Matching rate of the optimized matching algorithm : the case of two and three-way cycle and chain kidney exchanges

Adrianna Wrona, 10661698

Abstract

Donor-recipient incompatibility is one of the greatest obstacles to living donor kidney transplant. The above problem can be resolved by arranging Kidney Paired Donation (KPD). KPD occurs when a pair, in which a living donor is incompatible with its initial recipient, participates in a kidney exchange with another donor-recipient pair. Throughout the past decade, KDP has experienced a rapid growth in popularity. Nevertheless, since the scale of the allocation procedure is still too small to meet an ever-growing kidney demand, the new optimization techniques need to be developed. This study analyzes two and three-way cycle exchanges; domino paired donation and non-simultaneous extended altruistic donations chains, and its impact on the KPD matching ratio. The major focus of this research is an effect of the sample size and the presence of a non-directed donor on the KPD participation rate of highly sensitized donors. For this reason the optimized algorithm, built in cooperation with 8Vance, was used. The simulation showed that the matching ratio increases together with the sample size and the length of the cycle or chain. Moreover, exchanges arranged in the chains allow for more indivisible trade to occur. Next, including non-direct-ed donors in the incompatible donor-recipient pool bustnon-direct-ed the matching rate by 5%. Besides, the optimization of the matching algorithm allowed for the more highly sensitized patients to take part in the exchanges than it would not be otherwise possible.

Statement of Originality

This document is written by Student Adrianna Wrona who declares to take full responsibility for the con-tents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

1. Introduction ... 3

2. Background & Literature ... 7

2.1 Types of Exchanges ... 7

2.2 Top Trading Cycles ... 14

2.3 Conditions for Compatibility... 15

2.4 Barriers to Kidney Paired Donation ... 17

3. Methodology ... 19 3.1. Research Design ... 19 3.2. Data Simulation ... 20 3.3. Matching Algorithm ... 22 3.3.1 An Optimized Algorithm ... 22 3.3.2 Acceptance Criteria ...

4.1 Two and Three-way KPD cycles ... 29

4.1.1 Table of Results ... 29

4.1.2 Interpretation of Results ... 30

4.2 DPD and and NEAD ... 32

4.2.1 Table of Results ... 32

4.2.2 Interpretation of Results ...

5. Conclusion & Further Research ... 35

References ... 39

Appendix ... 41

1. Introduction

More than 100,000 adults and 1,500 pediatric candidates are waiting for a kidney transplant is the US alone (Hart, Smith, & Skeans, 2014). Each year, the number of patients registering on the waiting list continues to outpace the availability of required organs. As well as reducing the need for a costly dial-ysis, kidney transplants have a positive impact on patients’ chances of survival. Individuals undergoing successful transplantation tend to live ten years longer than dialyzed patients (Wolfe, Ashby, & Milford, 1999). Moreover, an organ obtained from a live donor functions, on average, twice as long as a kidney from a deceased donor (Hart, Smith, & Skeans, 2014).

Since the supply of kidneys from deceased donors does not cover its demand, and it is illegal to trade or-gans, most of the hope for those with malfunctioning kidney relies on the altruism of the loved ones that has to be supposted by donor-recipient compatibility (Roth, 2015). Blood (ABO) and tissue type (HLA) incompatibility are seen as the key obstacles that disenable a full utilization of the available living donors (Montgomery, 2011). The number of successful transplants is further reduced by the patients’ immune systems developing antibodies that reject potentially matching kidneys (Montgomery, 2011).

Matching Rate of the Optimized Matching Algorithm: the case of two and three-way

cycle and chain kidney exchanges.

The development of an efficient kidney allocation system constitutes an impactful application of economic theory. Kidney Paired Donation (KPD), also referred to as kidney exchange or live donor paired exchange, provides a way for the patient with an incompatible living donor to increase their chance for a faster and better-matched result and spare them the struggles of dialysis (Montgomery, 2011).

KPD occurs when a pair, in which a living donor is incompatible with its initial recipient, participates in a kidney exchange with another donor-recipient pair (Gentry, Montgomery, & Segev, 2011).The crucial fea-ture that distinguishes KPD from other types of matching is the requirement for the mutual compatibility between a donor from the first pair with the recipient from the second pair and the recipient from the first pair with the donor from the second pair (Gentry, 2011). A lack of two-way compatibility does not exclude the presence of larger (three-way, four-way etc.) cycles or chains where an initially unallocated donor can donate to a candidate from either a waiting list or a recipient that appears in the incompatible pairs pool in the future. Although the idea for an organ exchange system was introduced by Felix Rapaport in 1986, KPD has only been experiencing a rapid growth throughout the past decade (Wallis, Samy, Roth, & Ress, 2011). Alvin E. Roth – a laureate of the Nobel Memorial Prize in Economic Sciences - is claimed to be one of the greatest contributors to that field. In 1991, the first kidney exchange was carried out in South Korea, followed by the University Hospital in Basel, and the New England Hospital, where in 2000 the first KPD was performed (Wallis, 2011).

According to the 1984 National Organ Transplant Act (NOTA), kidney transplants are legal, and after a 2010 KDP program pilot testing, run by the United Network for Organ Sharing, the transplantation com-munity accepted a paired kidney exchange as an “ethically acceptable” practice (Ashlagi, Gilchrist, Roth, & Rees, 2011).

Interest in KPD has significantly increased among researchers, clinicians, and software engineers, resulting in multiple matching algorithms having been developed.

Moreover, the high potential impact of KPD on the lives of thousands waiting for an organ transplant has provoked an introduction of kidney paired exchange programs in countries such as the Netherlands, South Korea, UK, US, Canada, and Romania.

Although hundreds of exchanges have been run since 2000, Gentry (2011) stressed, the scale of the alloca-tion procedure is still too small to meet an ever-growing kidney demand. The major unrealized potential of the KPD is still to be addressed, and new optimization techniques are to be developed.

Realizing the hypothetical benefits of the kidney exchanges, this study reviews the concept of the KPD. The key components of an efficient system such as matching algorithms and the allocation design for the two and three-way exchanges including the altruistic or non-directed donors are studied. Moreover, the issue of the tradeoff between the quality and quantity is addressed and consequently the barriers to the optimized kidney allocations are examined.

The research question of this paper is: To what extent does the use of the optimized matching algorithm

influence the matching rate in the case of two and three-way cycle and chain KPD exchanges? How do the sample size and the presence of NDD influence highly sensitized recipients taking part in those ex-changes?

