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?