kidney exchange - current challenges itai ashlagi
TRANSCRIPT
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Kidney exchange - current challenges
Itai Ashlagi
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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”
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Simple two-pair kidney exchange
Donor 1Blood type
A
Recipient1Blood type
B
Recipient2Blood type
A
Donor 2Blood type
B
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4
Factors determining transplant opportunity
• Blood compatibility
• Tissue type compatibility
Panel Reactive Body –percentage of donors that will be tissue type incompatible to the patient
O
A B
AB
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B-A
B-AB A-AB
VA-B
A-O B-OAB-O
O-B O-A
A-B
AB-B AB-A
O-AB
O-OA-A B-B
AB-AB
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.
“Under-demanded” pairs
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B-A
B-AB A-AB
VA-B
A-O B-OAB-O
O-B O-A
A-B
AB-B AB-A
O-AB
O-OA-A B-B
AB-AB
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.
“Under-demanded” pairs
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Hospitals became players
• Often hospitals withhold internal matches, and contribute only hard-to-match pairs to a centralized clearinghouse.
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a3
a2
cd
a1
e b
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PMPa PMPb PMPc0%
10%
20%
30%
40%
50%
60%57%
22% 21%
31%
9% 9%
All In Centers
Not All In Centers
National Kidney Registry (NKR) Easy to Match Pairs Transplanted 9/1/13 – 3/25/14
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Transplanted internally and through NKR
% O donors
% O to O(from all O donor transplants)
% O to low PRA recipients A,B,AB (from such transplants)
NKR 40 92 33
Internal 55 73 88
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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 randomizedQuestion: Does there exist an (almost) efficient individually rational allocation?
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Current mechanisms aren’t Individually rational for hospitalsAshlagi and Roth (2011):
1. Centers are better off withholding their easy to match pairs
2. “Theorem”: design of an “almost” efficient mechanism that makes it safe for centers to participate in a large random pools.
O-A
A-O
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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…)
(Sometimes A-O can be hard to match if A is very highly sensitized)
O-A
A-O
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Loss is Small - Simulations
No. of Hospitals 2 4 6 8 10 12 14 16 18 20 22
IR,k=3 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
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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)
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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
Why? many very highly sensitized patients
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PRA distribution in historical data
PRA – “probability” for a patient to pass a “tissue-type” test with a random donor
0-5 5-10 10-15
15-20
20-25
25-30
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70
70-75
75-80
80-85
85-90
90-95
95-100
0%
5%
10%
15%
20%
25%
30%
35%
40%
NKRAPD
PRA Range
Per
cen
tage
95-96 96-97 97-98 98-99 99-1000%
2%
4%
6%
8%
10%
12%
14%
16%
NKRAPD
PRA Range
Per
cen
tage
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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?
Dynamic matching
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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?
Policies implemented by kidney exchanges
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Matching over time
Simulation results using 2 year data from NKR*
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
1 5 10 20 32 64 100 260 520 1041300
350
400
450
500
550
2-ways3-ways2-ways & chain3-ways & chain
Waiting period between match runs
Matches
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Matching over time (Anderson,Ashlagi,Gamrnik,Hil,Roth,Melcer 2014)
1D 1W 2W 1M 3M 6M 1Y250255260265270275280285290295
Matches
Simulation results using 2 year data from NKR*
1D 1W 2W 1M 3M 6M 1Y100
120
140
160
180
200
220
240
Waiting Time
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
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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
Pools with hard-to-match pairs
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If , then easy to achieve average waiting time
• patient-donor pools presently consist of many hard to match pairs
We consider
Homogenous (sparse) pools
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• Two-cycle formed between any two nodes w.p.
• Under greedy, in steady state, cycle formed at each time w.p. , so
• Not hard to show that for any policy
Only two-cycles:
Theorem[Anderson,Ashlagi,Gamarnik,Kanoria 14]: For greedy achieves
and no policy can achieve better waiting times than greedy.
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What about
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• If batch size is then
• We want to eliminate most of the batch, so triangles needed
• Hence, need
Can show that batch size gives
How does greedy compare?
Batching for
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1 2 4 8 16 32 62 1280
10
20
30
40
50
60
70
Size of batch
W
3-cycles: Simulation results for p = 0.08
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3-cycles: Simulation results for p = 0.05
1 2 4 8 16 32 62 1280
20
40
60
80
100
120
Size of batch
W
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• Batching with maximal packing of cycles is monotone
• Shows that greedy is optimal up to a constant factor
Greedy is “optimal”
Theorem[Anderson, Ashlagi,Gamarnik,Kanoria 14]: For we have• Greedy achieves • For any monotone policy
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• Suppose nodes in the system at
• Want to show negative drift over next few time steps
• Worst case is empty
Consider next arrivals. For appropriate show:
• Most new arrivals form cycles containing old nodes, leading to, whp,
3-cycles: Proof idea that greedy is good
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What about
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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 ?
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Greedy is “optimal”
Theorem[Anderson, Ashlagi,Gamarnik,Kanoria]: For a single unbounded chain• Greedy achieves • For any policy
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-cycles -cycles Chains
Lower bound on
Summary of findings
• 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
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In a heterogeneous with (E)asy and (H)ard to match patients batching can “help” in 3-ways but not in 2-ways!
Easy and Hard to match pairs
With who to wait? How much?
Can we do better than batching?
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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.
L
H
41
At each time step 1,2,…, n, one node arrives.
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Waiting a small period of time when 3-way cycles may be beneficial (Ashlagi, Jaillet, Manshadi 13)
h1
l2
l1
l3
time
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When the batch size is “small” there is little room for mistakes if you match greedily
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.
Distribution of High PRA patients in the pool is different from the population PRA.
arrived batch
residual graph
Intuition for 2-way cycles
time
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– Unver (2010)
– Ashlagi, Jaillet,Manshadi (2013)
– Akbarpour, Li, Gharan (2014)
– Dickerson et al (2012)
…..
Growing literature on dynamic matching
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Kidney exchange in the US
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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
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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
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Can collaboration between exchange programs be beneficial?
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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
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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… “
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First 3-way exchange between APD and NKR (Summer 2013)
Donor Patient PRA
A AB 48
AB AB 99
A A 0
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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!?
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• 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]
Computational challenges
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Decision variable for each potential cycle and chain with length at most 3.
Maximize weighted # transplantss.t. each pair is matched at most once
Works well in practice because length is bounded by 3
Early optimization formulation
55
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MAX weighted # transplants Max Pair gives only if receives s.t.
No cycles with length >b
• The last constraint is added only iteratively (when a long cycle is found
• Most instances solve quite fast.
Algorithms and software for kidney exchanges Integer Programming based algorithm for finding optimal cycle and chain based exchanges.
Formulation I:
56
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• Separation problem is solved efficiently.• Almost always finds optimal solution within 20
minutes
Algorithms and software for kidney exchanges
Formulation II inspired by the Prize-Collecting-Travelling-Salesman-Problem
Add cutset constraint for every subset of incompatible pairs and every pair
𝑺𝒗
flow into flow into
57
NDD
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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
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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
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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%!
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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!
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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!
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Software we developedExchange software
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• 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
Titers information can be entered
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• 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
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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
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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…
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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.
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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?