Live Donor Organ Exchange: Beyond Kidneys

Transcription

1 Haluk Ergin Tayfun Sönmez M. Utku Ünver UC Berkeley Boston College Boston College

2 Introduction Kidney Exchange became a wide-spread modality of transplantation within the last decade. More than 500 patients a year receive kidney transplant in the US along through exchange, about 10% of all live-donor transplants. In theory live donor organ exchange can be utilized for any organ for which live donation is feasible.

3 Institutions Human organs cannot received or given in exchange for "valuable consideration" (US, NOTA 1984, WHO) However, live donor kidney exchange is not considered as "valuable consideration" (US NOTA amendment, 2007) Livers and lungs are two of the other organs for which live donation is feasible.

4 Live Donor Lobar Donation for Livers and Lungs Livers: One donor donates usually left lobe of her liver (about 2/5th of her liver) to a patient. Both pieces of liver regenerate. Lungs: Two donors each donate to a single patient a lobe of their lungs (less than 1/4th of total lung volume) to a donor. Lung lobes enlarge but do not regenerate. Liver exchange has started being conducted in Korea.

5 Contribution We introduce a new transplant modality to the attention of scientific community: Live-donor lobar lung exchange We model liver and lung exchange as matching problems to characterize the maximum number of patients that can be saved under different institutional constraints and find simple algorithms to find optimal exchanges. We analyze the effects of patient-donor size constraints on liver exchange. We show that the requirement of size compatibility can increase the overall number of transplants: decreases the direct donation, increases exchange, and overall effect can be positive. We simulate gains from exchange for livers and lungs to show that an organized lung exchange program can almost triple the number of patients who receive live donor lung donation, much more than both kidney and liver donation.

6 Literature Kidney Exchange: Among many Rapaport [1986] proposed the idea Ross et al. [1997] proposed ethical implementation grounds Roth, Sönmez, Ünver [2004, 2005, 2007] introduced optimization, matching, and market design techniques Segev et al. [2005] simulated gains, approval of the optimization techniques among doctors Saidman et al. [2006] proposed non-simultaneous NDD chains Abraham, Blum, Sandholm [2007] designed an efficient algorithm for the NP-complete computational problem Rees et al. [2010] proof of concept of non-simultaneous NDD-chains Ünver [2010] dynamically optimal clearinghouses Sönmez & Ünver [2014,2015] compatible pairs in exchange Roth, Sönmez, Ünver[2005] and Ashlagi & Roth [2014] multi-hospital exchange programs

7 Literature Liver Exchange: Only three papers Hwang et al. [2010] proposed the idea and documented the practice in Korea since 2003 Chen et al. [2010] documented the program in Hong Kong Dickerson & Sandholm [2014] simulated gains from liver exchange and proposed joint liver+kidney exchange Lung Exchange: Ours is the first

8 Living Donor Liver Donation Liver Transplantation Living Donor Liver Transplantation Live donor liver donation is more common in Asian countries where Duedeceased to cultural di erences, donation is living at donor a minimum liver transplantation due to cultural is considerably more common than deceased donor liver transplantation differences and illegality of "brain death" in Asian countries. Annual liver transplant activity per million population Figure from Chen et al Nature Reviews Gastroenterology & Hepatology /54

9 Lobar Liver Donation

10 Lobar Liver Donation

11 Lobar Liver Donation: Compatibility Blood-type compatibility is required (like kidneys). Tissue-type compatibility is not required (unlike kidneys). Size compatibility is needed (not a big problem for kidneys). A patient requires a graft relatively large to survive.

12 Right Lobar Liver Transplant Liver Transplantation Increased Right lobar Use transplant of Right has Lobe been over utilized Time for size compatibility despite Sinceits theheightened left lobe of the donormortality is often toorisk. small for the patient, A high rightprofile lobe transplantations 2001 deathhave of aincreased live right overlobe time, donor despiteinfivefold the US mortality risk. decreased live donation steadily not only for livers, but for other This is possibly at odds with the oath all physicians pledge to keep: organs including kidneys in the US. Primum non nocere First, cause no harm. Figure from Shimada et al The Journal of Medical Investigation /54

13 Liver Exchange Liver exchange first done in Korea, followed by Hong Kong and Turkey. Liver exchange can have two benefits: It can increase the number of transplants. It can decrease right lobar transplants (and increase donor safety) through matching with respect to size.

