Can improved blood subtyping improve kidney exchange?
Tayfun Sönmez, M. Utku Ünver, Özgür Yılmaz
November 15, 2013
Abstract
Keywords : Mechanism design; Matching; Kidney exchange
Journal of Economic Literature Classification Number: C71, C78, D02, D63; I10
1
1
Introduction
2
Background
• Eighty percent of blood type A and AB individuals are subtype A1 and A1 B, respectively. The
other 20% of these blood types are subtype “non-A1 ,” most often A2 (or A2 B), but occasionally
a more rare subtype (e.g. A3 , Aint , etc.). Blood type “A, non-A1 ” individuals express only about
20% of all normal level of type A antigen on their RBCs and organs.
(Ref. Guidance for ABO Subtyping Organ Donors for Blood Groups A and AB)
• A2 and A2 B Matching
In order for a blood type B candidate to be eligible to be matched to a blood type A2 or A2 B
potential donor, or for a blood type O candidate to be eligible to match to a blood type A2
potential donor in the OPTN KPD Program, all of the following conditions must be met:
1. The candidate must have an IgG antibody titer value less than 1:8
2. The candidates Transplant Hospital must report to the OPTN Contractor the candidates
titer value and date of the test.
(Ref. Article 13.6.3.2, UNOS kidney exchange policy brief.)
3
A Model of Kidney Exchange
Let N denote the set of patient-donor pairs. The compatibility of any pair is determined with respect
to certain criteria and policies adopted by the transplant centers. These policies determine the set of
incompatible pairs who will participate the kidney exchange pool. Thus, the kidney exchange pool
depends on the policy and we denote the exchange pool (the incompatible patient-donor pairs) under
policy P by EP . A commonly used policy is the standard ABO blood type compatibility, or
simply ABO compatibility. According to this policy, there are four blood types A, B, AB, and O,
(representing the presence of proteins A or B, or both, or neither, respectively) and a patient may not
receive the kidney of a donor whose blood contains one of the A, B proteins that the patient does not
have. We denote the kidney exchange pool under ABO compatibility by EABO
Recently, the national kidney allocation system is revised to incorporate new features of expanded
access for certain blood type candidates who can accept kidneys from donors with blood-subtypes
2
of type A and AB. Based on the success rates of the transplantations from these blood-subtype
donors, it has been discussed to expand the compatibility structure and use it for live donation and
kidney exchange. One such policy of expanded access is such that in addition to the standard ABO
compatibility, A2 subtype of A blood-type donors and A2 B subtype of AB blood-type donors can
donate to B patients. We refer to this compatibility structure as A2 -to-B compatibility, and
denote the kidney exchange pool under A2 -to-B compatibility by EA2 →B . Another policy is to allow
transplantation from A2 subtype donors to O type patients. We refer to this policy as A2 -to-O
compatibility, and denote the kidney exchange pool under A2 -to-O compatibility by EA2 →O .
Assumption 1 (Upper-bound assumption)
No patient is tissue-type incompatible with another patient’s donor.
Assumption 2 (Large population of incompatible patient-donor pairs under ABO compatibility)
(i) For the exchange pool EABO , pairs of types O-A, O-B, O-AB, A-AB and B-AB are on the long
side of the exchange in the sense that at least one pair of each type remains unmatched in each feasible
set of exchanges.
(ii) There are at least two pairs of type A-A. The same is also true for each of the types B-B, AB-AB,
and O-O.
The number of A-O pairs in the exchange pool is determined by the probability of a patient having
a positive crossmatch probability with a donor. From the national patient characteristics, the ratio of
tissue-type incompatible A-O pairs is less than 20% of all A-O pairs (see PRA distribution in Table
1 and the part on low, medium and high PRA patients on page 839 in RSÜ (2007)). Also, eighty
percent of blood type A are subtype A1 . Thus, the exchange pool EA2 →O is such that the number
of O-A1 pairs is considerably greater than the number of tissue-type incompatible A-O pairs. Note
that tissue-type incompatible O-A2 pairs are also in the pool, but these pairs are not underdemanded
since an A2 donor is like an O donor and thus an O-A2 pair can be matched with a pair of the same
type or of type O-O. Thus, the pairs of type O-O and O-A2 are matched with each other, implying
that O-A2 pairs are not underdemanded.
