Kidney Exchange with Immunosuppressants · Kidney Exchange with Immunosuppressants Eun Jeong Heo Sunghoon Hong Youngsub Chun Vanderbilt University KIPF Seoul National University July

Kidney Exchange with Immunosuppressants

Eun Jeong Heo Sunghoon Hong Youngsub Chun

Vanderbilt University KIPF Seoul National University

July 26, 2018

Workshop on Matching, Search and Market Design

National University of Singapore

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Introduction

In kidney transplantation problem, there are different ways of receivingtransplants depending on the availability of donors

• If a patient has a compatible donor,she receives the kidney directly from her own donor

• Otherwise, she can receive

(1) transplants from the deceased donors

(2) transplants through kidney exchange program

(3) transplants using immunosuppressants from incompatible donor

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Introduction







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Introduction







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Kidney Transplantation in Korea 2009-2016

A centralized system for organ procurement in Korea:

data from Korean Network for Organ Sharing (KONOS)

YearPatients

in waitlistsTotal

transplantsDeceased donor

transplantsLiving donortransplants

2009 4,769 1,238 488 750

2010 5,857 1,287 491 796

2011 7,426 1,639 680 959

2012 9,245 1,788 768 1,020

2013 11,381 1,761 750 1,011

2014 14,477 1,808 808 1,000

2015 16,011 1,891 901 990

2016 18,912 2,236 1,059 1,177

Table 1. Kidney transplantation in Korea

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Kidney Transplantation in Korea 2009-2016Among these living donor transplants,

YearLiving donortransplants

in total

Transplantswithin ABO

compatible pairs

Transplantsthrough

exchanges

Transplantswithin ABO

incompatible pairs

2009 750 675 (90.0%) 40 (5.3%) 35 (4.7%)

2010 796 689 (86.6%) 29 (3.6%) 78 (9.8%)

2011 959 828 (86.3%) 18 (1.9%) 113 (11.8%)

2012 1,020 827 (81.1%) 0 (0.0%) 193 (18.9%)

2013 1,011 795 (78.6%) 4 (0.4%) 212 (21.0%)

2014 1,000 783 (78.3%) 5 (0.5%) 212 (21.2%)

2015 990 772 (78.0%) 10 (1.0%) 208 (21.0%)

2016 1,177 901 (76.6%) 4 (0.3%) 272 (23.1%)

Table 2. Living donor kidney transplantation in Korea

As can be seen,

more suppressants, less kidney exchanges

in fact, almost no kidney exchange in Korea.

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Introduction

What are immunosuppressants? – without technical details

• Medications that decrease body’s reactions to the foreign organ

– It has been used to overcome (minor) tissue-type incompatibility

for (almost) all transplants to avoid transplant rejection

• More recently, it is developed for incompatible kidney transplants

by removing (tissue-type & blood-type) immunological constraints

– Pro: quite successful in terms of long-term survival rate(5-year survival rate: ABOi 95.5% and ABOc 96.7%)

no need to wait for a compatible donor

– Con: higher medical cost and small possibility of side effects

• Nevertheless, incompatible transplant is obviously better than no transplant

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Introduction

• Incompatible kidney transplants using suppressants have been increasing

in many countries (e.g. South Korea, Japan, Germany, Sweden and so on)

– Public health insurance covers the cost partially in most countries

– e.g. National Health Insurance in Korea covers up to 80% of total cost

• For incompatible transplants, the immunosuppressive protocols combine

Plasmapheresis and Rituximab

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Introduction

It is also conducted in the States (but not as extensive as other countries):

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Introduction

It is also conducted in the States (but not as extensive as other countries):

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Introduction

• In the current situation,

– there is no (centralized) system to utilize this new possibility;

– pairs decide to use it individually.

• We investigate how to use suppressants in an efficient way while keeping thesame number of transplants.

⇒ We propose to combine the kidney exchange and the suppressants.

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Introduction: A Motivating Example

• Suppose that only ABO blood-types matter

• Reminder: blood-type compatibility

O

A B

AB

e.g. donor O can give her kidneyto all patients

• A pair of patient and donor: X – Y

where X : patient’s blood type and Y : donor’s blood type

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• Consider three pairs of patient-donor: A-B, B-AB, and O-AB

(1) No direct transplants from own donors:

A-B B-AB O-AB

× × ×

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(2) No “trading cycle” exists: no kidney exchange possible

A-B B-AB O-AB

×

× ×

×

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(2) No “trading cycle” exists: no kidney exchange possible

A-B B-AB O-AB

×

× ×

×

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(3) Suppressants:

Patients use suppressants to receive transplants from own donors.