14 Lobar Lung Donation Lung Transplantation Living Donor Lobar Lung Transplantation The right lung is divided into three lobes, whereas the left lobe is divided into two lobes. LDLLT involves donation of a lower lobe from each of two blood type and size compatible living donors. Finding two compatible donors is di cult, suggesting that gains from lung exchange might be considerable! Figure from Date et al. Multimedia Manual of Cardiothoracic Surgery 2005 Size compatibility and blood-type compatibility are required. No consensus on tissue-type compatibility, many transplant centers do not check. 16/54

15 Lung Exchange Finding two compatible donors is difficult. Lung exchange can substantially increase the number of transplants.

16 Possible Two&Three-way Lung Exchanges Two-Way:

17 Possible Two&Three-way Lung Exchanges Two-Way: Three-Way:

18 Umbrella Model for Organ Exchanges Each patient in need of an organ has k attached donors If all of them are compatible with her, she receives from them; Otherwise, she participates in exchange Preferences: Dichotomous over compatible donors Compatibility: Blood-type: Kidneys, Lungs, Livers Tissue-type: Kidneys, possibly Lungs Size: Lungs, Livers (roughly: each patient can get grafts from donors that are at least as heavy/tall as herself; the constraint could be more detailed for livers) Number of Required Donors: k k = 1 : Kidneys, Livers k = 2 : Lungs Model 0: Kidneys Roth, Sönmez, Ünver [2005]

19 Model 1: Lung Exchange As first approximation we abstract away from size compatibility. Blood types: O, A, B, AB Blood-type incompatibility: Tissue-type incompatibility: X Size incompatibility: X Number of donors: 2 There is an equivalent interpretation: A, O are the most common blood types, making up of 85% of the population. In this interpretation, suppose there are two types of agents large (l) and small (s), l can only receive from l, s can receive from both s and l; while patients and donors can have only A or O blood types.

20 Compatibility Partial Order O( Ol" 11" A( B( AB( ( Al( Os( ( As" 01( 10( 00" No(Size(Comp( No(B(AnBgen( Compa&bility,Par&al,Order, Binary,Par&al,Order,on,Unit,Square, Compatibility: 2 dimensional binary partial order on unit square: Model 1a: A blood antigen is the first dimension, B blood antigen is the second dimension. For X {A, B} No X antigen 1 Has X antigen 0 Model 1b: Size replaces antigen B in dimension 2 in the partial order. l No B antigen s Has B antigen

21 Lung Exchange Problem - Model 1a Set of blood types B = {O, A, B, AB} = {11, 01, 10, 00} set of compatibility types. A patient-donors triple is denoted by the blood types of its patient and donors respectively as X Y Z = X Z Y B 3 Set of triple types B 3 Definition A lung exchange problem is a vector of non-negative integers E lung = {n(x Y Z ) X Y Z B 3 } such that for all X Y Z B 3 (1) n(x Y Z ) = n(x Z Y ) and (2) Y X and Z X = n(x Y Z ) = 0.

22 Two-way Lung Exchange ng Exchange Lung Exchange Lemma (Participation Lemma for Two-way Exchanges) In any given lung exchange problem, the only types that could be part of a two-way exchange are ts can participate in two-way lung exchange if their donors titioned such that two donorsacan Ydonate B to and first Bpatient Y A maining two donors can donate to the second patient. for all Y {O, A, B}. ny given lung exchange nly types that could be ay exchange are B Y 0 A where O}. A-A-B A-O-B A-B-B B-B-A B-O-A B-A-A

23 Sequential Two-way Lung Exchange Algorithm Step 1: Match the maximum number of A A B and B B A types. Match the maximum number of A B B and B A A types. Step 2: Match the maximum number of A O B types with any subset of the remaining B B A and B A A types. Match the maximum number of B O A types with any subset of the remaining A A B and A B B types. Step 3: Match the maximum number of the remaining A O B and B O A types.