Assumption 3 (Large population of incompatible patient-donor pairs under A2 -to-O compatibility)
For the exchange pool EA2 →O , pairs of types O-A1 , O-B, O-AB, A-AB and B-AB are on the long side
3
of the exchange in the sense that at least one pair of each type remains unmatched in each feasible set
of exchanges.
Since tissue-type compatibility is independent from blood-type compatibility, the argument for the
large population of incompatible patient-donor pairs under A2 -to-O compatibility holds for A2 -to-B
compatibility as well. This implies that pairs of type B-A1 B are on the long side of the exchange.
Moreover, since an A2 B donor is like a B donor, a pair of type B-A2 B can be matched with another
pair of the same type or of type B-B. This implies that tissue-type incompatible B-A2 B pairs are not
underdemanded.
Assumption 4 (Large population of incompatible patient-donor pairs under A2 -to-B compatibility)
For the exchange pool EA2 →B , pairs of types O-A, O-B, O-AB, A-AB and B-A1 B are on the long side
of the exchange in the sense that at least one pair of each type remains unmatched in each feasible set
of exchanges.
Assumption 5 The number of A-B pairs is greater than the number of B-A pairs.
3.1
Maximal-size Two-way Exchange
A pair of type X-Y denotes a pair of X-blood type patient and Y-blood type donor. Let (X-Y) be the
set of pairs of type X-Y. Let #T denote the cardinality of the set T. For a subtype Y, let (X − Y )c be
the set of tissue-type compatible X-Y pairs and (X − Y )i be the set of tissue-type incompatible X-Y
pairs.
For any nonnegative number k, let bkc denote the integer part of k, i.e. the greatest integer no
larger than k. For a set T , let oddT be equal to one if the the cardinality of T is odd, and zero if it
is even.
Proposition 1 If the policy changes from ABO compatibility to A2 -to-O compatibility, then
(i) the number of transplants via direct donation increases by #(O − A2 )c ,
(ii) the number of transplants via exchange increases by #(O − A2 )i + Θ,
(iii) the total number of transplants increases by #(O − A2 ) + Θ,
where Θ = 2 min{odd(A−A) , odd(O−O) } + odd(O−O)∪(A−A) odd(O−A2 )i − odd(O−A2 )i ∈ {−1, 0, 1, 2}
4
Proposition 2 If the policy changes from ABO compatibility to A2 -to-B compatibility, then
(i) the number of transplants via direct donation increases by #(B − A2 )c + #(B − A2 B)c ,
(ii) the number of transplants via exchange decreases by
2#(B − A2 )c − #(AB − A2 B) − #(B − A2 B)i − Λ,
(iii) the total number of transplants decreases by
#(B − A2 )c − #(AB − A2 B) − #(B − A2 B) − Λ,
where Λ = odd(AB-AB) + odd(B-B) − odd(AB-A1 B) − odd(B-A2 B)i ∪(B-B) ∈ {−2, −1, 0, 1, 2}.
3.2
Maximal-size Two-way and Three-way Exchange
Proposition 3 Consider the two-way and three-way exchange regime. If the policy changes from
ABO compatibility to A2 -to-O compatibility, then
(i) the number of transplants via direct donation increases by #(O − A2 )c ,
(ii) the number of transplants via exchange increases by #(O − A2 )i ,
(iii) the total number of transplants increases by #(O − A2 ).
Proposition 4 Consider the two-way and three-way exchange regime. Let
∆ = #(B-O) + #(AB-A) + #(B-A) − #(A-B).
If the policy changes from ABO compatibility to A2 -to-B compatibility, then
(i) the number of transplants via direct donation increases by #(B − A2 )c + #(B − A2 B)c ,
(ii) the number of transplants via exchange changes by
#(AB − A2 B) + #(B − A2 B)i − 2#(B − A2 )c + max{0, min{∆, #(B-A2 )c + #(A-A2 B)}}
(iii) the total number of transplants changes by
#(AB-A2 B) + #(B-A2 B) − #(B-A2 )c + max{0, min{∆, #(B-A2 )c + #(A-A2 B)}}.
5
4
Discussion
• Where and when are these tests (blood-subtype tests, and IgG antibody titer value tests) are
done? (The IgG antibody titer values are reported by the candidate’s transplant hospital to the
OPTN Contractor.)
