A dynamic graph model of kidney exchange

A dynamic graph model of kidney exchange

Yashodhan KanoriaMicrosoft Research New England & Columbia

Joint work with Ross Anderson, Itai Ashlagi and David Gamarnik

MIT

Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange

Over 90,000 patients on the waiting list for cadaver kidneys in the U.S. today

In 2011:• 33,581 patients were added to the kidney waiting

list, and 28,625 patients were removed from the list.

• 11,043 transplants of cadaver kidneys performed.• 4,697 patients died while on the waiting list and

2,466 others removed from the list as “Too Sick to Transplant”.

• 5,771 transplants of kidneys from living donors in the US.

Kidney transplants


Donor 1Blood type X

Recipient 1Blood type Y

Donor 2Blood type Y

Recipient 2Blood type X

2-way kidney exchange

Kidney exchange


Donor 1 Recipient 1

Donor 2 Recipient 2Donor 3 Recipient 3

Pair 3

3-pair exchange (6 simultaneous surgeries)

Pair 1

Pair 2


Compatibility graph

1 2 3

5

7

6

8

4 9


• 4-way and larger exchanges have been successfully demonstrated

• However, significant challenges in conducting very large exchanges

Multi-way exchanges


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?


Suppose, all donor-patient pairs have same probability of being compatible 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


If , then easy to achieve average waiting time

• But hospitals withhold easy to match pairs from exchanges (Ashlagi et al. 2011)

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

We consider


• Two-cycle formed between any two nodes w.p. • Greedy exchange achieves • Not hard to show that for any policy • Hence, greedy achieves order optimal

Proposition: Greedy is optimal up to constants for

Only two-cycles:


What about


• If batch size is then • We want to eliminate most of batch, so triangles

needed• Hence, need

Can show that batch size gives

How does greedy compare?

Batching for


• Greedy removes 2 & 3 cycles as soon as available

• For a typical time , number of waiting nodes • Residual graph contains no 2 & 3 cycles, less

dense than ER• Optimistically contains 2 edges

Greedy for


• Residual graph optimistically contains 2 edges • Probability that 2 or 3-cycle formed is in steady

state• Probability of 3-cycle formation ~

Need to make this • Probability of 2-cycle formation ~

Need to make this • So 3-cycle formation dominates, and ,

heuristicallySeems like greedy may not do to badly

Greedy for


1 2 4 8 16 32 62 1280

10

20

30

40

50

60

70

Size of batch

W

Simulation results: p = 0.081𝑝1.5 ≈44.2


Simulation results: p = 0.05

1 2 4 8 16 32 62 1280

20

40

60

80

100

120

Size of batch

W

1𝑝1.5 ≈90


Simulation results: p = 0.02

1 2 4 8 16 32 62 128200

210

220

230

240

250

260

270

280

Size of batch

W

1𝑝1.5 ≈350


Optimal batch size is 1 (i.e., greedy beats batching)

Under greedy for small

What can we prove?

Summary of simulation results


• Batching with maximal packing of cycles is monotone

• Shows that greedy is optimal up to a constant factor

Open problem: get rid of the constant factor slack, and consider all possible policies

Main result

Theorem: 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 emptyConsider next arrivals. For appropriate show:• Few new arrivals persist till • Few triangles formed internal to new arrivals• So most new arrivals form cycles containing

old nodes, leading to, whp,

Proof idea: greedy is good


Consider graph of compatibility G between all nodes that ever arrive to the system.

A policy is monotone if:Fix all edges in G except for . Presence of only makes and disappear (weakly)

earlier.

Definition: monotone policies


• Proof by contradiction. Assume .• w.p. at least . Assume this.• Under monotone policy, • Probability of immediate triangle formation for node is• Whp, no more than edges formed between and .

Assume this.• Probability forms triangle with next arrivals • With probability node lives longer than

Proof that no monotone policy can beat greedy


We analyzed a dynamic graph/graph structured queue:showed that greedy is nearly optimal. Suggests that greedy should work well in kidney exchanges.Caveats:• Greedy proved optimal only up to constant factors• Only consider monotone policies

Conclusion

Conjecture: For greedy gives ,and no policy can do better.


• General result on ER-type graph structured queues with removal of given constant sized substructures?

• Kidneys: Multitype model with only some hard-to-match patients?Can we do better than greedy?

Future work


Thank you!