24 Lung Exchange Sequential Two-way Algorithm Two-way Lung Exchange Algorithm A-A-B B-B-A A-A-B B-B-A A-A-B B-B-A A-O-B B-O-A A-O-B B-O-A A-O-B B-O-A A-B-B B-A-A A-B-B B-A-A A-B-B B-A-A Step 1 Step 2 Step 3 27/54

25 Optimal Two-way Lung Exchange Lung Exchange Theorem (Optimal Two-way Lung Exchange) Given Optimal a lung Two-way exchange Lung problem, Exchange the sequential two-way lung exchange Theorem algorithm 1: Given maximizes a lung exchange the number problem, of thetwo-way sequentialexchanges. two-way The maximum lung exchange number algorithm of transplants maximizes the through number of two-wayexchanges is 2 min{nexchanges. 1, N 2, N 3, The N 4 } maximum where: number of transplants through two-way exchanges is 2 min{n 1, N 2, N 3, N 4 } where: N 1 = n(a N 1 = A n(a B) + A n(a B)+n(A O OB) + B)+n(A n(a B B) B) N 2 = n(a N 2 = O n(a B) O+ n(a B)+n(A B BB) + B)+n(B B A)+n(B + n(b O OA) A) N 3 = n(a N 3 = A n(a B) + A n(a B)+n(A O OB) + B)+n(B O A)+n(B + n(b A AA) A) N 4 = n(b N 4 = B n(b A) + B n(b A)+n(B O OA) + A)+n(B A A) A-A-B B-B-A A-A-B B-B-A A-A-B B-B-A A-A-B B-B-A A-O-B B-O-A A-O-B B-O-A A-O-B B-O-A A-O-B B-O-A A-B-B B-A-A A-B-B B-A-A A-B-B B-A-A A-B-B B-A-A N 1 N 2 N 3 N 4 28/54

26 Larger Lung Exchanges Participation Lemma can be generalized to larger exchanges. In addition to the earlier types, some types with O blood type patients can be matched! Lemma (Participation Lemma for All Exchanges) Fix a lung exchange problem and n 2. Then, the only types that could be part of an n-way exchange are O Y A, O Y B, A Y B, and B Y A for all Y {O, A, B}. Furthermore, every n-way exchange must involve one A and one B patient.

27 Larger Lung Exchanges We will make the following assumption for the remaining results on lung exchange. Assumption (Long Run Assumption) Regardless of the exchange technology available, there remains at least one unmatched patient from each of the two types O O A and O O B.

28 Optimal Two & Three-way Lung Exchange Proposition Consider a lung exchange problem that satisfies the long run assumption, and suppose n = 3. Then, there exists an optimal matching that consists of exchanges summarized in the following figure where: (1) A regular (non-bold/no dotted end) edge between two types represents a 2-way exchange involving those two types. Lung Exchange Optimal (2) AThree-way bold edgelung between Exchange two types represents a 3-way exchange involving those two types and a O O A or O O B type. Lemma 3: Consider a lung exchange problem that satisfies the long run assumption, and suppose n = 3. Then, there exists an optimal matching that consists of exchanges summarized in the following figure where: (3) An edge with a dotted end represents a 3-way exchange involving two types from the dotted end, and one type from the non-dotted end. (1) A regular (non-bold/no dotted end) edge between two types represents a 2-way exchange involving those two types (2) A bold edge between two types represents a 3-way exchange involving those two types and a O O A or O O B type. A-A-B A-O-B A-B-B B-B-A B-O-A B-A-A

29 Three-way exchanges (in the Proposition) with A O B types (Kind 2 in Proposition) A O B A O B B A B and B A A O O A O O B with 1 A B B and 2 B A A types (Kind 3 in Proposition) A B B B B A B B A Symmetrically defined for B O A and B A A types

30 Sequential Two & Three-Way Lung Exchange Algorithm Step 1: Carry out the 2 & 3-way exchanges in Proposition among A A B, A B B, B B A, and B A A types to maximize the number of transplants subject to the following constraints ( ): (1) Leave at least a total of min { n(a A B) + n(a B B), n(b O A) } A A B and A B B types unmatched. (2) Leave at least a total of min { n(b B A) + n(b A A), n(a O B) } B B A and B A A types unmatched.