• If the policy is switched to A2 -to-O or A2 -to-B compatibility, the maximal matching is obtained
by utilizing the pairs of type A-A2 , AB-A2 B. In general, tissue-type incompatible pairs might
not have incentives to test for the A2 subtype. A clear example is the pairs of type AB-AB. If
the number of such pairs is even, then any pair in this group is matched. So, they do not have
any incentive to test for the donors’ subtype. But, as we have seen in the proof of Proposition
2, although there is a decrease in the total number of transplants, to have a pool of pairs of
type AB-A2 B mitigates this efficiency loss. Thus, testing for subtypes needs to be incentivized.
Or maybe, as the general idea we have, the tests are provided by the pilot program via its
contractors. Well, the same question: where and when are these tests done?
• As the A2 -to-B compatibility policy results in efficiency losses, symmetrically, to propose a switch
from the ABO compatibility to a more restrictive policy (for example, the “only-the-same-type”
policy, i.e. a donor can donate only to a patient with the same blood-type) will bring all the
pairs of type A-O, AB-O, B-O, AB-A, and AB-B, to the exchange pool, and this will increase
the number of transplants. The question is: why can’t we switch from ABO to a more restrictive
policy, while we can switch from A2 -to-B to a more restrictive policy (i.e. ABO)? For that we
need incentives (See Sönmez, Ünver (2013)). But we should mention this such that the way we
bring this subtype policies up should exclude this possibility or the interpretation that we can
propose anything we want without incentives.
6
5
Appendix
Proof of Proposition 1:
(i) Consider the A2 -to-O compatibility policy. Since A2 subtype donor can donate to an O-type patient unless there is tissue-type incompatibility, each tissue-type compatible pair of type O-A2 does
not participate the exchange and the donor of the pair donates to the patient via a direct donation.
(ii) Each pair of type A-O is compatible with each pair of type O-A1 and by the assumption that pairs
of type O-A1 are on the long-side of the market, each pair of type A-O makes a two-way exchange
possible. Thus, the fact that tissue-type compatible pairs of type O-A2 does not participate the exchange, does not change the number of exchanges done with pairs of type A-O.
By the upper-bound assumption, each tissue-type incompatible pair of type O-A2 can be matched
with a pair of type O-A2 , type O-O, or type A-A2 . Similarly, each pair of type A-A2 can be matched
with a pair of type O-O. Thus, whenever both #(A − A) and #(O − O) are odd, while a pair of type
A-A and a pair of type O-O remain unmatched under ABO compatibility, a pair of type A-A2 can
be matched with a pair of type O-O under A2 -to-O compatibility, increasing the number transplants
by two via this exchange. Moreover, since each pair of type O-A2 can be matched with another pair
of the same type, the number of transplants via exchange by the pairs of type O-A2 increases by
2b
#(O-A2 )i
c.
2
Whenever both #(A − A) and #(O − O) are even, the number of transplants increases via exchanges
between two pairs of type O-A2 and the increase in the number is equal to 2b
#(O-A2 )i
c.
2
Whenever both # ((A − A) ∪ (O − O)) and #(O − A2 )i are odd, a pair of type O-A2 can be matched
with a pair of type A-A2 or type O-O, implying that each pair of type O-A2 , A-A, and O-O is matched
under A2 -to-O compatibility and that the number of transplants via exchange increases by the number
#(O − A2 )i + 1.
Whenever # ((A − A) ∪ (O − O)) is odd and #(O − A2 )i is even, each pair of type O-A2 is matched
with another pair of type O-A2 , implying that the number of transplants increases by #(O-A2 )i .
Thus, the number transplants via exchange increases by the number of pairs of type #(O − A2 )i plus
a residual which depends on whether the number of pairs of types O-A2 , O-O, and A-A2 are odd or
even. It is easy to check that this residual given in all the cases above is equal to Θ in the proposition.
(iii) The total number of transplants increases by the sum of the increase in the number of transplants
7
via direct donation and via exchange.
Proof of Proposition 2:
(i) Consider the A2 -to-B compatibility policy. Since A2 subtype donor can donate to a B-type patient
unless there is tissue-type incompatibility, each tissue-type compatible pair of type B-A2 does not
participate the exchange and the donor of the pair donates to the patient via a direct donation.