This study highlights the importance of optimized prioritization criteria being used in the kidney alloca-tion programs. Acknowledging the presence of highly sensitized patients and the different treatment of this group results in higher quality matches. Moreover, the positive influence of large pools on the quantity of matches is showen. Based on the performed simulation, it is suggested that national registries are used in-stead of regional pools when introducing KPD programs.

The paper is organized as follows. In section 2, a literature review is provided. This section lists different types of exchanges being used in KPD programs, explains conditions for donor-recipient compatibility and briefly provides the reason for the optimized top trading cycles algorithm being used throughout the research. Section 3 addresses methods used in this study along with insights on the developed algorithm. In Section 4, the results of two and three-way exchanges organized in cycles and chains are presented and analyzed. Finally, section 5 concludes the findings of the research and proposes further research.

2. Background & Literature

2.1 Types of Exchanges

In 1974 Shapley & Scarf (1974) published a paper that treated indivisible goods and the ways to trade them without using money. An indivisible good is defined as a commodity one does not have positive utility from possessing more than one of, yet one has a high ordinal preference towards a particular item. Since the monetary exchange is not allowed, the redistribution of the commodity is based on the strict preferences/ needs of the traders (Shapley, & Scarf, 1974). The model introduced by Shapley & Scarf didn't treat a par-ticular market. However, the market for houses is often given as an example. Soon after, Al Roth applied the abovementioned modality to the organ-donor market. In this environment, the traders are incompatible donor-recipient pairs and each pair is willing to trade their kidney in exchange for a kidney they are in need of (Roth, 2015).

Kidney exchange can be organized based on cycles or chains. Cycle exchanges always take place simultane-ously and they are characterized by a closed loop, while chains can be arranged into simultaneous exchanges (domino paired donations) and non-simultaneous exchanges (non-simultaneous extended donations) so that the loop can close in the future (Roth, 2015). Thus, the major difference between the cycles and chains con-cerns the need for a closed loop. The major difference between the different types of chains is the timing of an operation. The types of exchanges mentioned above are visually represented in figure 1.

R1 D1 R2 D2 D1 R2 D2 R3 R1 D1 R1 W D1 B1 R2 R3 D4 R5 B2

Chain

Cycle

Sim

ultaneous

Non-sim

ultaneous

Domino paired donation (DPD)

Non-simultaneous extended altruistic donation (NEAD)

Fig. 1 Models representing types of exchanges discussed in this study

The above figure is a visual representation of the types of exchanges described in section 2.1. On the horizontal axis, an exchange organization method is shown, while vertical axis treats the timing of an operation. Moreover, R is a recipient, D is a donor. The number that follows those letters (for example 1,2,3) repressents the number of a pair a donor and a recipient is a part of. NDD represents non-directed donor; W is a waiting list recipient, and B is a bridge donor. In addition, the dotted line represents donor-recipient incompatibility, and the continuous line represents compatible candidates.

Kidney trade, proposed by Roth (2015), is a trade arranged in cycles. The simplest cycle is the two-way exchange between the incompatible donor-recipient pairs in which each donor fulfilled the criteria of com-patibility with the recipient from the other pair (Roth, 2015). Each donor of a pair donates to the recipient of the other pair. A lack of two-way compatibility does not exclude the presence of longer cycles. In fact, the chances of identifying three-way cycles are higher than two-way cycles, and they are producing higher matching outcomes (up to 66% match ratio). The latter requires each pair's donor to donate to the next pair's recipient with the last donor closing the cycle by donating to the first recipient (Roth, 2015). Gentry et al. (2011) add that as long as the recognition of the three-way exchanges is beneficial for the efficiency of the system, it does not necessarily hold true for larger cycles. The main reason is the risk of an unpredicted pos-itive cross-match (antibodies that an organism develops towards other organisms) which rises with every ad-ditional pair included in the simultaneous exchange, and which, when the case arises may destroy a planned process, leaving some of the candidates without a promised organ (Gentry, 2013). The largest simultaneous kidney exchange took place at Johns Hopkins in 2008 and included eight incompatible recipient-donor pairs (Roth, 2015).

The requirement of reciprocal compatibility can be greatly diminished when a non-directed, also referred to as an altruistic or Samaritan donor, enters the equation (Gentry, 2011). The presence of a third (non-directed) party, that is outside the incompatible donor-recipient pairs pool and decides to enter the matchmaking pro-cess, constitutes a factor distinguishing trade arranged in cycles from trade arranged in chains. Yhe presence of non-directed donors may be perceived as a blessing significantly maximizing the performance of a matching algorithm. The major advantage of an altruistic donor entering the incompatible pairs pool is that it does not require a reciprocal match (Montgomery, 2011). The presence of non-directed donors (NDDs) can result in two types of exchanges with the timing of an operation being the major difference between them – simultaneous ex-changes (domino paired donation) and non-simultaneous exex-changes (non-simultaneous extended donation).

Domino paired donations are defined as simultaneous transplants brought about by a non-directed donor and terminated by passing a kidney to the waiting list. An organ from the NDD is usually first allocated in the KPD pool (Gentry, 2011). He follows that doing so better addresses the altruistic intentions of a NDD. It enables the creation of the chains and as a result increases the number of successful transplants. The above process can be described in the following way. A non-directed donor donates to the hard-to-match recipient whose donor can later donate to the other recipient, and since an NDD does not have its own recipient it does not make any reciprocal demands, the chain continues to grow. The paired donor terminating a domino-paired donation gives its organ to the matching recipient from the waiting list (Gentry , 2011) . Domino-paired ar-rangement is particularly beneficial for the hard-to-match candidates regarding both donors and recipients, especially given that a single altruistic donation enables at least two subsequent transplants (Gentry, 2011) . The requirement of the simultaneous donor nephrectomy is crucial for the safety of the arrangement. It is implemented in the cases when a paired donor withdraws after its recipient underwent a kidney trans-plant, leaving another recipient, unfairly, without a donor. Moreover, Gentry et al. (2009) explain that the reason behind the simultaneity rule is to ensure that each paired donor has a freedom to change one's mind about the kidney donation any time before the surgery starts. It is only possible if all the matched donors are undergoing anesthesia at the same time, so that the donor chainging his or her mind about the donation, stops the entire domino. It ensures that no donor gives away the kidney without simultanuously getting kidney in exchange and protects from the case that donor donates and its recipuent is left without the compatible kidney (Gentry, 2009). Taking into consideration the simultaneous nature of domino-paired donation the logistics behind the surgeries might become challenging. First, for each couple taking part in the exchange, two teams of surgeons have to be operating. Second, all the exchanges need to be performed at the same time to avoid a hazard that one of the donors drops a chain and as a consequence breaks it (Montgomery, 2011). Although the condition of simultaneity should be widely complied with while performing domino-paired donations, there are instances when the restriction is relaxed (Gentry, 2011).