31 Two & Three-Way Lung Exchange Algorithm Lung Exchange Step 2: Carry out the maximum number of 3-way exchanges in Proposition Two & Three-Way involvinglung A Exchange O B types Algorithm and the remaining B B A or B A A types. Step 2: Carry out the maximum number of 3-way exchanges in CarryLemma out the 3 involving maximum A Onumber B typesof and3-way the remaining exchanges B B ina or Proposition B A involving A types. B O A types and the remaining A ACarry Bout orthe Amaximum B Bnumber types. of 3-way exchanges in Lemma 3 involving B O A types and the remaining A A B or Step 3: CarryAoutB the B maximum types. number of 3-way exchanges in Proposition Step 3: Carry involving out the the maximum remaining numberaof 3-way O exchanges B andinb O A types. Lemma 3 involving the remaining A O B and B O A types. A-A-B B-B-A A-A-B B-B-A A-A-B B-B-A A-O-B B-O-A A-O-B B-O-A A-O-B B-O-A A-B-B B-A-A A-B-B B-A-A A-B-B B-A-A Step 1 subject to (*) Step 2 Step 3 33/54

32 Optimal Two & Three-Way Lung Exchange Theorem (Optimal Two & Three-way Lung Exchange) Given a lung exchange problem satisfying the long run assumption, the sequential two & three-way lung exchange algorithm maximizes the number of transplants through two and three-way exchanges.

33 Sufficiency of 6-way Exchange Theorem (6-way Sufficiency Theorem) Consider a lung exchange problem satisfying the long run assumption. Then, there exists an optimal matching which consists only of exchanges involving at most 6-way exchanges.

34 Lack of Sufficiency of Less than 6-way Exchanges Example There are 3 blood type O patients and 6 blood type O donors, 2 blood type B patients and 4 blood type B donors, and 1 blood type A patient and 2 blood type A donors. Hence, for optimality, each patients receives a lung lobe from two donors of exactly his own blood type, and all are matched. Triple types are: 1. A O B needs to be in the same exchange as both Patients 2 & 3 2. B O A 3. B O A 4. O O B needs to be in the same exchange as one of Patients 1, 2, 3 5. O O B needs to be in the same exchange as one of Patients 1, 2, 3 6. O O B needs to be in the same exchange as one of Patients 1, 2, 3 The blue argument along with the red arguments imply that a 6-way exchange is necessary.

35 Optimal Exchange Theorem (Maximum Number of Patients Matched) The number of patients matched in an optimal matching is given by m I + min{n(a O B), s B } + min{n(b O A), s A }, where I {0, 1}, and m := m A + m B where m A := min{p A, d A+d O 2, s B } : #A patients that can be matched s B := 2n(B O A) + n(b A B) + 2n(B A A) m B and s A symmetrically defined. s B : Max. #A patients that can be potentially matched with the help of B patients p A : #A patients d X : #X donors

36 Welfare Gains from Lung Exchange Average Numbers of Patients Matched Exchange Technology Patient Size Direct 2& 2&3& 2&3& Numbercc Constraint Donation 2-way 3-way 4-way 4&5-way Unrestricted 10 X X X

37 Model 2: Liver Exchange Model Liver exchange differs from kidney exchange in two key ways: The lack of tissue-type compatibility, and the presence of size compatibility. In the absence of size compatibility the scope for liver exchange would be very limited: The only viable exchange would be between a blood type A patient with blood type B donor and a blood type B patient with blood type A donor. Model 2: A Liver Exchange Model: Donor number: k = 1. Blood-type compatibility: Tissue-type compatibility: X Size compatibility: with two types large (l) and small (s)

38 Liver Exchange Model {O, A, B, AB} {l, s} : Set of individual types }{{}}{{} B S Compatibility: A donor can donate to a patient if and only if (1) the patient is blood type compatible with the donor, and (2) the donor is not strictly smaller than the patient. Ol Liver Donation Partial Order on B S Os Bl Al Bs As ABl ABs

39 An Equivalent Representation Consider the following two partially ordered sets: (1) The liver donation partial order on B S, and (2) the standard partial order over the corners of the three-dimensional cube {0, 1} 3. Ol 111 Os Bl Al Bs As ABl ABs 000

40 An Equivalent Representation Ol 111 Os Bl Al Bs As ABl ABs 000 Note that (B S, ) and ({0, 1} 3, ) are order isomorphic, where the order isomorphism associates each individual type τ B S with the following vector X {0, 1} 3 : X 1 = 0 τ has the A antigen X 2 = 0 τ has the B antigen X 3 = 0 τ is small

41 X Y ( {0, 1} 3) 2 such that Live Donor X Organ Y Exchange: and Y Beyond X. Kidneys Liver Exchange Problem For notational transparency, we will work with the equivalent representation ({0, 1} 3, ). Definition A liver exchange { problem is a vector of nonnegative integers E liver = n(x Y ) : X Y ( {0, 1} 3) } 2 such that X Y ( {0, 1} 3) 2 Y X = n(x Y ) = 0. }{{} no compatible pairs Here n(x Y ) denotes the number of pairs of type X Y. Lemma In any liver exchange problem, the only types that could be part of a two-way exchange are

42 Possible Two-Way Liver Exchanges

43 Two-way Liver Exchange Algorithm Consider the following sequential liver exchange algorithm: Step 1: Match the maximum number of X Y and Y X types for all X, Y {0, 1} 3. Step 2: Match the maximum number of , , and types, without matching them to each other. Step 3: Match the maximum number of , , and types among each other.