The same argument holds for the tissue-type compatible pairs of type B-A2 B. Thus, the number of
transplants via direct donation increases by the number #(B − A2 )c + #(B − A2 B)c .
(ii) Each pair of type B-A is compatible with each pair of type A-B and by the assumption that the
number of pairs of type A-B is greater than the number of pairs of type B-A, each pair of type B-A is
matched with a pair of type A-B under the ABO compatibility policy. Under the A2 -to-B compatibility
policy, since the tissue-type compatible pairs of type B-A2 do not participate the exchange, the pairs
of type B-A are still on the short side of the market. Actually, the set of blood-type incompatible B-A
pairs and the tissue-type incompatible B-A2 pairs is even shorter. Thus, the number of exchanges
decreases by the number of tissue-type compatible pairs of type B-A2 , and the number of transplants
via exchange by these two pairs, one pair of type A-B and one pair of type B-A, decreases by the
number 2#(B-A2 )c .
Under the A2 -to-B compatibility policy, while the number of transplants via pairwise exchanges
with B-A and A-B pairs decreases due to the fact that the tissue-type compatible pairs of type BA2 do not participate the exchange, there are new possible matches, which are not allowed under
the ABO compatibility policy. These possibilities can be seen by the well-known Gallai-Edmonds
Decomposition Lemma, which characterizes the structure of maximal matchings in our model.
Lemma 1 (Gallai-Edmonds Decomposition Lemma)
Let EA2 →B be a set of pairs. In each maximal matching, a pair, which is not underdemanded and is
compatible with an underdemanded pair, is matched with an underdemanded pair.
By the upper bound assumption, A2 -to-B compatibility implies that each pair of type AB-A2 B can
be matched with a pair of type B-A1 B or AB-A1 B. Since each pair of type AB-A2 B is on the short side
of the market and is compatible with an underdemanded pair (e.g. any pair of type B-A1 B), GallaiEdmonds Decomposition Lemma implies that it is matched with an underdemanded pair. While pairs
8
are of type B-A1 B are on the long side of the market (hence underdemanded), pairs of type AB-A1 B,
are underdemanded only if the number of pairs of type AB-A1 B is odd. In this case, at most one such
pair remains unmatched in a maximum matching and a pair of type AB-A2 B can be matched with
such an (underdemanded) pair of type AB-A1 B. Since the pairs of type B-A1 B are on the long side of
the market, for any maximum matching where a pair of type AB-A2 B and a pair of type AB-A1 B are
matched, one can obtain another maximum matching where that pair of type AB-A2 B is matched with
a pair of type B-A1 B, and a pair of type AB-A1 B remains unmatched. Thus, we can assume without
loss of generality that each pair of type AB-A2 B is matched with a pair of type B-A1 B. Since under
the ABO compatibility, each pair of type AB-AB is matched only with another pair of type AB-AB,
A2 -to-B compatibility increases the number of transplants via exchange, which includes a pair of type
AB-AB, by the number of pairs of type AB-A2 B. Moreover, each pair of type AB-A1 B is matched
with a pair of the same type. Also, each pair of type B-A2 B can be matched with a pair of type B-B.
Thus, by the upper-bound assumption, each pair in the set of pairs of types B-B and (B-A2 B)i can
be matched with a pair from the same set. Thus, the number of transplants via exchange from the
ABO compatibility to the A2 -to-B compatibility changes by the number
2#(AB-A2 B)+2b
#(B-A2 B)i + #(B-B)
#(AB-AB)
#(B-B)
#(AB-A1 B)
c+2b
c−2b
c−2b
c−2#(B-A2 )c .
2
2
2
2
Since 2b #T
2 c = #T − oddT and #(AB-AB) = #(AB-A1 B) + #(AB-A2 B), this number is equal to
#(AB-A2 B) + #(B-A2 B)i − odd(AB-A1 B) − odd(B-A2 B)i ∪(B-B) + odd(AB-AB) + odd(B-B) − 2#(B-A2 )c
Let Λ = odd(AB-AB) + odd(B-B) − odd(AB-A1 B) − odd(B-A2 B)i ∪(B-B) and note that Λ ∈ {−2, −1, 0, 1, 2}.