A-B B-AB O-AB

This is how suppressants are currently used in South Korea.

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(4) Suppressants: a better way of using suppressants

Find a chain and transform it into a cycle.

A-B B-AB O-ABA-B B-AB

A-B B-AB O-AB

All patients can receive transplants even though only two patientsuse suppressants. We use suppressants to “complete” cycles

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A-B B-AB O-ABA-B B-ABA-B B-AB O-AB


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A-B B-AB O-ABA-B B-ABA-B B-AB O-AB


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More generally, figure out chains and transform them to cycles

(each node: patient – donor pair)

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There are many ways to choose (cycles and) chains:

we propose a “better” choice in terms of fairness and efficiency.

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LiteratureKidney exchange:

Roth, A. E., Sonmez, T., and Unver, U. M., “Kidney Exchange,” QuarterlyJournal of Economics, 119 (2004), 457-488.

Roth, A. E., Sonmez, T., and Unver, U. M., “Pairwise Kidney Exchange”Journal of Economic Theory, 125 (2005), 151-188.

Roth, A. E., Sonmez, T., and Unver, U. M., “Efficient Kidney Exchange:Coincidence of Wants in Markets with Compatibility-Based Preferences,”American Economic Review, 97 (2007), 828-851.

Sonmez, T. and Unver, U. M., “Kidney Exchange: Past, Present, andPotential Future,” Slides Presented at the Eighth Biennial Conference onEconomic Design, Lund, Sweden, 2013.

Sonmez, T. and Unver, U. M., “Altruistically Unbalanced Kidney Exchange,”Journal of Economic Theory, 152 (2014), 105-129.

Sonmez, T. Unver, U. M., and Yılmaz, O., “How (Not) to Integrate BloodSubtyping Technology to Kidney Exchange,” mimeo, 2016.

Andersson, T. and Jorgen, K., “Kidney Exchange over the Blood GroupBarrier,” mimeo, 2016

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Model

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Model

• N ≡ {1, . . . , n}: the set of pairs

• Each pair i = (patient i – donor i) ⇒

Ri

N+i

N−i

• A problem (or a compatibility profile): R ≡ (Ri )i∈N

• A matching µ specifies which patient is matched to which donor

– A patient can be matched to an incompatible donor only if it is her own

– A patient receives a transplant if matched to a compatible donor

– M(R): the set of matchings at R.

– M∗(R): the set of maximal matchings at R.

• A priority over N: i � j ⇔ patient i has a higher priority than patient j .

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ModelPriority-based maximal matchings. Let 1 � 2 � · · · � n.

Let M0 ≡M∗(R) and for each k ∈ {1, · · · , n},

Mk ≡{{µ ∈Mk−1 : k ∈ N(µ)} if there is µ ∈Mk−1 such that k ∈ N(µ);

Mk−1 otherwise,

and lastly, let M∗�(R) ≡Mn (Roth et al. 2005).

Note that M∗�(R) is the subset of maximal matchings at which pairs withthe highest possible priorities are matched.

This set can be identified by using a linear programming or by usingmaximum weight matchings in a properly defined graph (Roth et al. 2005;Okumura 2014).

Also note that, from the definition, the matchings in M∗�(R) are “essentiallysingle-valued”: each pair is indifferent across all matchings in M∗�(R).

We choose a matching from M∗�(R) and set it as a benchmark when weextend the model to include suppressants.

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Model

We represent each R ∈ RN as a graph, g(R)

e.g.

R1 R2 R3

2 1, 3 2, 3

1, 3 2 1

• • •1 2 3

• Three cycles: 1→ 2→ 1, 2→ 3→ 2, 3→ 3

• Four chains: 1, 2, 1→ 2→ 3, 3→ 2→ 1

⇒ 1 and 2→ 3→ 2 are jointly feasible chain and cycle

a

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Model

• Start with the set of incompatible pairs N

• k ∈ Z+ : # of patients using suppressants (we try to minimize the usage)

• If i uses suppressant,

Ri

N+i

N−i

changes to

Ri

N

∅

e.g.