Pair 1

Pair 2

Pair 3

Pair 4

Pair 6

Pair 7

Pair 5

Altruistic donor

Altruistic donors: cycles plus chains



• One altruistic donor at every stage(initially a volunteer, later a donor whose patient already got a kidney)

• A node arrives at each forms link with each existing node independently with probability

• Can eliminate any chain starting with altruistic donor. Last node in chain becomes new altruistic donor

Question: What is the optimal policy? Greedy or batch?

Model


• d

Batch produces matching upper bound


Ongoing work: what about greedy?


Future work


Pair 1

Pair 2

Pair 3

Pair 4

Pair 6

Pair 7

Pair 5

Altruistic donor

Altruistic donors: cycles plus chains

33


Previous efficiency results

37

In a really large market efficiency is gained with short cycles:

Roth, Sonmez & Ünver, AER 2007 – if there are no tissue type incompatibilities, no need for exchanges of size >4

Ünver, ReStud 2009 - efficient dynamic kidney exchange assuming no tissue type incompatibilities - exchanges of size > 4 are not needed

Ashlagi & Roth 2010, in large random exchange pools, no need for exchanges of size>3

Toulis and Parkes 2011, similar results


n hospitals, each of a size c>0 D(n) - random compatibility graph:1. n pairs/nodes are randomized –compatible pairs are

disregarded2. Edges (crossmatches) are randomized

Random graphs will allow us to ask two related questions:What would efficient matches look like in an “ideal”

large world?

Random Compatibility Graphs

38


Theorem (Erdos-Renyi) G(n,p) contains a perfect matching with probability approaching 1 as n grows for even n when p>log n/n.

- Random graph on n nodes with edge probability p

“Proof”: Say . Use Use greedy algorithm. Probability of failure in step k is

As long as Probability of failure at any step is

Matchings in random graphs

B-A

B-AB A-AB

VA-B

A-O B-O

AB-O

O-B O-A

A-B

AB-B

AB-A

O-AB

O-OA-A B-

B AB-

AB

Efficiency in a large poolTheorem (Ashlagi and Roth, 2011): In almost every large random graph (directed edges are created with probability p) there is an efficient allocation with exchanges of size at most 3.

“Under-demanded” pairs


Non-simultaneous extended altruistic donor chains (reduced risk from a broken link)

41

A. Conventional 2-way Matching

R1 R2

D1 D2

B. NEAD Chain Matching

R1 R2

D1 D2LND

A. Conventional 2-way Matching

R1 R2

D1 D2

R1 R2

D1 D2


R1 R2

D1 D2LND


R1 R2

D1 D2LND

Since non-directed donor chains don’t require simultaneity, they can be longer…


The First NEAD Chain (Rees, APD)

42

Recipient PRA

* This recipient required desensitization to Blood Group (AHG Titer of 1/8).# This recipient required desensitization to HLA DSA by T and B cell flow cytometry.

MI

O

AZ

July2007

O

O

62

1

Cauc

OH

July2007

A

O

0

2

Cauc

OH

Sept2007

A

A

23

3

Cauc

OH

Sept2007

B

A

0

4

Cauc

MD

Feb2008

A

B

100

5

Cauc

MD

Feb2008

A

A

64

7

Cauc

NC

Feb2008

AB

A

3

8

Cauc

OH

March2008

AB

A

46

10

AA Recipient Ethnicity

MD

Feb2008

A

A

78

6

Hisp

# *

MD

March2008

A

A

100

9

Cauc

HusbandWife

MotherDaughter

DaughterMother

SisterBrother

WifeHusband

FatherDaughter

HusbandWife

FriendFriend

BrotherBrother

DaughterMother

Relationship



In a really large market efficiency is gained with short cycles…

Are NEAD chains effective?

44

B-A

B-AB A-AB

VA-B

A-O B-O

AB-O

O-B O-A

A-B

AB-B

AB-A

O-AB

O-OA-A B-

B AB-

AB

Efficiency in a large pool

altruistic donorAn altruistic donor can increase the

match size by at most 3


• The large graph model with constant p (for each kind of patient-donor pair) predicts that only short chains are useful.

• But we now see long chains in practice.• They could be inefficient—i.e. competing with

short cycles for the same transplants.• But this isn’t the the case when we examine the

data.

A disconnect between model and data:

46


We have formulated and solved on real data

One donor added

Long cycles and altruistic donors help!