Moreover, given that in any community, the number of pairs including an AB-type donor or patient
is substantially lower compared to the number of pairs not including and AB-type donor or patients,1
this number is negative. Thus, the number of transplants via exchange decreases by the number
2#(B-A2 )c − #(AB-A2 B) − #(B-A2 B)i − Λ
(iii) The total number of transplants changes by the decrease in the number of transplants via ex1
See the related data on page XXX.
9
change minus the number of increase in the transplants via direct donation, and the result follows
from the fact that (B-A2 B) = (B-A2 B)i + (B-A2 B)c .
Proof of Proposition 3:
(i) Since direct donation is not affected by the exchange regime, the number transplants via direct
donation increases by #(O-A2 )c , as it is under the two-way exchange regime.
(ii) Three-way exchanges prevent any potential efficiency loss due to the number of pairs of type A-A
being odd and one pair of this type remaining unmatched. The same is also true for the pairs of type
O-O and type O-A2 . Thus, each tissue-type incompatible pair of type O-A2 can be matched with
another pair of the same type. If the number of tissue-type incompatible pairs of type O-A2 is even,
then each pair of type O-A2 is matched with another pair of the same type via a two-way exchange;
if it is odd, then three of these pairs are matched via a three-way exchange and the remaining pairs
are matched via two-way exchanges. Thus, the number of transplants via two-way exchange increases
by #(O-A2 )c − 3odd(O-A2 )i and the number of transplants via a three-way exchange increases by
3odd(O-A2 )i . Since switching to the A2 -to-B compatibility policy does not affect other two and threeway exchanges, the total number of transplants via exchange increases by #(O-A2 )i .
(iii) The result follows directly from (i) and (ii).
Proof of Proposition 4:
(i) By part (i) of Proposition 2 and the argument in part (i) of the proof of Proposition 3, the number
of transplants via direct donation increases by #(B − A2 )c + #(B − A2 B)c .
(ii) Case 1: ∆ ≤ 0. The gains due to subtype compatibilities are the shrinkage of the long-side of the
market by the pairs of types AB-A2 B and (B-A2 B)i . Also, only #(B-O) + #(AB-A) of the A-B type
pairs which are not matched to B-A type pairs can be matched via three-way exchange. The rest of
these pairs of type A-B remain unmatched. When switched to the A2 -to-B compatibility policy, not
only these pairs continue remaining unmatched but also #(B-A2 )c of the previously (under the ABO
compatibility) matched pairs become unmatched. Thus, since there are not enough B-O and AB-A
pairs to match with the unmatched A-B pairs via three-way exchange, under the two-way and threeway exchange regime, switching to the A2 -to-B compatibility policy does not increase the number of
10
transplants via three-way exchange but it decreases the number of transplants via two-way exchange
by 2#(B-A2 )c .
Case 2: ∆ > 0. Under three-way exchange, the pairs of type B-O types are matched via pairs of
types A-B and O-A (both on the long-side of the market), and the pairs of AB-A types are matched
via pairs of types A-B and B-AB (both on the long-side of the market) to increase the number of
exchanges compared to two-way exchanges. Under the A2 -to-B compatibility policy, any maximal
matching includes these three-way exchanges. But, since ∆ > 0, there are pairs of types B-O or ABA, which are not matched via a three-way exchange because there are not enough number pairs of type
A-B, which are not matched to pairs of type B-A. The A2 -to-B compatibility makes a new three-way
exchange possible: a pair of type A-A2 B, a pair of type B-O and a pair of type O-A. Thus, a pair of
type B-O which is not matched via a three-way exchange under the ABO compatibility policy, can
be matched with two pairs from the long-side of the market under the A2 -to-B compatibility policy.
Under the ABO compatibility policy, all the A-B pairs are already matched either with a B-A pair in
a two-way exchange or with B-O and O-A pairs (or AB-A and B-AB pairs) via three-way exchange
But, under A2 -to-B compatibility, since each pair in the set (B-A2 )c is compatible and leaves the
exchange pool for a live donation, the number of pairs of type A-B available for a three way exchange
increases by #(B-A2 )c . Thus, the total number of pairs of types A-B and A-A2 B available for threeway exchange is equal to #(B-A2 )c + #(A-A2 B). The number of these new three-way exchanges is
bounded above by ∆ > 0. Thus, the number of transplants via three-way exchange increases by
min{∆, #(B-A2 )c + #(A-A2 B)}.