Some of those cases are described by Lee et al. (2009). Researchers cite the Koran experience, where donors often undergo non-simultaneous nephrectomies. As a result, the transplants are carried out in the time-span of up to couple of days of one another, with some cases of a paired donor donating before a non-directed donation being recorded.

The more extreme example of disregarding a requirement of simultaneity is the non-simultaneous ex-tended donation chain. Removing the condition of simultaneity can lessen the complicity of a dom-ino. Non-simultaneous Extended Altruistic Donation (NEAD) is characterized by the presence of a bridge donor (Roth, 2015). In this type of exchange the last kidney in a chain is not allocated at the waiting list but, instead, a terminating donor becomes a bridge donor who is expected to continue the chain of exchanges on the later date – from days to months after the paired recipient received an or-gan – when a new recipient becomes logistically available. The phenomenon of bridge donors aris-es due to the praris-esence of difficult to match units (Gentry, 2011). They note that bridge donors are of-ten characterized by the blood type AB, and almost never the O type. The simulations have shown that the bridge donors are often spurned in competition with a simultaneous KPD strategy (Gentry, 2011). Furthermore, Montgomery (2011) notice, since an altruistic donor donates first, the risk of each pair's recipients not receiving the kidney is minimized. He further explains that this is because each recipient obtains an organ before his or her paired donor donates. Following this logic, no-body is disadvantaged when one of the hypothetical donors backs out. However, the system's effi-ciency is no longer maximized because the hypothetical future exchange will no longer be possible.

Notwithstanding, the idea that one non-directed donor can cause a large chain to appear is of great significance. Moreover, the above modality constitutes a major advantage for the smaller hospi-tals that do not have conditions for multiple simultaneous surgeries. Correspondingly, the restric-tive compliance with the domino-paired donations conditions removes their potential capabilities.

Additionally, appropriate restriction and prioritization is used in the matching algorithm, the identified chains are longer and used more often than standard KPD with incompatible pairs pools only (Roth, 2006). In the event of bridge donor withdrawal from an agreement its potential contribution is gone. Otherwise, when no bridge donor is lost, the NEAD outperforms the simultaneous KPD by about 3% in terms of the matching ratio (Roth, 2006)

An alternative method of increasing the possible match rate is through including the set of compatible pairs in the incompatible pairs pool (Gentry, 2011). The aforementioned solution might benefit both types of recipients. Recipients from the compatible pair have a chance of obtaining better quality matches, meaning a younger or 0 HLA mismatch kidney. Simulations run by Montgomery (2011) show that 45% of the com-patible pairs entering the KPD pool managed to obtain as much as a ten-years-younger kidney or 0 HLA mismatch. Although 15% of fit entrants do not experience any significant gain, their presence often facili-tates an incompatible pair's transplant. First, compatible pairs supply the KPD pool with normative blood types. According to the standard rules of blood matching, blood type O can be a donor to every other blood type whereas it cannot be a recipient of any non-zero blood type. Following the aforesaid conventions, it should not come as a surprise that the O type donors would be relatively under-represented in the incom-patible pairs database (Gentry, 2011). The O blood type skewing is seen as one of the major obstacles of KPD (Gentry, 2011). A statistical KPD pool is characterized by >50% of O blood type recipients and less than 37% O blood type donors. Thus, Gentry (2011) follows, the way to deal with the blood imbalance is the allocation of the compatible pairs into the incompatible pairs pool to bust a number of O donors and let more exchanges take place. The rise of the O ABO increases the possibility of the first match happening and although the blood type advantage tends to disappear after the first domino is performed it still enhances the overall match rate.

Moreover, including an easy-to-match pair in the hard-to-match pool allows for the creation of longer chains and cycles that wouldn't be otherwise possible (Montgomery, 2011)

Taking the above arguments into consideration, the simulation performed by the researcher indicates that including a pool of compatible pairs in KPD pools can improve the match rates by more that 25%.

Simulations carried out by Montgomery (2011) indicate that the match rate is positively related to the sample size. The author points out that an international matching expansion would lead to the collection of larger databases, as a result, improving an impact of KPD. The Korean and Dutch national programs confirmed the aforementioned relation.

Next to the quantity of the matches a system can produce, its quality should also be taken into consider-ation. To achieve the highest efficiency of a match, the matching priorities should be imposed in the algo-rithm. Those priorities could be the quantity of matches, the best HLA matches, the age of a kidney, the shortest travel distance, etc. Variables can be distinguished by assigning different weights to each of them (Montgomery, 2011). Again, Montgomery's (2011) simulation shows that such an optimization may lead to up to 15% increase in the matching rates. Even the application of the optimization strategies to the ex-tensive database containing incompatible pairs does not guarantee a match rate higher than 50%. This is because the recipients displayed in the incompatible pool are by definition hard-to-match; they are likely to be unmatched throughout a couple of the database updates (Roth, 2015). In his paper, Gentry (2011) con-firm the possible matching percentages obtained by Montgomery (2011). He adds that the highest match rates are achieved for the cross-match positive pairs with the %PRA being in between low and moderate.

2.2 Top Trading Cycles

Along with introducing the concept of indivisibility, Shapley and Scarf (1974) proposed a method of run-ning an indivisible goods trade. The concept, developed by David Gale, is called Top Trading Cycle (TTC). Shapley and Scarf (1974) applied an algorithm to the housing market. First allocating algorithm, that served kidney exchanges between the incompatible pairs, was built by Roth, Abdulkadroglu, and Sonmez (Roth, 2015). Identification of TTC constitutes one of the first steps of finding efficient allocation. Those cycles are characterized by the fact that no recipients and donors could go on their own and find the cycle of trades that they like best without making others worse off (Roth, 2003). In his initial work on kidney exchange Roth et al. (1977) identified the efficient exchanges in a way that revealing a full set of preferences should be a strictly dominant strategy for both patients and the surgeons to arrive at the Pareto Efficiency. TTC are the only modality that ensures individual rationality, Pareto-efficiency and strategy-proofness on the domain of strict preferences (Unzu & Molis, 2011). That is why an algorithm based on TTC is often used in the market where the strict-core allocation is to be identified. The market for kidneys can be seen as such a market and they explained that the easiest way to understand the mechanism behind a TTC is to imagine a directed graph involving two types of players (recipients and donors) (Unzu & Molis, 2011). Researchers emphasize that each of them directs towards an agent of a strict preference and directed graphs fulfilling such charac-teristics always form at least one cycle and never let two cycles intersect. As a consequence, they argued, the TTC algorithm always arrives at the assignment.