R2

1, 3

2

→R2

1, 2, 3 • • •1 2 3

• When patients in S use suppressants, R changes to RS ≡ ((Ri )i∈S ,R−S).

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Model

For each S ⊆ N and each RS ≡ ((Ri )i∈S ,R−S), let M(RS) be the set ofmatchings at RS at which each patient i ∈ S is matched to an incompatibledonor at Ri .

If patient i is matched to a compatible donor j at Ri , she receives acompatible transplant from donor j at µ.

When patient i is matched to donor j who is incompatible at Ri butcompatible at RS

i , she receives an incompatible transplant from donor j .

At each µ ∈M(RS), let C (µ) be the set of pairs whose patients receivecompatible transplants and I (µ) be the set of pairs whose patients receiveincompatible transplants.

Note that |I (µ)| ≤ K should always hold and patient i receives anincompatible transplant if and only if i ∈ S .

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Model

Timing

for each problem, R

Suppressants are allocated

(choose recipients of suppressants)

?They become compatible with all donors

update compatibility profile

?Matching between patients and donors

σ: a recipient choice rule

ϕ: a matching rule

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Model

Timing

for each problem, R

Suppressants are allocated

(choose recipients of suppressants)

?They become compatible with all donors

update compatibility profile

?Matching between patients and donors

σ: a recipient choice rule

ϕ: a matching rule

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Model

– A recipient choice rule, σ, that selects at most K patients from N.

– A matching rule, ϕ, that selects a set of matchings from M(Rσ(R)).

• A solution is a pair (σ, ϕ) such that

– For each R, choose σ(R)

– R changes to Rσ(R).

– Apply ϕ to this new profile and obtain matchings

ϕσ(R) ≡ ϕ(Rσ(R)): the set of matchings chosen by the solution.

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Properties of Solutions

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Properties of Solutions: (1) Efficient Matching

“For any compatibility profile, no other matching making

everyone weakly better off & at least one strictly better off”

• Pareto efficiency: For each R and each µ ∈ ϕσ(R), there is no µ ∈M(Rσ(R))that Pareto dominates µ at R.

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Properties of Solutions: (2) Fairness

“By introducing suppressants, all patients become weakly better off”

• Responsiveness: For each R, all patients weakly prefer each matching inϕσ(R) to each matching in M∗�(R).

Remark 1. Responsiveness can be trivially satisfied by choosing σ(R) = ∅ andϕσ(R) ∈M∗�(R) for all problems: simply by wasting all suppressants, peopleremain indifferent

⇒ reasonable to prevent such waste

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Properties of Solutions: (3) Efficient Use of Suppressants

“The recipients of suppressants should be chosen to maximize the total number oftransplants as well as the number of compatible transplants.”

• Maximality: For each solution (σ, ϕ), each R, each µ ∈ ϕσ(R), and eachµ ∈ ϕσ(R), |C (µ)| ≥ |C (µ)| and |N(µ)| ≥ |N(µ)|.

Remark 2. Maximality implies Pareto efficiency. Suppose that (σ, ϕ) satisfies theformer but not the latter. There are R and µ ∈M(Rσ(R)) that Pareto dominatesa matching µ ∈ ϕσ(R) at R. This implies that |C (µ)| ≥ |C (µ)| and|N(µ)| ≥ |N(µ)| with at least one strict inequality, a contradiction to maximalityof (σ, ϕ).

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Proposition 1. No solution jointly satisfies responsiveness and maximality.

Proof. Consider the following compatibility profile and the priority 1 � 2 � 3 � 4:

R =

R1 R2 R3 R4

3 3 1, 2 1

1, 2, 4 1, 2, 4 3, 4 2, 3, 4

•• • •12 3 4

When K = 0, the priority-based maximal matching is such that pairs 1 and 3are matched, while pairs 2 and 4 remain unmatched.

Suppose instead that K = 1. The solution satisfying maximality shouldassign a suppressant to patient 1 and match pairs 1 and 4 and pairs 2 and 3,respectively.

Note that patient 1 receives a compatible transplant from donor 3 whenK = 0, but then receives an incompatible transplant from donor 4 whenK = 1, getting worse off.