Why? many very highly sensitized patients

48

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

49

PRA – “probability” for a patient to pass a cross-match test (tissue type) with a random donor


Short cycles leave many highly sensitized patients unmatched


A real graph

Graph induced by pairs with A patients and A donors 38 pairs, only 5 can be covered by some cycle


Jellyfish structure of the compatibility graph: highly connected low sensitized pairs, sparse hi-

sensitized pairs

52


Cycles and paths in random dense-sparse graphs

• n nodes. Each node is L w.p. q<1/2 and H w.p. 1-q

• incoming edges to L are drawn w.p.


L

H

53


Cycles and paths in random sparse (sub)graphs (v=0, only highly sensitized patients)

H

Theorem. (a) The number of cycles of length O(1) is O(1). (b) But when pH is a large constant there is cycle with length O(n)

“Proof” (a):

54To be logistically feasible, a long cycle must be a chain, i.e. contain a NDD


Cycles and paths in random sparse graphs (v=0)

H

Theorem. (a) The number of cycles of length O(1) is O(1). (b) But when pH is a large constant there is path with length O(n)

Since cycles need to be short (as they need to be conducted simultaneously) but chains can be long (as they can be initiated by an altruistic donor,) the value of a non-directed donor is very large!

55


Case v>0. Why increasing cycle bound helps?

L

H

Theorem. Let Ck be the largest number of transplants achievable with cycles · k. Let Dk be the largest number of transplants achievable with cycles · k plus one altruistic donor. Then for every constant k there exists ½>0

Furthermore, Ck and Dk cover almost all L nodes.


Fact: in almost every random directed random graph D(n,c/n) every tree of constant size appears linearly many times and there are no constant size cycles

Lemma: Let p(n)=c/n. Almost every random bipartite graph G(qn,(1-q)n,p(n)) has a maximum matching of a linear size z(c,q)qn, 0<z(c,q)<1

Some more on random graphs

57

qn nodes

(1-À)n nodes


Definition: u,v1,v2,…,vk is a good cycle if:• u is L and all other nodes are H• the only L node that has an edge to v1,v2,…,vk is u• the only H node that u has an edge to is v1

• No edges from v1,v2,…,vk to other H nodes• No edges from v2,…,vk to u

Claim: there are linearly many good cycles of length k+1

L

Hv1

v2v3

u


L

H

Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graph

k=3


L

H

Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graph

k=3


L

H

Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graphStep 2: find maximum number of disjoint edges from L to beginnings of paths

k=3


L

H

Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graphStep 2: find maximum number of disjoint edges from L to beginnings of paths

k=3


L

H

Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graphStep 2: find maximum number of disjoint edges from L to beginnings of pathsStep 3: there is a linear number of edges that close good cycles from the last nodes of the established paths

k=3


L

H

Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graphStep 2: find maximum number of disjoint edges from L to beginnings of pathsStep 3: there is a linear number of edges that close good cycles from the last nodes of the established paths

k=3


L

H

So far: linearly many good cycles of length k+1 Final step: Start from an allocation Qk and construct from it a Qk+1

allocation that adds a linear term (using the good cycles of length k+1)

k=3


L

H



Add a good cycle if it is disjoint from Qk or delete the cycle that contains u in Qk and add it …

k=3

v2 v3v1

u


L

H




k=3

v2 v3v1

u


L

H




k=3

v2 v3v1

u


Long chains benefit highly sensitized patients (without harming low-sensitized patients)

69


NYTimes February 18, 2012. 60 lives, 30 kidneys


What is the tradeoff between waiting and number of matches?

Dynamic matching in dense graphs (Unver, ReStud,2010).

What about dynamics?


Matching over time

72

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

73




Matches – high PRA

1 5 10 20 32 64 100 260 520 104190

110

130

150

170

190

210

230

2-ways3-ways2-ways & chain3-ways & chain


Matching over time

74

1D 1W 2W 1M 3M 6M 1Y250255260265270275280285290295

Matches


1D 1W 2W 1M 3M 6M 1Y100120140160180200220240

Waiting Time




Matching over time

75

Simulation results using 2 year data from NKR

1D 1W 2W 1M 3M 6M 1Y4045505560657075

Matches – high PRA

1D 1W 2W 1M 3M 6M 1Y210

230

250

270

290

Waiting – high PRA


Match the pair right away?

A H-node forms an edge with each node u of U with probability ξ/n. A L-node forms an edge with each node u of U with probability π

76

Arriving pair

Lemma: the online algorithm matches almost all pairs when p is a constant and n is large enough (even with just 2-way cycles)

Online:match the arrived node to a neighbor; remove cycles when

formed.

Either a sparse finite horizon modelor an infinite horizon model and analyze steady state


Dynamic matching in dense-sparse graphs




L

H

77


Dynamic matching in dense-sparse graphs




L

H

78

At each time step 1,2,…, n, one node arrives.

