K idney exchange - current challenges

1. Kidney exchange - current challenges Itai Ashlagi

2. 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” What are the design issues?

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

4. Factors determining transplant opportunity O • Blood compatibility • Tissue type compatibility • Panel Reactive Body –percentage of donors that will be tissue type incompatible to the patient A B AB

5. 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. A-A O-AB O-O B-B B-A AB-AB O-B AB-A AB-B A-B • VA-B B-AB A-AB A-O B-O O-A AB-O “Under-demanded” pairs

6. 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-AB B-A A-A O-O B-B O-B AB-AB AB-A AB-B A-B • VA-B B-AB A-AB A-O B-O O-A AB-O “Under-demanded” pairs

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

8. a1 a3 b e c a2 d

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

10. Transplanted internally and through NKR

11. n hospitals, each of a size bounded by c>0 . pairs/nodes are randomized –compatible pairs are disregarded Edges (tissue type compatibility) are randomized Random Compatibility Graphs Question: Does there exist an (almost) efficient individually rational allocation?

12. Ashlagi and Roth (2011): 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. Current mechanisms aren’t Individually rational for hospitals A-O O-A

13. 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) Incentive hard to match pairs! A-O O-A

14. Loss is Small - Simulations

15. 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)

16. 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

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

18. 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

19. 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

20. Matching over time Simulation results using 2 year data from NKR* Matches 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

21. Matching over time (Anderson,Ashlagi,Gamrnik,Hil,Roth,Melcer2014) 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

22. 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

23. If , then easy to achieve average waiting time patient-donor pools presently consist of many hard to match pairs We consider Homogenous (sparse) pools

24. Only two-cycles: • 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 Theorem[Anderson,Ashlagi,Gamarnik,Kanoria 14]:For greedy achieves and no policy can achieve better waiting times than greedy.

25. What about

26. 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

27. 3-cycles: Simulation results for p = 0.08

28. 3-cycles: Simulation results for p = 0.05

29. 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

30. 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

31. What about

32. 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 ?

33. Greedy is “optimal” • Theorem[Anderson, Ashlagi,Gamarnik,Kanoria]:For a single unbounded chain • Greedy achieves • For any policy

34. 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

35. 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 2-ways! With who to wait? How much? Can we do better than batching?

36. 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

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

38. Intuition for 2-way cycles 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 time

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

40. Kidney exchange in the US

41. 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 Transplants through kidney exchange in the US

42. Methodist San Antonio KPD program (since 2008) - includes compatible pairs • 210KPD 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

43. Can collaboration between exchange programs be beneficial?

44. Benefits of merging patient-donor pools: over 3 years of data (with duplicates removed) 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

45. Collaboration might be useful GaretHil (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… “

46. First 3-way exchange between APD and NKR (Summer 2013)

47. 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!? Innovation has come from having multiple kidney exchange programs

48. 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]

49. 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

50. Algorithms and software for kidney exchanges Integer Programming based algorithm for finding optimal cycle and chain based exchanges. Formulation I: 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.