This violates responsiveness.

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This impossibility result is disappointing, but recall that we are mainly interestedin responsive solutions. We can define a weaker requirement than maximality byapplying the same idea to responsive solutions only.

• Constrained Maximality: For each responsive solution (σ, ϕ), each R, eachµ ∈ ϕσ(R), and each µ ∈ ϕσ(R), |C (µ)| ≥ |C (µ)| and |N(µ)| ≥ |N(µ)|.

Remark 3. In contrast to Remark 2, there is no logical relation between Paretoefficiency and constrained maximality.

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Pairwise Cycles and Chains (PCC) Solution

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Pairwise Cycles and Chains Solution

Step 1. Choose any matching from M∗�(R) and denote it by µ∗.

Step 2. Define R∗ as follows:

2.1 for each i ∈ N(µ∗) and each j ∈ N, if patient i is incompatible withdonor j but patient j is compatible with donor i , let these pairs mutuallyincompatible at R∗;

2.2 for each i /∈ N(µ∗) and each j ∈ N, if patient i is incompatible withdonor j , but patient j is compatible with donor i , let these pairs mutuallycompatible at R∗;

2.3 for each i /∈ N(µ∗), let patient i be compatible with her own donor i atR∗;

2.4 for all other pairs i , j ∈ N, let patient i be compatible with donor j at R∗

if and only if patient i is compatible with donor j at R.

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Step 3. Identify

X ≡ argmax{µ∈M(R∗):N(µ∗)⊆N(µ),|I (µ)|≤K}|N(µ)|,

X ≡ argminµ∈X |I (µ)|,

and choose µ ∈ X .

Step 4. Let σ(R) ≡ I (µ) be the recipients and {µ ∈ X : I (µ) = I (µ)} be theset of matchings chosen by the PCC solution.

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Step 1. Choose any matching from M∗�(R) and denote it by µ∗.

In Step 1, we identify a priority-based maximal matching µ∗ in the absence ofsuppressants and set it as a default matching.

Note that to satisfy responsiveness, the patients that are matched at µ∗

should receive compatible transplants when suppressants become available.

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Step 2. Define R∗:

2.1 for each i ∈ N(µ∗) and each j ∈ N, if patient i is incompatible with donor jbut patient j is compatible with donor i , let these pairs mutuallyincompatible at R∗;

2.2 for each i /∈ N(µ∗) and each j ∈ N, if patient i is incompatible with donor j ,but patient j is compatible with donor i , let these pairs mutually compatibleat R∗;

2.3 for each i /∈ N(µ∗), let patient i is compatible with her own donor i at R∗;

2.4 for all other pairs i , j ∈ N, let patient i be compatible with donor j at R∗ ifand only if patient i is compatible with donor j at R.

The modified profile R∗ in Step 2 is a key to achieving the threeaforementioned requirements.

(2.1): We delete all directed edges from the pairs in N(µ∗) to the pairsoutside N(µ∗). By deleting them, we remove the possibility that any pairin N(µ∗) is matched to an incompatible donor. This guaranteesresponsiveness of the final matching.

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Step 2. Define R∗:

2.1 for each i ∈ N(µ∗) and each j ∈ N, if patient i is incompatible with donor jbut patient j is compatible with donor i , let these pairs mutuallyincompatible at R∗;

2.2 for each i /∈ N(µ∗) and each j ∈ N, if patient i is incompatible with donor j ,but patient j is compatible with donor i , let these pairs mutually compatibleat R∗;

2.3 for each i /∈ N(µ∗), let patient i is compatible with her own donor i at R∗;

2.4 for all other pairs i , j ∈ N, let patient i be compatible with donor j at R∗ ifand only if patient i is compatible with donor j at R.

(2.2): We change all the remaining directed edges into undirected edges.

(2.3): We add self-directed edges for all nodes outside N(µ∗) and maintainall undirected edges as initially.

By doing (2.2)-(2.4), we get ready to find a largest set of self- and 2-waymatches.

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Step 3. Identify

X ≡ argmax{µ∈M(R∗):N(µ∗)⊆N(µ),|I (µ)|≤K}|N(µ)|,

X ≡ argminµ∈X |I (µ)|,

and choose µ ∈ X .