Heterogeneous Dynamic Model (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.

79

pc/n

𝑝2

Chunk Matching in a heterogeneous graph

80

At time steps Δ, 2Δ, …, n:

Find maximum matching in H-L; remove the matched nodes.

Find maximum matching in L-L; remove the matched nodes.


81

Theorem (Ashlagi, Jalliet and Manshadi): When matching only 2-way or 2+3-way cycles:

1. If Δ = o(n), M(Δ) = M(1) + o(n)

2. Δ = αn, then M(Δ) = M(1) + f(q)n

for strictly increasing f()>0.

Chunk matching finds a maximum matching at time steps Δ, 2Δ, …, n.M(Δ) - expected number of matched pairs at time n.

Denser Poolsξ:

82

Theorem: 1. If Δ < 1/,

M(Δ) = M(1) + o(n)2. If Δ = α/

M(Δ) = M(1) + f(q)nfor strictly increasing f()>0.

Need to wait less time to gain… If the graph is dense (large) – no need to wait at all…

Proof Ideas Special structure: Sparse H-L and dense L-L.

(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.

Compare the number of H-L matchings.

83

pξ/n

𝑝2

Proof IdeasIn H-L graph, Δ = o(n):

No edge in the residual graph.

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.

Decision of online and chunk matching are the same on depth-one trees. M(Δ) = M(1) + o(n).

84

arrived chunk

residual graph

In H-L graph, Δ = αn: Find f(α)n augmenting paths to the matching obtained by online. Given M the matching of the online scheme:

Chunk matching would choose (l1,h1) and (l2,h2). M(Δ) = M(1) + f(α)n,

85

Proof Ideas

h1

l2 l1

h2


86

Theorem (Ashlagi, Jalliet and Manshadi):

MC(1) = M(1) + f(q)n

Chunk matching finds a maximum matching at time steps Δ, 2Δ, …, n.M(Δ) - expected number of matched pairs at time n when matching only 2-ways MC(Δ) - expected number of matched pairs at time n when matching 2-ways and allowing one unbounded chain.


Merging NKR and APD

5 10 20 50 100 250480500520540560580600

Pairs matched

5 10 20 50 100 250160

180

200

220

240

PRA >= 80 matched

5 10 20 50 100 2505060708090

100110120

PRA >= 97 matched

5 10 20 50 100 250202530354045505560

PRA >= 99 matched


• Current pools contain many highly sensitized patients and (long) chains are very effective.

(Partially since hospitals don’t share all their easy to match pairs.)• In those highly sensitized pools, number of

matches increase significantly only when waiting “for a long” time between match runs -> use more chains!

• Many more matches from pooling, especially highly sensitized patients.

Conclusions

89


Merging exchange programs

NKR Korea APD Korea SA Korea APD NKR APD SApairs 222 81 196 81 173 81 196 222 196 173

matched 30 16 49 16 59 16 49 30 49 59

Average PRA 61.9 7.1 57.8 7.1 59.1 7.1 57.8 61.9 57.8 59.1

PRA OD 75 43 85.6 43 75.2 43 85.6 75 85.6 75.2Pairs 192 65 147 65 114 65 147 192 147 114

matched 6,6 4,5 4,8 3,7 0,5 0,1 0,13 0,15 2,13 4,22

PRA>80 5,5 1,1 3,6 0,0 0,5 0,0 0,11 0,9 2,11 2,18OD 4,4 1,1 1,4 0,0 0,5 0,0 0,9 0,8 2,9 2,15O

donors 2,2 0,0 1,4 0,0 0,3 0,0 0,8 0,6 2,9 1,9

PRA OD 95,95 97,97 100,96.8 -,- -,98.8 -,- -,97.7 -,96.8 100,97.9 97.5,97.3


2000

2001

2002 2003 2004 2005 2006 2007 2008 2009

2010

#Kidney exchange transplants in US*

2 4 6 19 34 27 74 121 240 304 422 (+203 +139)*

Deceased donor waiting list (active + inactive) in thousands

54

56

59 61 65 68 73 78 83 88 89.9

Kidney exchange is progressing, but progress is still slow

*http://optn.transplant.hrsa.gov/latestData/rptData.asp Living Donor Transplants By Donor Relation•UNOS 2010: Paired exchange + anonymous (ndd?) + list exchange

In 2011: 11,043 transplants from deceased donors 5,769 transplants from living donors

s

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A dynamic graph model of kidney exchange