policy do even better?
Matching over time
Simulation results using 2 year data from NKR*
Matches
550
500
2-ways
3-ways
2-ways & chain
3-ways & chain
450
400
350
300
1
5
10
20
32
64
100
260
520
1041
Waiting period between match runs
In order to gain in current pools, we need to wait probably “too” long
*On average 1 pair every 2 days arrived over the two years
Matching over time
(Anderson,Ashlagi,Gamrnik,Hil,Roth,Melcer 2014)
Simulation results using 2 year data from NKR*
Matches
Waiting Time
295
290
285
280
275
270
265
260
255
250
240
220
200
180
160
140
120
100
1D
1W
2W
1M
3M
6M
1Y
1D
1W
2W
1M
3M
6M
In order to gain in current pools, we need to wait probably “too” long
*On average 1 pair every 2 days arrived over the two years
1Y
Pools with hard-to-match pairs
Suppose every directed edge is present iid with same
probability  ⇒ nodes form directed Erdos-Renyi graph
Graph-structured queuing system:
• At each time , a node  arrives
• Node  forms edge with each node in the system
independently with probability 
• If cycle of size ≤  is formed, it may be eliminated
Objective:
Minimize average waiting time = Average(#nodes in system)
Call this 
Homogenous (sparse) pools
If  = Θ 1 , then easy to achieve average waiting time
 1
• patient-donor pools presently consist of many hard to
match pairs
We consider  → 0
Only two-cycles:  = 2
• Two-cycle formed between any two nodes w.p. 2
• Under greedy, in steady state, cycle formed at each time
w.p. ½, so  =  ∼ 1/2
• Not hard to show that for any policy  = Ω
1
2
Theorem[Anderson,Ashlagi,Gamarnik,Kanoria 14]:
For  = 2, greedy achieves
ln 2
1
gr = 2 +  2


and no policy can achieve better waiting times than
greedy.
What about  = 3?
Batching for  = 3
• If batch size is , then E #triangles = 2

3
 3 ∼ 3  3
• We want to eliminate most of the batch, so ~ /3 triangles
needed
• Hence, need
3 3 ≿ n
⇒
 ≿
Can show that batch size  = Θ
1
1.5
1
1.5
gives  = Θ
How does greedy compare?
1
1.5
3-cycles: Simulation results for p = 0.08
70
60
50
W
40
30
20
10
0
1
2
4
8
16
Size of batch
32
62
128
3-cycles: Simulation results for p = 0.05
120
100
80
W
60
40
20
0
1
2
4
8
16
Size of batch
32
62
128
Greedy is “optimal”
Theorem[Anderson, Ashlagi,Gamarnik,Kanoria 14]:
For  = 3, we have
• Greedy achieves  = Θ
1
1.5
• For any monotone policy  = Ω
1
1.5
• Batching with maximal packing of cycles is
monotone
• Shows that greedy is optimal up to a constant
factor
3-cycles: Proof idea that greedy is good
• Suppose  ≥ /1.5 nodes in the system at 
• Want to show negative drift over next few time steps
• Worst case  is empty
Consider next  =
1
1.5
arrivals. For appropriate ,  show:
• Most new arrivals form cycles containing old nodes,
leading to, whp,
+ ≤  − 1/( ′ 1.5 )
What about ?
Altruistic/non-directed donors
Bridge
donor
• Altruistic kidney donors facilitate asynchronous chains.
• One altruistic donor at time 0
How much do such altruistic donors improve ?
Greedy is “optimal”
Theorem[Anderson, Ashlagi,Gamarnik,Kanoria]: For a
single unbounded chain
• Greedy achieves  = Θ
• For any policy  = Ω
1

1

Summary of findings
2-cycles
3-cycles
Chains
gr
ln 2
2
1
Θ 1.5

1
Θ

Lower bound
on 
ln 2
2
1
Ω 1.5

1
Ω

• Greedy policy (near) optimal in each case
• 3-cycles substantially improve 
• Altruistic donors ⇒ chains lead to further large
improvement
• Most exchanges occur via chains > 3-cycles > 2-cycles
Easy and Hard to match pairs
In a heterogeneous with (E)asy and (H)ard to match
patients batching can “help” in 3-ways but not in 2ways!
With who to wait? How much?
Can we do better than batching?
Dynamic matching in dense-sparse graphs
• n nodes. Each node is L w.p. v<1/2 and H w.p. 1-v
• incoming edges to L are drawn w.p.
• incoming edges to H are drawn w.p.
At each time step 1,2,…, n, one node arrives.
L
H
41
Waiting a small period of time when 3-way cycles may
be beneficial (Ashlagi, Jaillet, Manshadi 13)
h1
l3
l1
l2
time
Intuition for 2-way cycles
When the batch size is “small” there is little room for mistakes if you
match greedily
arrived batch

Tissue-type compatibility: Percentage Reactive Antibodies (PRA).