44 Liver Exchange Algorithm: Step

45 Liver Exchange Algorithm: Step

46 Liver Exchange Algorithm: Step

47 Optimal Two-way Liver Exchange Theorem Given a liver exchange problem, the sequential liver exchange algorithm maximizes the number of two-way exchanges.

48 A Model on Impact of Liver Size Constraints on Donation In this section, we consider a model with a continuum of agents. Baseline model (no liver size constraint) Baseline population: Λ 1 = { λ 1 (X Y ) : X Y B 2} λ 1 (X Y ): Mass of patient-donor pairs with blood types X Y in Λ 1 Next, suppose that there exist l 2 possible sizes 1,..., l, i.i.d. across agents with probabilities p 1,..., p l. Induced model (with liver size constraint) Induced population: Λ 2 = { λ 2 (Xs Ys ) : Xs Ys (B {1,..., k}) 2} λ 2 (Xs Ys ) Mass of pairs where the patient has blood type X, size s = λ 1 (X Y )p s p s : the donor has blood type Y, size s

49 A Model on Impact of Liver Size Constraints on Donation Compatibility: Baseline model: Blood type compatibility only Induced model: Blood type compatibility + size compatibility Donor size should be at least as large as patient size Long Run Assumption on Λ 1 : X, Y B, Y X = λ 1 (X Y ) λ 1 (Y X ) Incompatible pairs accumulate over time while compatible pairs leave after a short while due to transplantation. Patient-donor types: Type I: X = Y (O O, A A, B B, AB AB) Type II: Y X (A O, B O, AB O, AB A, AB B) Type III: X Y (O A, O B, O AB, A AB, B AB) Type IV: X Y and Y X (A B, B A) p = k l=1 p l( k i=l p i ): Odds that a random patient-donor pair is size compatible

50 Impact of Liver Size Constraints on Donation Theorem Given the long run assumption, the number of transplants through direct donation and two-way exchange in populations Λ 1 and Λ 2 are given as follows: Λ 1 : λ 1 (Type I ) + λ 1 (Type II ) }{{} + 2 min{λ 1 (A B), λ 1 (B A)} }{{} transplants via direct donations transplants via 2-way exchanges Λ 2 : p λ 1 (Type I ) + p λ 1 (Type II ) }{{} transplants via direct donations 2(1 p )λ 1 (Type II ) + 2 min{λ 1 (A B), λ 1 (B A)} }{{} transplants via 2-way exchanges Therefore the removal of liver size constraints (1) increases transplants from direct donation, (2) decreases transplants from exchanges, and (3) #Transplants(Λ 1 ) #Transplants(Λ 2 ) λ 1 (Type I ) λ 1 (Type II ) +

51 Welfare Effects of Liver Size Constraints on Donation Patient Survival Data (Lo et al. 1999): Graft Weight Ratio 0.4 ( Graft/Body Weight Ratio 0.08): 95% Graft Weight Ratio < 0.4 ( Graft/Body Weight Ratio < 0.08): 40% Donor Mortality(Chan et. al [2012]): Left Lobe Living Donor Liver Transplantation: 0.1% Right Lobe Living Donor Liver Transplantation: 0.5% Liver Lobe Weight Ratio (Florman & Miller [2006]): Left Lobe Weight / Liver Weight: 40% Right Lobe Weight / Liver Weight: 60%

52 Welfare Effects of Liver Size Constraints on Donation Based on these numbers, most donors feel obliged to donate their more risky right liver lobe, so that graft weight ratio exceeds the threshold 40%. Chan et. al [2012] argue that reducing the 40% threshold will not only increase living donor liver donation but also reliance on the left liver lobes for liver transplantation. While this is clearly correct in the absence of liver exchange, it may fail to hold in its presence.

