1. Kidney exchange - current challenges Itai Ashlagi

2. 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”

3. Simple two-pair kidney exchange Donor 1 Blood type A Recipient1 Blood type B Recipient2 Blood type A Donor 2 Blood 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 4

5. B-A B-AB A-AB V A-B A-O B-O AB-O O-B O-A A-B AB-B AB-A O-AB O-O A-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

6. B-A B-AB A-AB V A-B A-O B-O AB-O O-B O-A A-B AB-B AB-A O-AB O-O A-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

7. Hospitals became players Often hospitals withhold internal matches, and contribute only hard-to-match pairs to a centralized clearinghouse.

8. a3a3 a2a2 c d a1a1 e b

9. National Kidney Registry (NKR) Easy to Match Pairs Transplanted 9/1/13 – 3/25/14

10. 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) NKR409233 Internal557388

11. 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?

12. Individually Rational Allocations (less than 1.5%). So the worst-case impossibility results don’t look at all like what we could expect to achieve in large kidney exchange pools (if individually rational mechanisms are adopted). 12

13. Current mechanisms aren’t Individually rational for hospitals Ashlagi 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

14. 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

15. Loss is Small - Simulations No. of Hospitals246810121416182022 IR,k=36.818.3735.4249.363.6881.4397.82109.01121.81144.09160.74 Efficient, k=36.8918.6735.9749.7564.3481.8398.07109.41122.1144.35161.07

16. 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)

17. B-A B-AB A-AB V A-B A-O B-O AB-O O-B O-A A-B AB-B AB-A O-AB O-O A-A B-B AB- AB Efficiency in a large pool altruistic donor An altruistic donor can increase the match size by at most 3 in large pools waiting list

18. Yash Kanoria (MSR-NE)A dynamic graph model of kidney exchange 18 The majority of the transplants are now done through chains

19. Literature about chains in the medical literature Rees et al..., NEJM 2009 – story about first long chain Gentry & Segev, AJT 2010 – long chains are not effective ? Ashlagi, Gilchrist, Rees & Roth, AJT 2011a - long chains are effective Gentry & Segev, AJT 2011a – honeymoon phase is over and long chains are not effective Ashlagi, Gilchrist, Rees & Roth, AJT 2011b - letter: honeymoon is still around for a while P2-D2P1-D1NDD P3

20. 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

21. PRA distribution in historical data PRA – “probability” for a patient to pass a “tissue-type” test with a random donor

22. Dynamic matching

23. 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

24. 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 Matches

25. Matching over time ( Anderson,Ashlagi,Gamrnik,Hil,Roth,Melcer 2014) 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

26. Pools with hard-to-match pairs

27. Homogenous (sparse) pools

31. 3-cycles: Simulation results for p = 0.08

32. 3-cycles: Simulation results for p = 0.05

33. Batching with maximal packing of cycles is monotone Shows that greedy is optimal up to a constant factor Greedy is “optimal”

34. Monotone policies

35. 3-cycles: Proof idea that greedy is good

37. Altruistic/non-directed donors Bridge donor

38. Greedy is “optimal”

39. Chains Summary of findings

40. 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?

41. 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.

42. Waiting a small period of time when 3-way cycles may be beneficial (Ashlagi, Jaillet, Manshadi 13) h1 l2 l1 l3 time

43. 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

44. Batch matching in a heterogeneous graph Theorem (Ashlagi, Jalliet and Manshadi): (i) When matching only 2-way cycles: 1. for Δ = o(n) (small batch size) M(Δ) = M(1) + o(n) 2. for Δ = αn, then (large batch size) M(Δ) = M(1) + f()n (ii) For 3-way cycles, there exist regimes in which even small batches (o(n)) are beneficial! Chunk matching finds a maximum matching at time steps Δ, 2Δ, …, n. M(Δ) - expected number of matched pairs at time n.

45. –Unver (2010) –Ashlagi, Jaillet,Manshadi (2013) –Akbarpour, Li, Gharan (2014) –Dickerson et al (2012) ….. Growing literature on dynamic matching

46. Kidney exchange in the US

47. 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

48. 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

49. Can collaboration between exchange programs be beneficial?

50. Benefits of merging patient-donor pools: over 3 years of data (with duplicates removed) NKR + APD + SA SA + APDNKR + APD NKR + SA All matches15% (3%) 11% (1.5%) 10% (3%)8% (2.5%) PRA >= 80 matches 28% (5%) 21% (5%)21% (4%)17% (25) PRA >= 9540% (10%) 25% (6%)27% (6%)22% (4%) PRA >= 9941% (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

51. 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… “

52. First 3-way exchange between APD and NKR (Summer 2013) DonorPatientPRA AAB48 AB 99 AA0

53. 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!?

54. 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

55. 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 Early optimization formulation 55

56. 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

57. 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 57 NDD

58. 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

59. 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

60. 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%!

61. 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!

62. 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!

63. Software we developed Exchange software

64. 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

65. 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

66. 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

67. 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…

68. 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.

69. 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?