When an agent is indifferent among the available options, the indifferences should be turned into strict or-ders (via tie-breakers) and then the standard TTC mechanism should be applied (Roth, 1982). Including the option of indifference can decrease the efficiency of an assignment (Unzu, 2011).

Notwithstanding, Top Trading Cycles are commonly used as a basis for the allocation algorithms for the KPD. Each allocation system can then differ in terms of accepting a first found match or implementing specific optimization criteria. First-accept match, is a matching method that does not respect optimization priorities while running a matching process (Segev, 2005). A standard cycle/chain starts with the first in-compatible donor-recipient pair in the pool and when an acceptable match is found the units are removed from the database, and at the same time from further consideration (Segev, 2005). The same happens to the next identified KPD and the process continues until there are no other possible matches.

The matching scheme described above is commonly used by the transplant centers implementing kid-ney-paired donation as an organ allocation method. However, it is worth remembering that kidneys allocat-ed that way do not result in the best quality of exchange. Therefore, there is room for an improvement, and implementation of optimization priorities may constitute one of these ways (Segev, 2005).

Unlike first-accept match, an optimized match takes into consideration optimizing priorities and applies them to the entire database at the same time (Segev, 2005) . Hence, each of the produced matches consti-tutes a feasible solution that is identified due to the score obtained in the process of assigning optimization priorities. Then, the cycle characterized by the highest score is chosen and units involved are removed from the database (Segev, 2005).

Every human being has six DNA proteins that build its tissue type (Duquesnoy, & Claas, 2007). Duques-noy, & Claas, (2007) explained that each individual inherits their HLA type from their parents – 50% from each parent. Following this logic, each sibling has a 50% chance of inheriting one mutual haplotype, a 25% chance of inheriting two of the same haplotypes and a 25% chance of not inheriting any common haplo-types. One common haplotype is already enough for the HLA of a donor and recipient to match (Said-man, Roth, Sonmez, Unver, & Delmonico, 2006). Although the chances of having common haplotypes are smaller in the case of non-spousal donor-recipient pairs, it is generally assumed that as much as 50% of the population has at least one common haplotype (“HLA Matching, Antibodies, and You”, 2016).

R/D O A B AB O 1 0 0 0 A 1 1 0 0 B 1 0 1 0

AB 1 1 1 1

Tab.1 Matrix of ABO compatibility Thus, donor and recipient are blood type compatible if a donor’s blood type allows for a donation to the recipient with a particular blood type. When the above condition is met, HLA matching comes into play. Two people are said to be a “match” when their tissue types are compatible with each other.

2.3 Conditions for compatibility

A successful match between the two individuals taking part in a kidney exchange is based on blood (ABO) and tissue-type (HLA) compatibility (Unver, 2009). Blood type is treated as a filtering condition that needs to be fulfilled for matching to start. Hence, each individual has one of the following blood types: O, A, B, and AB. O is a universal donor that is compatible with all other blood types; A can donate to A and AB; B can donate to B and AB; and AB can donate only to AB (Garratty, Glynn, & Mcentire, 2004). Table 1. rep-resents a blood type compatibility between the donor and recipient. For the simplicity of this research ABO rhesus is ignored. In real life, however, it can have a small effect on the development of antibodies and, as a consequence, negatively impact a blood compatibility.

Although ABO and HLA compatibility already brings high hopes for the potential donor-recipient match, one more test needs to be conducted in order to exclude a possible positive cross-match between the involved individuals. A positive cross-match means that the recipient’s blood produces antibodies (PRA) against the donor tissue’s antigens (“HLA Matching, Antibodies, and You”, 2016). A Positive cross-match happens with no more than 10% of cases; it excludes the possibility of an otherwise feasible transplant (Saidman, 2006).

Tab.2 % PRA and positive cross-match probability

% PRA value % Distribution Probability of a positive cross-match

<10% 70% 5%

10-80% 20% 45%

>80% 10% 90%

The performance of a cross-match test has to be done before any transplant takes place (Saidman, 2006). It is run on each of the matched paired units and only when the cross-match between the donor and its paired recipient is negative can the exchange take place (Saidman, 2006). Otherwise, they said, the indi-viduals paired by the system are again included in the matching pool. Nevertheless, an identification of a positive cross-match can only be performed in the hospital. Only there can the reaction one individual towards the other’s individual blood be tested. As a consequence, it cannot be simulated in this research. To sum up, donor and recipient are said to be compatible when they show ABO and HLA compatibility that is supported by a negative cross-match.

2.4 Barriers to Kidney Paired Donation

Among the greatest challenges that the market designers are facing is addressing the constraints of KPD. Kidney exchange can be vulnerable to logistical and operational obstacles. An exchange involving two pa-tient-donor pairs requires four simultaneous surgeries with four independent teams happening in roughly similar locations to transport organs quick enough (Gentry, 2011).

Nevertheless, even while facing those constraints, it should still be effective and encouraging to organize an exchange (Roth, 2015). Second challenge concerns meeting the needs of the institutions and the individuals who are subject to adjusting to the new market design (Roth, 2015). Some hospitals are not willing to share data on their transplant candidates whereas some transplant candidates are not willing to travel to undergo a surgery.

On the other hand, diverse institutional rules governing the processes of kidney allocation could be re-spected by the allocating software (Roth, 2015). As follows, the system's tough match can resolve different institutions' views on the priority allocation by producing efficient matches based on earlier specified ranks.

As mentioned before, the match success rate is positively related to the pool size. It is possible for small (15

units) samples to produce no matches at all, whereas large pools such as national registries could match up to 50% of incompatible pairs (Sergev, 2005). Hence, a lack of large registries constitutes the next barrier to the optimal allocation.

The population density a country is characterized by might be one of the factors determining whether the allocation system is successful (in the country) or not (Klerk, Witvliet, Haase-Kromwijk, Claas, & Weimar, 2008) recognized the above pattern in the densely populated Netherlands and South Korea (each of them about 1,270 people per square mile) that experience high allocation rates.

In contrast, the USA, with a population density of only about 88 people per square mile faces difficulties finding an incompatible pair within the reasonable geographical distance. Moreover, there are still coun-tries that impose legal barriers towards KPD (Gentry, 2011). These are mostly the councoun-tries following the living donation rules that predate a concept of above-mentioned modality, he added. Although legal barri-ers had a fundamental influence on the pace of implementing KPD in both the UK and the US, an UNOS proposal to create a national registry of incompatible donor-recipient pairs was passed by the US congress in 2007, as a result exempting KPD from NOTA. Next, while applying precaution rules of allowing a do-nation only from a donor showing general good health, kidney exchange does not increase mortality, end-stage renal disease nor hypertension of any of the involved parties (Sergev, 2010). From the maximization point of view, it is important to build software that could enable larger exchanges (Roth, 2015).