Step 4. Let σ(R) ≡ I (µ) be the recipients and {µ ∈ X : I (µ) = I (µ)} be the setof matchings chosen by the PCC solution.

In Step 3, we choose matchings at R∗ subject to the number of patientsreceiving incompatible transplants does not exceed K and all patients inN(µ∗) are matched.

If there are multiple matchings, we choose a matching that maximizes thenumber of transplants.

The choice is finalized in Step 4.

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Remark 4. A special case of this problem is when the K constraint does not bindand all patients receive transplants (e.g., K ≥ n). Formally, this is when there areS ⊆ N with |S | ≤ K and µ ∈M(RS) such that S ∩ N(µ∗) = ∅ and |N(µ)| = n.For this case, the PCC solution chooses a “minimax” matching in Step 3, which isa maximal matching that minimizes the use of suppressants, still subject toresponsiveness. Therefore, all patients will be matched and the number ofcompatible transplants will be maximized.

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Example

Example. Suppose that K = 3. Consider the following compatibility profile R andthe priority 1 � 2 � · · · � n.

R1 R2 R3 R4 R5 R6 R7 R8

3, 4 3 1, 2 ∅ 1, 2 ∅ ∅ 5, 7

N \ {3, 4} N \ {3} N \ {1, 2} N N \ {1, 2} N N N \ {5, 7}

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Example

•

•

• •

•

•

••

1

2

3 4

5

6

78

(a) Profile R

•

•

• •

•

•

••

1

2

3 4

5

6

78

(b) Profile R∗

Figure: Profiles R and R∗: in (b), we draw dashed edges between pairs that are mutuallycompatible at R∗, but are not mutually compatible at R. 44 / 80

Example

Step 1. We choose a matching from M∗�(R). In this example, the only possiblematching is µ∗(1) = 3 and therefore N(µ∗) = {1, 3}.

Step 2. We derive R∗ from R. Since N(µ∗) = {1, 3}, we delete a directed edgefrom 1 to 5 and transform the following directed edges to undirected edges: edgesfrom 4 to 1, from 2 to 5, from 5 to 8, from 7 to 8. We also add self-directededges for each node other than 1 and 3. We represent new edges as dotted onesin Figure 1(b). We therefore have the following profile R∗.

R∗1 R∗2 R∗3 R∗43, 4 2, 3, 5 1, 2 1, 4

N \ {3, 4} N \ {2, 3, 5} N \ {1, 2} N \ {1, 4}

R∗5 R∗6 R∗7 R∗82, 5, 8 6 7, 8 5, 7, 8

N \ {2, 5, 8} N \ {6} N \ {7, 8} N \ {5, 7, 8}

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Example

Step 3. We identify X ≡ {µ ∈M(R∗) : {1, 3} ⊆ N(µ), |I (µ)| ≤ K , and|N(µ)| ≥ |N(µ′)| ∀ µ′ ∈M(R∗)}. We find that X = {µ14, µ16, µ26, µ27, µ35, µ36}such that

µ1i (1) = 3

µ1i (2) = 5

µ1i (7) = 8

µ1i (i) = i

i = 4 or 6

µ2i (1) = 4

µ2i (2) = 3

µ2i (5) = 8

µ2i (i) = i

i = 6 or 7

µ3i (1) = 4

µ3i (2) = 3

µ3i (7) = 8

µ3i (i) = i

i = 5 or 6

Since all µ ∈ X , |I (µ)| = 3 = K , we choose one of the matchings identified aboveand set it as µ. For instance, let µ ≡ µ16.

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Example

Step 4. We finalize the choice. The PCC solution assigns suppressants to{2, 6, 7} and choose the final matching µ16.

•

•

• •

•

•

••© ©

©1

2

3 4

5

6

78

PCC matching

This figure describes the final choice, where the circled pairs are provided withsuppressants. �

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Example

Remark 4. In choosing µ in Step 3, the priorities can be used as in Section 2.Without loss of generality, let 1 � 2 � · · · � n. Let X0 ≡ X and for eachk ∈ {1, · · · , n}, let

Xk ≡{{µ ∈ Xk−1 : k ∈ N(µ)} if there is µ ∈ Xk−1 such that k ∈ N(µ);

Xk−1 otherwise,

and let µ ∈ Xn.