PRA determines the likelihood that a patient cannot receive a kidney from a blood-type compatible donor.

PRA < 79: Low sensitivity patients (L-patients).

80 < PRA < 100: High sensitivity patients (H-patients).

Most blood-type compatible pairs that join the pool have H-patients.
residual graph

Distribution of High PRA patients in the pool is different from the population PRA.
time
Growing literature on dynamic matching
– Unver (2010)
– Ashlagi, Jaillet,Manshadi (2013)
– Akbarpour, Li, Gharan (2014)
– Dickerson et al (2012)
…..
Kidney exchange in the US
Transplants through kidney exchange in the US
• UNOS kidney exchange (National pilot)
>90 transplants
>45% of the transplants done through chains
• Methodist Hospital at San Antonio (single center)
>240 transplants
• National Kidney Registry (largest volume program):
>1,000 transplants
>88% transplanted through chains!
>15% of transplanted patients with PRA>95!
>25% transplanted through chains of length >10
Alliance for Paired Donation
>240 transpants
> 170 through chains
Methodist San Antonio KPD program
(since 2008) - includes compatible pairs
• 210 KPD transplants done (this slide is from May 2013)
– Thirty-Three 2-way exchanges
– Twenty-three 3-way exchanges
– Two 6-recipient exchanges
– One 5-recipient chain
– One 6-recipient chain
– One 8-recipient chain
– One 9-recipient chain
– One 12-recipient chain
– One 23-recipient chain
Can collaboration between exchange
programs be beneficial?
Benefits of merging patient-donor pools: over 3
years of data (with duplicates removed)
NKR +
APD +
SA
SA + APD NKR +
APD
NKR + SA
All matches
15%
(3%)
11%
(1.5%)
10% (3%)
8% (2.5%)
PRA >= 80
matches
28%
(5%)
21% (5%)
21% (4%)
17% (25)
PRA >= 95
40%
(10%)
25% (6%)
27% (6%)
22% (4%)
PRA >= 99
41%
(9%)
35% (7%)
63% (10%) 16.6%
(5%)
3 years of data from each program: match each week, separately
about 8 pairs each of nkr and apd per week and 4 for sa , resampling
arrival time in actual clinical data 15% more from full match (still one
week, so more pairs) 3% run each program separately, but every 2
months merge remaining pairs
Collaboration might be useful
Garet Hil (NKR): “Consistent with Al’s presentation....the NKR
has begun a program to provide the attached list of
donors….upon request to other paired exchange
programs in the hope that we can begin facilitating
exchange transplants across programs.
Mike Rees (APD): “It would be great if we could begin to
collaborate… I don't understand how to move forward though.
As I understand it, all of these donors have unmatched
recipients in the NKR system whose information is not
provided… “
First 3-way exchange between APD and NKR
(Summer 2013)
Donor
Patient
PRA
A
AB
48
AB
A
AB
A
99
0
Innovation has come from having multiple kidney
exchange programs
•
APD
– Non-simultaneous chains
– International exchange
•
San Antonio
– Compatible pairs
– Novel cross matching
•
NKR
– Immediately reoptimizing whole match after a rejection
– Prioritizing via both patient and donor difficulty in matching
– Recruiting NDD’s (credit system)
– Maybe frequent flyer program!?
Computational challenges
• Unbounded cycles and chains [Easy but not logistically
feasible]
• Only 2-way cycles [Easy, Edmonds maximum matching
algorithm]
• Bounded cycles and unbounded chains [NP-Hard]
Early optimization formulation
Decision variable for each potential cycle and
chain with length at most 3.
Maximize weighted # transplants
s.t. each pair is matched at most once
Works well in practice because length is bounded by 3
55
Algorithms and software for kidney exchanges
Integer Programming based algorithm for finding optimal cycle and chain based exchang
Formulation I:
MAX weighted # transplants
Max  
Pair gives only if receives s.t.
 ∈ 
∈()  ≤ ∈()  ≤ 1
Use NDDs once
∈()  ≤ 1
 ∈ {0,1}
No cycles with length >b
∈ 
 ∈ 
≤  −1
∀   > 
• The last constraint is added only iteratively (when a long cycle is found
• Most instances solve quite fast.
56
Algorithms and software for kidney exchanges
Formulation II inspired by the Prize-Collecting-TravellingSalesman-Problem
Add cutset constraint for every subset  of incompatible pairs and
every pair  ∈ 