The patients that do not fit a two-way and three-way exchanges might be allocated when four or more pairs are involved. This approach is conceptually difficult and requires complicated matching algorithms. It won’t be examined in this research.

3. Methodology

3.1. Research Design

The organizational and facility constraints are the reasons why the short two and three-way exchanges will be a focus of this research. Besides, as Abraham, Blum, & Sandholm (2007) mentioned in their research, smaller cycles affect less candidates in the case of a cycle failure. Reasons for a cycle malfunction can be di-verse. Starting from the new incompatibilities that were not recognized earlier, through the agent withdraw-ing from an exchange or failwithdraw-ing on fulfillwithdraw-ing medical recommendations to still participate in a donation, to the geographical constraints arising from the time and effort needed organize such an exchange (Abraham, 2007). Notwithstanding, organization of large cycles is also complicated for the matching program to iden-tify and it is rarely being performed in real life.

First, optimized two and three-way exchanges are identified in the pool containing only incompatible do-nor-recipient pairs. In order to test the impact of a pool size on the match rate, matching is run on the two pools with sample sizes of 50 resp. 100 pairs.

Next, the exchanges involving non-directed donors will be examined. In this case, altruistic donors and the waiting list recipients will enrich incompatible donor-recipient pairs’ pools. When DPD occur, the last do-nor from the identified chain ends up donating to the waiting list. Otherwise, if no compatible waiting list recipient is there for the last donor to donate, this last donor is expected to donate in the future when a com-patible with him recipient is identified in an updated pool. In this situation NEAD exchanges are formed.

Allowing for four, abovementioned, types of exchanges to occur, eight matching outcomes will be identified (each type of exchange is run on two sample sizes). The results will then be compared and an effect of KPD arranged both in cycles and chains will be addressed.

3.2. Data Simulation

Since databases containing confidential medical records are not publically available, simulated data will be used for this research. Simulation of the database can result in more applicable results than the clinical donor-recipient information (Gentry, 2011) . This is due to the fact that KPD in a relatively new and inno-vational modality, so the sparse data about the transplant candidates does not necessarily reflect the charac-teristics of a broad population.

Database containing the race, age, gender, ABO, HLA, %PRA, relations between the donor and recipient and the reason for a incompatibility was carefully generated based on the statistics provided by the U.S. Or-gan Procurement and Transplantation Network and the Scientific Registry of Transplant Recipients (Table 3). The sample of 100 incompatible donor-recipient pairs, 50 recipients from a waiting list and five non-di-rected donors was randomly generated according to statistics.

Each individual in the database has its unique race, age, gender, ABO, HLA, %PRA. The relation between the recipient and incompatible donor is also included in the dataset and the reason for their incompatibility is stated. It is assumed that 65% of the donor-recipient pairs are related and 75% are sharing the same race. Since, blood type distribution differs between the races, statistics concerning %ABO distribution by race are also implemented in the simulated dataset (Table 4). Each of the individual has its own set of two HLA series, that out of simplicity are called: ABC, DEF, GHI, JKL, LMN, PRS, TUW, XYZ. PRA and conse-quently positive cross-match probability is assigned based on the statistics presented before in Table 2. An individual characterized by PRA>80% is assumed to be highly sensitized. Only 10% of the sample shows this condition and those individuals are receiving a prioritization in the matchmaking process. The list of the donor-recipient respecting abovementioned criteria was generated and can be reviewed in Appendix 1.

Criteria Recipient (%) Donor (%)

Age 18-34 11 20 35-49 29 30 50-64 42 36 >65 18 14 Sex Female 41 37 Male 59 63 Race White 38 67 Black 35 13 Hispanic 18 14 Asian 8 5 Other/Unknown 1 1 Kidney Transplant

History First transplant 85 90

Re-transplant 15 10 Blood Type A 28 40 B 17 16 AB 3 7 O 52 37 PRA/CPRA <1% 55 72 1<20% 14 8 20<80% 15 14 80<98% 7 4 98-100% 8 2 Unknown 1 0

Waiting time <1year 31

1<2 years 22

2<3 years 16

3<4 years 11

4<5 years 7

>5 years 13

Will accept ECD or

KDPI>85% KIDNEY 100%

HLA mismatches 0 2

1 50

2 23

Race/ABO O A B AB White 45 40 11 4 Black 50 26 20 4 Hispanic 57 31 10 2 Asian 40 28 25 7 Others 55 35 8 2

Tab.4 % Distribution of ABO by race

3.3. Matching Algorithm

3.3.1 An Optimized Algorithm

Identification of the best matches from a pool of incompatible pairs depends on an algorithm that, respecting the acceptance criteria, decides on feasible solutions. TTC is a concept used in this research. An algorithm is enriched in the prioritization criteria that attempt to identify the best match respecting the characteristics of a sample. Hard-to-match (highly sensitized) individuals may never be able to produce highest score match with a donor. That is why, in order to bust exchanges involving highly sensitized recipients, those individuals are assigned extra points to start with. Moreover, in order to achieve the best quality match, each individual (unless highly sensitized who’s strict preference is any matching kidney) can only receive kidney from the do-nor in the same or younger age group. An algorithm modeled that way is optimized. An optimized algorithm analyses each candidate, identifies possible compatibility, recognizes potential exchanges and assigns a rank score to each of them depending on the prioritization criteria applied to the particular pool. In other words, it compares all the candidates present in a database to one another to derive a Pareto efficient solution.

To sum up, an optimized algorithm is used in this research. It is programmed to respect the criteria listed below. A crucial constraint imposed by the algorithm is that each pair can take part in no more than one exchange.

3.3.2 Acceptance Criteria

Conditions under which there is a hypothetical possibility for the two donor-recipient pairs to be compati-ble.

(1) Blood type compatibility as a crucial condition for all the candidates.

Tab.5 Blood type acceptance conditions

Donor’s ABO Matching Recipient

O O, A, B, AB

A A, AB

B B, AB

AB AB

(2) Each non-sensitized candidate receives a kidney from the donor of one’s age group or younger. This condition does not apply to the sensitized recipients.