(iii) The result follows directly from (i) and (ii).
5.1
Maximal-size Two-way, Three-way, and Four-way Exchange
Proposition 5 Let µ be a maximal matching when there is no restriction on the size of exchanges that
can be included in a matching. Then, there exists a maximal matching ν which consists only of twoway, three-way and four-way exchanges, under which the same set of patients benefit from exchange
as in matching µ.
Proof. Let µ be a maximal matching made of n-way or smaller exchanges with n ≥ 5. We will
construct a maximal matching ν made of (n − 1)-way or smaller exchanges such that it matches the
11
same set of patients as matching µ.
Let E = (P1 − D1 , P2 − D2 , . . . , Pn − Dn ) be an n-way exchange in µ. Since n ≥ 5 and there are
four blood types, A, B, O, and AB, there are at least two receiving agents in exchange E who are of
the same type. Suppose neither of these receiving agents is matched with the donating agent of the
other pair under exchange E. Without loss of generality, suppose these receiving agents are P1 and
Pn−1 . Under exchange E, Dn−2 donates to Pn−1 and Dn donates to P1 . Since P1 and Pn−1 are of
the same types, Dn−2 is compatible with P1 and Dn is compatible with Pn−1 . Thus, there are two
exchanges E 0 = (P1 − D1 , P2 − D2 , . . . , Pn−2 − Dn−2 ) and E 00 = (Pn−1 − Dn−1 , Pn − Dn ), sizes of n − 2
and two, respectively, with the same set of patients matched as in µ.
Suppose at least one of these two receiving agents is matched with the donating agent of the
other pair under exchange E. Without loss of generality, suppose these two agents are P1 and P2 .
Since P1 and P2 are of the same type, Dn is compatible with P2 as well, the n-way exchange E ? =
(P2 − D2 , . . . , Pn − Dn ) is feasible. Now we show that the pair P1 − D1 can be included in an exchange
without affecting pairs that are matched under µ.
Suppose D1 is ABO type compatible with P2 . Thus, type Y of D1 is ABO compatible with type X
of P1 . If X and Y are different types, then by Assumption 2, any pair of type Y − X is on the long side
of the market and there exists such a pair unmatched in µ. Thus, P1 − D1 can be matched with such a
pair and we can obtain a matching ν with a higher number pairs that are matched, contradicting that
µ is maximal. Thus, it must be that X = Y . In this case, any pair of type X − X must be matched in
µ, since otherwise, P1 − D1 can be matched with such an unmatched pair, increasing the number of
pairs matched and contradicting that µ is maximal. Let Pn+1 − Dn+1 be an X − X type pair matched
via an exchange Ê. If the exchange Ê involves less than four pairs, then P1 − D1 can be appended to
Ê just before or just after Pn+1 − Dn+1 . Suppose Ê is a larger exchange. Then, Pn+1 − Dn+1 can be
removed from Ê to form a two-way exchange with P1 − D1 , and by definition of ABO compatibility,
the donating agent to Pn+1 can donate to the patient, to whom Dn+1 donates in Ê.
Suppose P1 is of type B and D1 is of type A2 .2 By following the same argument above, all pairs of
type B-A2 and B-B are matched in µ, otherwise P1 −D1 can be matched with one of these pairs and this
contradicts the maximality of µ. Thus, there exists an exchange E 0 including this pair Pn+1 − Dn+1 of
type B-B. If E 0 involves less than four pairs, P1 − D1 can be appended to E 0 just before Pn+1 − Dn+1 .
2
The analysis that follows is the same if the type D1 is A2 B.
12
Suppose E 0 is a larger exchange. By definition of ABO compatibility, the agent who donates to Pn+1
can also donate to the patient, to whom Dn+1 donates. Thus, the pair Pn+1 − Dn+1 can be removed
from the exchange E 0 to form a two-way exchange with the pair P1 − D1 and we are done.
Thus, if µ involves an exchange with n pairs with n ≥ 5, then we can find another maximal
matching ν made of (n − 1)-way or smaller exchanges such that it matches the same set of patients
as matching µ. Since the argument relies only on the fact that n ≥ 5, inductively, we can construct
a maximal matching made of four-way or smaller exchanges such that it matches the same set of
patients as matching µ.
13