In Example 4, if we use the priority ordering 1 � 2 � · · · � n, we haveX0 = X1 = X2 = X3, X4 = X3 \ {µ16}, X5 = X4 \ {µ36}, andX8 = X7 = X6 = {µ26}.

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Main Result

TheoremThe PCC solution satisfies Pareto efficiency, responsiveness, and constrained

maximality.

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No restriction on the Length of Exchanges

We assume that only pairwise exchanges can be made due to the restrictionin operating transplants.

This algorithm can be modified to allow any length of exchanges.

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Simple Counterfactual Analysis

to quantify how much we can reduce the suppressants

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Counterfactual Analysis

• The use of suppressants for incompatible transplants is quite new:

→ little data about how suppressants are used in practice

• Good data is available in South Korea:

– the Korean Network for Organ Sharing (KONOS) data

– all living donor kidney transplants operated during 2011-2014

– ABO blood-type profiles of patient-donor pairs

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donor →

patient

↓Type A B O AB Total

A 210 35 80 24 349

B 28 172 77 24 301

O 38 39 164 5 246

AB 34 37 20 33 124

Total 310 283 341 86 1020

Table 3. KONOS data of living kidney transplantation in 2012

• Note that all these pairs already received transplants from living donors.

• Assuming that compatibility is determined only by blood types,

- compatible pairs: A-A, B-B, O-O, AB-AB as well as X-O, AB-A, AB-B

- incompatible pairs: A-B, B-A, O-A, O-B, O-AB, A-AB, B-AB

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donor →

patient

↓Type A B O AB Total

A 210 35 80 24 349

B 28 172 77 24 301

O 38 39 164 5 246

AB 34 37 20 33 124

Total 310 283 341 86 1020

Table 3. KONOS data of living kidney transplantation in 2012

• Note that all these pairs already received transplants from living donors.

• Assuming that compatibility is determined only by blood types,

- compatible pairs: A-A, B-B, O-O, AB-AB as well as X-O, AB-A, AB-B

- incompatible pairs: A-B, B-A, O-A, O-B, O-AB, A-AB, B-AB

• We collect all incompatible pairs.

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• We collect all incompatible pairs:

Type A-B B-A A-AB B-AB O-A O-B O-AB Total

# of pairs 35 28 24 24 38 39 5 193

Table 4. Blood-types of incompatible pairs in 2012

• These incompatible pairs should have received transplants

either through exchanges or by using suppressants

• In 2012, there were no kidney exchange at all

⇒ # incompatible pairs in the data = # suppressants used in 2012

⇒ That is, 193 patients used suppressants to receive transplants

[How much can we reduce the use of suppressants,

while all 193 patients still receive transplants?

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• We collect all incompatible pairs:


# of pairs 35 28 24 24 38 39 5 193

Table 4. Blood-types of incompatible pairs in 2012

• In this setting, we have

2-cycle: A-B � B-A

3-chain: O-A → A-B → B-AB, O-B → B-A → A-AB

2-chain:A-B → B-AB O-A → A-B O-A → A-AB

B-A → A-AB O-B → B-A O-B → B-AB

1-chain: A-B, B-A, A-AB, B-AB, O-A, O-B, O-AB

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The Minimum Chains Algorithm

Step 1. Identify and remove all 2 cycles:

find all (A-B � B-A) and remove them from the pool.

Step 2. Identify and remove all 3-chains:

find all 3-chains among the remaining pairs and remove them.

either O-A → A-B → B-AB, or O-B → B-A → A-AB

Step 3. Identify and remove all 2-chains:

find all 2-chains among the remaining pairs and remove them.

Step 4. Figure out all remaining 1-chains

⇒ Lastly, compute the number of chains identified from Step 2 to Step 4.

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The Minimum Chains Algorithm2-cycles → 3-chains among remaining → 2-chains among remaining → 1-chains

Table 5. The min algorithm for the set of incompatible pairs in 2012


# pairs 35 28 24 24 38 39 5 193

2-cycle 28 (7) 28 (0) 56

3-chain 7 (0) 7 (17) 7 (31) 21

2-chain 24 (0) 24 (7) 48

2-chain 17 (0) 17 (22) 34

1-chain 7 (0) 7

1-chain 22 (0) 22

1-chain 5 (0) 5

(28 2-cycles, 7 3-chains, 41 2-chains, and 34 1-chains : 82 chains)

• # of suppressants could have decreased from 193 to 82 (reduction of 57%):

⇒ Total reduction 111 = 56 reduction from cycles + 55 reduction from chains

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• Other years show similar counterfactual results.