Tab.6 Age-wise kidney acceptation conditions for both non-sensitized and sensitized recipients

Non-sensitized Sensitized Donor Matching Recipient Matching Recipient

18-34 18-34; 35-49; 50-64; >65 18-34; 35-49; 50-64; >65 35-49 35-49; 50-64, >65 18-34; 35-49; 50-64; >65 50-64 50-64; >65 18-34; 35-49; 50-64; >65

>65 >65 18-34; 35-49; 50-64; >65

(3) Each patient can only accept a kidney from the recipient towards who he or she has zero or one HLA mismatches. The less mismatches the better the quality of a match and consequently more points are assigned. Due to the hard-to-match characteristics, sensitized recipients are rewarded with the bonus points (prioritization) in relation to the non-sensitized candidates. The score rubric is only a representation of pos-sible priorities criteria and can easily be adjusted in accordance to the system’s prioritization needs.

Tab.7 Score rubric for non-sensitized and sensitized candidates based on the HLA overlap

Sensitized\HLA Overlap 0 1 2

No 0 1 2

(4) Each patient is willing to travel and accepts every kidney that fulfills the acceptance criteria. If this was not the case some of the identified exchanges would not be possible although an optimal match is iden-tified.

The process of matching involves distinguishing between the filters and matching phase. Filter is the char-acteristic that must always hold for the matching to start. In case of kidney transplants and based on the acceptance criteria listed above, blood type (ABO) and age group are strong filters. No recipient (unless highly sensitized) can receive a kidney from neither the donor from an incompatible blood type nor a donor representing an older age group. When those conditions are unfulfilled, an algorithm removes the units from a database and proceeds with the HLA matching performed on the remaining pairs. PRA assists both filtering and matching stage. In the process of filtering it informs about the candidates with %PRA>80% and removes those with no possibility to find a match in the underlining pool. In the process of matching, on the other hand, it informs about the prioritization condition that a sensitized recipient (%PRA>80%) has a right to and subsequently assigns relevant bonus points.

The processes behind the matching program can be best represented by figure 2.

Profile Filtering - ABO -Age -PRA Matching -HLA -PRA % Match

Fig.2 Visual representation of the processed behind an algorithm used in this study

3.3.3 Developing an Algorithm

The software was developed during my joint internship in FreedomLab and 8Vance. The Algorithm was programed in Python and database form the Microsoft Excel was placed in JSON. Respecting the optimized acceptance criteria listed above, the algorithm proceeds based on the commands presented in figure 3.

Fig.3 Visual representation of the most important commands an algorithm is built of Given D Find matching R Matches present? Given graph of connections between the units Check reciprocal compatibility Unit used in the cycle?

Remove from the database Find the best chain fore this unit YES

YES NO

In order to check whether an algorithm works, the sample of five donor-recipient pairs has undergone test trial. Such a small size was chosen on purpose. The values of the matches and possible combinations where known in advance. This gives an opportunity to assess whether an algorithm indicates the best matching outcome.

3.3.4 Test trial

The sample of five donor-recipient pairs was organized in a way that every person blood type is compatible, and the age of each of the individuals is the same. This way, the filters are passed and the algorithm can proceed with the HLA matching. Each of the donor-recipient units is internally incompatible by definition and four of those pairs demonstrate a compatibility with either donor or recipient from the other unit. Each individual’s HLA series were obtained through constricting an overlap matrix with each type of HLA on both vertical and horizontal axis. Each donor-recipient unit has assigned number form one to five. As presented in Table 8, a recipient from incompatible pair number 1 is represented by R1 and donor by D1. HLA series for R1 is ABC, DEF and for D1 it is DEF, GHI (although they could potentially match they are incompatible by definition). A donor-recipient pair number 5 was incompatible with all other pairs and, in consequence, should not be detected by an algorithm as a match.

ABC DEF GHI JKL LMN PRS TVW XYZ

ABC R1\D4 DEF D1\R2 GHI D2\R3 JKL D3\R4 LMN R5 D5 PRS TVW XYZ

Tab.8 HLA overlap matrix

The above representation was expected to produce the cycle presented in Figure 4. Unit represents a single incompatible donor-recipient pair.

Unit1 R1:ABC,DEF D1:DEF,GHI Unit2 R2:GHI,DEF D2:GHI,JKL Unit3 R3:GHI,JKL D3:JKL,LMN Unit4 R4:JKL,LMN D4:ABC,DEF

Fig.4 Cycle expected to be identified during the test trial of an algorithm used in this study

Data of ten non-sensitized individual having O blood type, 30 years old and one of the HLA combinations indicated in the above matrix was saved as JSON file and then exposed to the matching.

The algorithm has produced an expected outcome. It confirmed that a cycle has closed, the value of a cycle is 8, there are four units involved in the cycle and the optimal solution is: "[u1-2->u2]->[u2-2->u3]->[u3-2->u4]->[u4-2->u1]".

After the test trial confirmed an ability of an algorithm to find an optimal solution, the simulated incompat-ible pairs pool underwent matching process. In total, six matching outcomes were obtained. Each of them used the same algorithm, only the matching pools were modified each time.

First, pools gathering only incompatible donor-recipient pairs underwent matching. Possible exchanges were found for the pool size of 50 and 100 incompatible donor-recipient pairs. On each of those pools, match-ing identifymatch-ing two-way exchanges and later three-way exchanges was imposed. Next, an incompatible do-nor-recipient pairs pool was enriched by the non-directed donors and the waiting list candidates. In this case chains were identified. No chain can involve more than three donor-recipient pairs, one non-directed donor, and one bridge donor or recipient from the waiting list. In other words on restriction on the length of a chain is imposed and the exchange can either end with a donation to a candidate from a waiting list or the non-simultaneous extended donation can occur. An unallocated bridge donor undergoes matching after the next database update.

Type of a cycle

Outcome Two-way Three-way

Incompatible D-R sample size 50 100 50 100

Number of identified matches 3 10 7 21

% Matching rate 6% 10% 14% 21%

Average value of the cycle 7 9 8 12

D /Gender F 1 4 4 9 M 2 6 3 12 R/Gender F 2 4 2 10 M 1 6 5 11 O 2 5 3 10 D/ABO A 1 4 3 8 B 0 1 1 2 AB 0 0 0 1 R/ABO O 1 3 3 5 A 2 5 2 10 B 0 2 1 4 AB 0 0 1 2 Number- sensitized 0 0 0 1

Matches by HLA mismatches 0 0 0 0 1

1 3 10 7 20

2 0 0 0 0

Matches by age group 18-34 0 2 1 5

35-49 2 2 3 6

50-64 1 5 2 7

>65 0 1 1 3

4. Results

4.1 Two and Three-way KPD cycles 4.1.1 Table of the Results

4.1.2 Interpretation of the Results

The results obtained in the process of the two and three-way matching, clearly show a positive benefit of the large pools. In the case of both two and three-way exchanges, doubling a donor-recipient pool size positively affected the matching rate. The two-way exchanges matched 6% of the pairs contained in the 50 pairs pool, and 10% of the pairs from the 100 pairs pool. A similar trend perseveres in the three-way exchanges with the 7% matching rate difference between the two pools. In the sample containing 50 pairs seven optimized matches were identified, while in the larger pool as many as 21 cycles were found. Moreover, the number of matches recognized in the pools containing both 50 and100 pairs doubled while allowing for three-way exchanges compared to two-way exchanges. Similarly, the matching rate is expected to grow further with increasing a pool size.

