K idney exchange - current challenges

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 1Blood type

A

Recipient1Blood type

B

Recipient2Blood type

A

Donor 2Blood type

B

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

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

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

Hospitals became players

• Often hospitals withhold internal matches, and contribute only hard-to-match pairs to a centralized clearinghouse.

a3

a2

cd

a1

e b

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

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

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?

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

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

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

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)

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

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

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

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

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

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

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

If , then easy to achieve average waiting time

• patient-donor pools presently consist of many hard to match pairs

We consider

Homogenous (sparse) pools

• 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.

What about

• 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

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

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

• 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

• 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

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 • For any policy

-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

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?

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.

Waiting a small period of time when 3-way cycles may be beneficial (Ashlagi, Jaillet, Manshadi 13)

h1

l2

l1

l3

time

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

– Unver (2010)

– Ashlagi, Jaillet,Manshadi (2013)

– Akbarpour, Li, Gharan (2014)

– Dickerson et al (2012)

…..

Growing literature on dynamic matching

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






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 AB 99

A A 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!?

• 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

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

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

• 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

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%!


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!


Adam Bingaman from San Antonio:

If you don’t have enough failures – you are not transplanting enough hard to match patients!

Software we developedExchange software

• 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








• Madison, WI

And also set tolerances

Output – users can observe Donor Specific Antibodies








• Madison, WI

Software is used by several centers:








• 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?

K idney exchange - current challenges

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