# pairs 2-cycle 3-chain 2-chain 1-chain # Suppressants

2011 131 17 2 30 31 63

2012 193 28 7 41 34 82

2013 216 29 10 38 52 100

2014 217 34 3 48 44 95

• During 2011-2014, the use of suppressants could have been reduced by 55%without decreasing transplants (from 757 to 340 in total)

• We obtain a similar result with the hypothetical population

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Proposition

The minimum chains algorithm finds the minimal suppressants

needed to ensure that all incompatible pairs receive transplants.

why? making 2-cycles with pairs from chains does not increase # of chains

AO BA ABB

AB ABA

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Proposition




AO BA ABB

AB ABA

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Proposition




AO BA ABB

AB ABA

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Proposition




AO BA ABB

AB ABA

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Proposition




AO

BA

ABB

AB

ABA

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Proposition




AO

BA

ABB

AB

ABA

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Proposition




AO BA ABB

BO AB ABA

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Proposition




AO BA ABB

BO AB ABA

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Proposition




AO

BA

ABBBO

ABA

AB

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Proposition




AO

BA

ABBBO

ABA

AB

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Proposition



why? making 3-chains with pairs from chains does not increase # of chains

AO BA

BO ABB

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Proposition




AO BA

BO ABB

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Proposition




AO BA ABB

BO

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Remark. Relation between the minimum chains algorithm and the extendedTTCC

(1) Different “status quo”: current practice (with suppressants) vs no-suppressants

(2) Information: blood-type profile of those who already received transplants

vs profile of all incompatible pairs in the pool

⇒ the two algorithms are not directly related, but the min chains algorithm is

one possible approximation of the extended TTCC given these conditions.

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Summary

• Implications of introducing suppressants to the kidney exchange program.

• The extended TTC solution:

Pareto efficiency + individual rationality + maximal transplants

• Counterfactuals: the use of suppressants can be significantly reduced fromthe current practice.

Open questions

• Sequential vs simultaneous assignment of suppressants

• Incentives for the heads of chains: cost sharing?

• Generalization of dichotomous preference assumption

e.g. trichotomous preference: Andersson and Kratz (2016)

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Summary of the paper

I. Theory part: formalize this idea in kidney exchange program

(1) we propose additional requirements: efficiency and fairness

(2) we cannot satisfy all requirements: two impossibility results

(3) for attainable requirements, we construct the extended TTC

II. Counterfactual analysis:

To quantify the improvement as per our proposal,we conduct counterfactual analyses using data from South Korea

(1) identify all transplants and the number of suppressants used in practice

(2) compute the minimum suppressants needed for the same transplants;

(3) compare them: we could have reduced 55% of the use of suppressants

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Simulation

ABO blood types Frequency (percent)

O 27.32

A 34.26

B 26.90

AB 11.52

Patient gender Frequency (percent)

Female 40.54

Male 59.46

Unrelated living donors Frequency (percent)

Spouse 32.90

Other 67.10

PRA types Frequency (percent)

Low PRA 70.19

Medium PRA 20.00

High PRA 9.81Table 6: Patient and living donor distributions used in simulations based on KONOS Annual

Reports (2014-2016) and Roth et al. (2007)

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Simulation

The PRA type of a patient is randomly determined with respect to thefrequencies of types in Table 6.

I A patient is given low PRA type with probability of 70.19 percent. As in Rothet al. (2007), each patient of low PRA type has a positive crossmatch (PCM)with a donor with probability of 5 percent.

I Each patient of medium PRA type has a positive crossmatch with a donorwith probability of 45 percent.

I Each patient of high PRA type has a positive crossmatch with a donor withprobability of 90 percent.

There is a higher chance of having a positive crossmatch between a femalepatient and her spousal donor. The gender of a patient, as well as thespousal relationship between a patient and a donor, is randomly determinedwith respect to the frequencies in Table 6. For each female patient with ahusband donor, if she is of low PRA type, medium PRA type, and high PRAtype, she has a positive crossmatch with her husband donor with probabilityof 28.75 percent, 58.75 percent, and 92.5 percent, respectively.