Also, an average score assigned to the identified exchanges is higher in the case of 100 sample for both two and three-way exchanges, and three-way cycles produced more optimized matches than two-way cy-cles. The above is represented by the average value of the cycle. The higher the score, the more optimized exchanges are identified. So, the quality of matches is expected to rise together with the size of the pool, especially when larger cycles are allowed for.

Observation 1: Large registries are producing more and better matches than the smaller records. Moreover, three-way cycles identify more and better exchanges than two-way cycles.

Next, the characteristics of the matched pairs are to be examined. As expected, the incompatible pairs with an O ABO donor are the group involved in the most exchanges. O ABO donors donate in 7 out of 13 two-way exchanges, and 13 out of 28 three-two-way exchanges recognized in the experiment. Overall, 50% of all the cycles identified in the simulation involved an O ABO donor that was HLA incompatible with its initial re-cipient. Moreover, the larger the pool, the more O ABO donors and potentially HLA compatible recipients, the greater the chances of O ABO donors initiating an exchange.

Observation 2: O ABO donors are universal donors whose presence in the incompatible donor-recipient pool can positively influence the amount of the potential matches being identified.

By the same token, the case of the highly sensitized patients needs to be considered. Despite the fact that they constitute only 10% of a sample, they are also the greatest beneficent of KDP. Exposition to many potential donors significantly increases the possibility of finding a negative cross-match donor, and, as a consequence chances for a successful transplant. No highly sensitized candidate found a match neither in any of two-way exchanges nor the three-way exchange with only 50 pairs involved. Three-way matching, run in the pool of 100 donor-recipient pairs, resulted in the two (20% of all highly sensitized recipients) highly sensitized recipients finding a donor. It is a huge impact, especially taking into consideration the fact that those recipients are hard to match by definition. The above outcome could only be achieved because the optimized algorithm has been used.

An effect of the bonus points assigned to the highly sensitized patients and the fact that they were the only group allowed to accept the kidney from the donor in the higher age-group played a significant role in the kidney assignment. Without these prioritization criteria, such a great participation of the highly sensitized patients did not occur. In fact, it significantly dropped.

Observation 3: An optimized algorithm, accompanied by a large donor-recipient pool, substantially busts the participa-tion of the highly sensitized recipients in the KPD. This modality positively influences hard-to-match individuals whose waiting time for a kidney is likely to be reduced. B

34 35 4.2 Domino Paired Donations and Non-simultaneous Extended Altruistic Donations

4.2.1 Table of Results

Type of a chain

Outcome DPD NEAD

Incompatible D-R sample size 50 100 50 100

Number of identified matches 8 38 8 39

% Matching rate 7% 25% 7% 25%

Average value of the chain 12 16 13 17

D/Gender F 3 14 4 12 M 5 24 4 26 R/Gender F 5 18 3 17 M 4 20 5 22 O 3 20 4 19 D/ABO A 4 14 2 16 B 1 3 1 1 AB 0 1 1 3

Number of Bridge Donors 0 0 3 8

Bridge Donors/ABO O 0 0 A 1 2 B 1 3 AB 1 4 R/ABO O 2 8 3 13 A 5 22 3 18 B 0 5 1 5 AB 1 3 1 3 Number-sensitized 0 3 1 4

Matches by HLA mismatches 0 0 0 0 1

1 8 38 8 38

2 0 0 0 0

Matches by age group 18-34 2 8 3 10

35-49 3 14 2 13

50-64 2 12 2 13

>65 1 4 1 3

Tab.10 Matches obtained in the DPD and NEAD chains

4.2.2 Interpretation of the Results

In this simulation, waiting list candidates and non-directed donors were included in the incompatible nor-recipient pool. First, allowing altruistic donors to take part in an initiation of the chains, instead of do-nating directly to the waiting list, increased the number of feasible transplants. Notwithstanding, the effect of using NDD in DPD and NEAD is comparable. About half of the exchanges ended up with a bridge do-nor being involved, whereas the other half ended with a donation to the waiting list. Introducing five NND in the pool of 100 incompatible pairs enriched by 50 candidates from the waiting list led to as much as 25% of recipients finding a potential donor. It is equivalent to 39 matches being identified. Regarding possible transplants it is almost double the matches recognized in the three-way exchanges on the incompatible do-nor-recipient pool. The effect of NDD also depends on the sample size.

Observation 4: Allowing NDD to participate in exchanges increases match ratio compared to the standard KPD. Never-theless, there is no significant change in potential transplants while allocating NDD in DPD compared to NEAD.

Furthermore, ABO types of NDD reflect the blood types of the U.S. population. Since the incompatible donor-recipient pool is O ABO scarce, the fact that almost 40% of the NDD are characterized by O ABO type has a positive effect on the possibility of identifying a match. Although a desirable blood type is lost af-ter the first possible match is identified a presence of O ABO donors is busting the exchanges. Again, NDD is involved in about 50% of recognized chains. In real terms, it is, again, more than cycle-organized trade. Furthermore, relaxing the condition of reciprocity increases occurrences of a hard-to-match DPD donor finding a compatible waiting list recipient. On the other hand, bridge donors in NEAD are almost never representatives of O blood type group.

The AB blood type characterizes more than 40% of the bridge donors. The above makes them hard to match in the next rounds, and they are expected to remain unmatched for the extended periods. To examine this effect, however, multiple updated pools need to undergo the matching process.

Observation 5: Thanks to fueling a database with an O ABO, NDD have the positive effect on the matches quantity. Participation in DPD is more time effective than NEAD whose hard to match donors are likely to remain unmatched for the prolonged periods of time.

Finally, the effect of a NND on the highly sensitized candidates is examined. Both in the case of DPD and NEAD hard-to-match recipients are positively influenced by the presence of an altruistic donor. While allowing the above modalities only one highly sensitized recipient was matched while employing the algo-rithm to the sample of 50 pairs. On the other hand, as many as four highly sensitized patients were matched with a compatible donor from the pool of 100 incompatible donor-recipient pairs. Concerning potential exchanges, DPD and NEAD allowed for an identification of twice as many transfers as three-way cycles. This output could only be achieved thanks to the optimized algorithm. The program took advantage of the presence of additional O ABO donors, that normalized an O ABO donor-scarce incompatible donor-recip-ient pool and assigned the prioritization criteria to those who are hard-to-match.