NDD
flow into  ≤ flow into 
• Separation problem is solved efficiently.
• Almost always finds optimal solution within 20
minutes
57
Existing challenges
• Incentives for participation
• Increase participation - only a small fraction of patients and
donor are enrolling in kidney exchanges!
• Pre-transplant “failures” – crossmatch, acceptance,
availability – congestion
How do things happen in practice:
• Transplant centers enter patients and donors data
including preferences (blood types, antibodies, antigens,
max age, etc.)
• The clearinghouse runs an optimization algorithm every
“period” and sends “offers” to centers involved in
exchanges
• Blood tests (crossmatches) for acceptable exchanges
are conducted.
• Exchanges that pass blood tests are scheduled and
conducted
Failures and how to deal with them?
We see failures…. offers rejected, crossmatch failures.
Antibodies are not binary!
Highly sensitized patients have a much higher crossmatch
failure rate then low sensitized patients.
Optimization literature: take failures as an input: Song et al,
2013, Dickerson et al. 2013, Blum et al 2013.
What is needed? collect better data. titers, preferences…
National Kidney Registry have dropped the (one-way) failure
rate from 20% to 3%!
Failures and how to deal with them?
UNOS and the APD have very high failure rates! Offers are
rejected, crossmatch failures (can reach over 30% per one-way)
Antibodies are not binary! Currently no good predictor for failures.
Highly sensitized patients have a much higher crossmatch failure
rate then low sensitized patients.
Optimization literature: take failures as an input: Song et al, 2013,
Dickerson et al. 2013, Blum et al 2013.
Needed: collect better data. titers, preferences…
National Kidney Registry have dropped the (one-way) failure rate
from 20% to 3%!
Centers have different capabilities!
Failures and how to deal with them?
Adam Bingaman from San Antonio:
If you don’t have enough failures – you are not transplanting
enough hard to match patients!
Exchange software
Software we developed
Titers information can be entered
• Rabin Medical Center, Israel
• Northwestern Memorial hospital, Chicago
• Methodist Hospital, San Antonio, TX
• Georgetown Medical Center, DC
• Samsung Medical Center, Korea
• Mayo clinic (Arizona)
• Cleveland clinic, OH
• Madison, WI
And also set tolerances
• Rabin Medical Center, Israel
• Northwestern Memorial hospital, Chicago
• Methodist Hospital, San Antonio, TX
• Georgetown Medical Center, DC
• Samsung Medical Center, Korea
• Mayo clinic (Arizona)
• Cleveland clinic, OH
• Madison, WI
Output – users can observe Donor
Specific Antibodies
• Rabin Medical Center, Israel
• Northwestern Memorial hospital, Chicago
• Methodist Hospital, San Antonio, TX
• Georgetown Medical Center, DC
• Samsung Medical Center, Korea
• Mayo clinic (Arizona)
• Cleveland clinic, OH
• Madison, WI
Software is used by several centers:
• Rabin Medical Center, Israel
• Northwestern Memorial hospital, Chicago
• Methodist Hospital, San Antonio, TX
• Georgetown Medical Center, DC
• Samsung Medical Center, Korea
• Mayo clinic (Arizona)
• Cleveland clinic, OH
• Madison, WI
But software is not enough to achieve good results…
Towards reducing failures
• What should centers observe?
• NKR has adopted since beginning of 2014 a policy that allows
centers to do “exploratory crossmatches” (so they see also
incompatible donors and inquire to do a blood test with some
incompatible donor).
• Centers are using this option in an increasing rate!
• This arguably saves online failures.
Summary and research directions
• Current pools contain many highly sensitized patients and
(long) chains are very effective (but how to utilize them?)
• Need to provide incentives to enroll easy-to-match pairs.
• Pooling can help highly sensitized patients.
• How to reduce pre-transplant failures?
• Why should sophisticated/large centers participate?
• How to attract more people from the waiting list?

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