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Simulation

If a patient-donor pair is ABO-compatible (ABOc) and no positivecrossmatch (PCM) occurs within the pair, the patient can receive atransplant directly from his or her own donor. Such a pair has no incentive toparticipate in an exchange pool. Thus, we assume that ABOc pairs having noPCM within the pairs do not participate in exchange pools. In other words, ifan ABOc pair participates, there is a positive crossmatch within the pair.

To construct exchange pools, we consider two cases, depending on theparticipation of ABOc pairs.

I In the first case, no ABOc pairs participate.I In the second case, ABOc pairs can participate. Each pool consists of ABOc

pairs with PCM and ABOi pairs.I We consider three subcases depending on the size of a pool, either 25, 50, or

100 pairs.I During the data generation process, for each pool, if an ABOc pair is

generated in the first case, or if an ABOc pair with no PCM is generated inthe second case, the pair is dropped automatically from the pool. This processis repeated until the required size of a pool is reached. We construct 500pools for each of the six cases.

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Simulation

ABOc par-ticipation

Poolsize

(1)2-cycles

and1-chains

(2)3-, 2-cycles,

and1-chains

(3)2-cycles,2- and

1- chains

(4)3-, 2-cycles,3-, 2- and1-chains

25 17.240 12.162 12.102

(2.849) (2.283) (2.313)

No 50 33.200 23.160 23.104

(4.002) (3.421) (3.482)

100 64.980 45.144 45.128

(5.483) (4.625) (4.640)

25 16.324 15.700 9.232 8.968

(2.797) (3.040) (1.948) (2.068)

Yes 50 26.060 23.776 16.064 15.888

(4.506) (4.823) (3.181) (3.272)

100 44.940 40.128 29.362 29.174

(6.849) (6.973) (4.963) (5.106)Table 7: Simulation results on the average minimum number of suppressants predicted by the

formulae; standard errors in parentheses

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SimulationBy comparing the average minimum numbers of chains across the fourcolumns, we observe that it can reduce the use of immunosuppressantssignificantly to allow the formation of 2-chains in addition to the formation ofcycles.

I Specifically, in an exchange pool of 100 ABOi pairs, when only 2-cycles areallowed, on average, 64.980 patients need to use immunosuppressants toreceive kidney transplants, as shown in column (1).

I However, when 2-chains are also allowed, on average, 45.144 patients need touse immunosuppressants to receive kidney transplants, as in column (3).

I Thus, the formation of 2-chains, in addition to the formation of 2-cycles, candecrease the use of immunosuppressants by about 19.836 (= 64.980 − 45.144)patients.

Similarly, in an exchange pool of 100 ABOc and ABOi pairs, the formation of2-chains can contribute to the decrease in the use of immunosuppressants byabout 15.578 (= 44.940− 29.362) patients. However, as in column (2), theformation of 3-cycles leads to a much smaller decrease in the use ofimmunosuppressants.Regardless of the participation of ABOc pairs and the size of exchange pools,as shown in columns (3) and (4), the formation of 3-cycles and 3-chains, inaddition to the formation of 2-cycles and 2-chains, does not lead to asignificant decrease in the use of immunosuppressants.

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Simulation

In Table 2, we also observe economies of scale with respect to the size ofexchange pools.

Regardless of the participation of ABOc pairs and the constraint on theformation of cycles and chains, as exchange pools become larger, the averageminimum numbers of chains decrease.

Suppose that there are 100 ABOc and ABOi pairs who can form only2-cycles and 2-chains, as in column (3). If they participate in four exchangepools with 25 pairs each, on average, about 36.928 (= 9.232× 4) patientsneed to use immunosuppressants to receive kidney transplants. In twoexchange pools with 50 pairs each, on average, about 32.128 (= 16.064× 2)patients need to use immunosuppressants. In one exchange pool with 100pairs, about 29.362 patients need to use immunosuppressants. Thus, it canhelp reduce the use of immunosuppressants to make larger exchange pools.

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Thank you

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Kidney Exchange with Immunosuppressants · Kidney Exchange with Immunosuppressants Eun Jeong Heo Sunghoon Hong Youngsub Chun Vanderbilt University KIPF Seoul National University July

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