53 Welfare Gains from Left Lobe Liver Exchange Average Numbers of Patients Matched Exchange Technology Patient Weight Direct Number Threshold Donation 2-way 2&3-way 2&3&4-way Unrestricted 20 0% % % % % % % % %

54 Welfare Gains from Left Lobe Liver Exchange Average Numbers of Patients Matched Exchange Technology Patient Weight Direct Number Threshold Donation 2-way 2&3-way 2&3&4-way Unrestricted %

55 Welfare Gains from Left Lobe Liver Exchange Average Numbers of Patients Matched Exchange Technology Patient Weight Direct Number Threshold Donation 2-way 2&3-way 2&3&4-way Unrestricted %

56 Welfare Gains from Left Lobe Liver Exchange Average Numbers of Patients Matched Exchange Technology Patient Weight Direct Number Threshold Donation 2-way 2&3-way 2&3&4-way Unrestricted %

57 Welfare Gains from Left Lobe Liver Exchange Average Numbers of Patients Matched Exchange Technology Patient Weight Direct Number Threshold Donation 2-way 2&3-way 2&3&4-way Unrestricted 100 0% % %

58 Conclusion We introduce a new transplant modality to the attention of scientific community: Live-donor lobar lung exchange We model liver and lung exchange as matching problems to characterize the maximum number of patients that can be saved under different institutional constraints and find simple algorithms to find optimal exchanges. We analyze the effects of patient-donor size constraints on liver exchange. We show that the requirement of size compatibility can increase the overall number of transplants: decreases the direct donation, increases exchange, and overall effect can be positive. We simulate gains from exchange for livers and lungs to show that an organized lung exchange program can almost triple the number of patients who receive live donor lung donation, much more than both kidney and liver donation.

59 Current Direction Incentive problems in liver exchange Implementation: Japan

60 Optimal Lung Exchange Lemma (Optimality Limit) In a given exchange problem, if all A Y B and B Y A for Y {O, A, B} can be matched perfectly in a matching µ in exchanges among themselves, then an optimal matching matches exactly n(a A B) + n(a B B) + 2n(A O B) +n(b A A) + n(b A B) + 2n(B O A) patients. Such an optimal matching can be formed inserting in every exchange in µ for any A O B or B O A triple, one O O A or O O B type triple.

61 Reducing Two Triples to One: Donor Supply-Demand We refer to this operation treating an A O B triple like an A A B or B A A triple Similarly for B O A (like B A A or B A B)

62 Optimal Lung Exchange Algorithm If we can find an algorithm simultaneously satisfying Obj. 1. match types A Y B, B Y A for all Y {O, A, B} with each other in two and three-way exchanges optimally, and Obj. 2. maximize the number of A O B and A B O that can be matched in any matching then we can insert for each A O B and B O A used one additional O O B or O O A using the above Reduction depending on how each A O B and B O A was treated in the above matching. This operation yields, by Optimality Limit Lemma above, an optimal matching only with 6 or less-way exchanges.

63 Optimal Lung Exchange Algorithm Since we will classify A O B as A B A or A B B and vice versa for B O A, inspect matching A B A, A B B, B A B, B A A: A-A-B B-B-A A-A-B B-B-A A-A-B B-B-A A-B-B B-A-A A-B-B B-A-A A-B-B B-A-A Step 1 Step 2 Step 3 We then find out how to classify A O B and B O A so that we maximize their matches and total matches subject to Obj. 1 and Obj. 2.

64 Optimal Lung Exchange Algorithm Case 1 if there are comparable A and B patients α A A-O-B A-B-A B-O-A B-A-B α B A-O-B A-B-A B-O-A B-A-B A-O-B A-B-A B-O-A B-A-B 1- α A A-O-B B-O-A 1- α B A-B-B B-A-A A-O-B A-B-B B-O-A B-A-A A-O-B A-B-B B-O-A B-A-A Step 1 Step 2 Step 3 Every patient is matched. Case 2.1 if there are too many A patients all A-O-B A-B-A B-A-B A-O-B A-B-A B-A-B A-O-B A-B-A B-A-B A-B-B B-O-A B-A-A all A-B-B B-O-A B-A-A A-B-B B-O-A B-A-A Step 1 Step 2 Step 3 Some A patients remain unmatched. At most one B-A-B or some A-B-A likes remain unmatched, but not both.

65 Sufficiency of 6-way Revisited Above construction also proves 6-way Sufficiency Theorem