Observation 6: Presence of NDD positively influences a matching ratio for the highly sensitized candidates. The success in finding potential matches is, again, positively related to the size of the donor-recipient pool.

5. Conclusion & Further Research

Throughout the past decade, KDP has experienced a rapid growth in popularity. Initially, morally and legal-ly constraint concept soon has become a widelegal-ly used tool for kidney exchanges. Development of an efficient organ allocation system constitutes an impactful application of economic theory. KPD, also referred to as kidney exchange or live donor paired exchange, provides the way, for the patients with an incompatible liv-ing donors, to increase their chance for a faster and better-matched result (Montgomery, 2011).

Moreover, KPD is relying on the concept of Pareto Efficiency. For the exchanges, it means that many trans-plant candidates will benefit from the participation in the trade, whereas no candidate is worse of by taking part in the program. For this reason, multiple complexprograms between the transplant centers are being introduced in the countries such as US, UK, the Netherlands, South Korea or Romania.

In this study, the optimized algorithm was developed and used. The process took place during my internship. The cooperation with 8Vance resulted in building a system that meets the needs of KPD. Imposing the op-timized prioritization is crucial for the efficient organization of KPD. As a result, hard-to-match candidates do not need to compete with other patients (Sergev, 2005). The prioritization criteria are designed to bust participation of the hard-to-match transplant candidates. The optimization criteria include assigning extra points to the highly sensitized recipients, supported by the different age requirements concerning a potential donor.

This modality significantly increases a highly sensitized recipient’s quality of life (Sergev, 2011). Moreover, it decreases the time of the dialysis one needs to otherwise undertake, and consequently, limits the burden of the healthcare system.

The research question of this study is: To what extent does the use of the optimized matching algorithm influence the matching rate in the case of two and three-way cycle and chain Kidney Paired Donation? How do the sample size and the presence of non-directed donor influence highly sensitized recipients taking part in those exchanges?

As the study shows, the ability to identify optimized matches is positively related to the size of the pool. Correspondingly, it should be of great importance that the transplant centers carefully considered the bene-fits of the regional pool vs. the national pool and decided on one that allows for more exchanges to happen. The research indicates that the large pools produce more matches than the small pools. So, the national pools should be preferred to the regional pools. Next, O ABO donors are largely underrepresented in the incompatible door-recipient pools. On the other hand, when they enter the system, they are automatically blood compatible with other potential recipients. After, the blood filter is passed (respecting the prioritiza-tion criteria), the only condiprioritiza-tion for the donor-recipient compatibility is the HLA match. The simulaprioritiza-tion conducted in this study showed that more than 50% of all the identified matches involve O donors. The presence of O blood type donors benefits the entire pool and increases the chances of finding a match. In general, this effect is also dependent on the database size. The larger the pool and the more O donors are present, the more exchanges can be identified.

Furthermore, the optimized algorithm employed at the large donor-recipient pool substantially busts kid-ney exchange participation of the highly sensitized recipients. As a consequence, it may decrease waiting time a hard-to-match individual would otherwise have to experience. Moreover, a period of a dialysis a highly sensitized patient goes through is shorted. Finally, the costs of one’s hospitalization are lessened, and the burden to the health system is diminished.

Next, allowing NDD to participate in the exchanges increases matching ratio compared to the standard KPD. Nevertheless, there is no significant change in potential transplants while allocating NDD in DPD compared to NEAD. Also, since NDD are reflecting the ABO distribution characteristics of the US popula-tion, they are supplying O ABO to the pool, normalizing pool’s blood distribution. In effect it enables even more exchanges to happen. Participation in DPD is more time effective than NEAD where hard-to-match donors identified throughout the process are likely to remain unmatched for prolonged periods of time. Finally, the presence of NDD positively influences the matching ratio for the highly sensitized candidates, and this effect is, again, stronger than in the case of trade between only incompatible donor-recipient pairs.

To sum up, the research has also shown that three-way exchanges are easier to identify, and they produce higher matching rate than two-way exchanges. Both DPD and NEAD, however, allow for more transactions to happen than any of the studied cycle-organized trades. Thus, including NDD donors at the incompat-ible donor-recipient pool positively affects the chances for a transplant. Next, the pool size influences the matching ratio. The larger the database, the higher the potential matching outcome. Moreover, an optimized algorithm used in this study did, indeed, bust the participation of the highly sensitized recipients.

An algorithm developed in this research worked on the simulated data. To improve the accuracy of the ob-tained matching rate, an actual medical record would be preferred. The above would also allow for the gen-eration of the greater sample to eventually reach the national scale and better examine an influence of the pool size on the nationwide matching success. Moreover, the study assumes that all the KPD participants are willing to travel. This is not the case in real life, and this condition needs to be taken into account while optimizing a matching algorithm. It can be improved by, for instance, rewarding a small distance between the donor’s and recipient’s residence with prioritization points (Gentry, 2011).

Furthermore, since most of the institutions are applying different prioritization criteria, the system should be able to adjust to respect all of them. Next, to better examine an effect of a bridge donor on the transplant success rate, matching should be performed on the multiple database updates. This would indicate how successful bridge donors are in starting a new chain of operations and what is an average time they need for finding a compatible recipient. Additionally, the problem of competition, politics, and financial constraints the hospitals and organ allocation centers are facing should be addressed. The above problem cannot be dealt with by the means of an algorithm. It can, however, significantly influence the number of undertaken surgeries, so it is important to address it.

With the above in mind, there is a lot of room for improvement regarding both the algorithm's optimization and the KPD’ s organizational structure. Although hundreds of exchanges have been run since 2000, the scale of the allocation procedure is still too small to meet an ever-growing kidney demand (Gentry, 2011). The major unrealized potential of the KPD is still to be addressed, and new optimization techniques are to be developed.

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Appendix 1

43 SEX RACE AGE ABO HLA %PRA R/D RELA- REA- SEX RACE AGE ABO HLA %PRA R/D RELA-

REA-R1 F W 18-34 A ABC, GHJ R51 M W 50-64 B PRS, XYZ

D1' M W 18-34 A DEF, LMN 0% SIBLINGS HLA D51 F W 50-64 AB DEF, PRS 28% SIBLINGS ABO

R2 F W 50-64 O DEF, ABC R52 M W 35-49 A LMN, XYZ

D2' M W 50-64 B PRS, XYZ 0% MER- ABO D52 F B 35-49 B LMN, PRS 0% MERRIAGE ABO

R3 F W 35-49 A ABC, PRS R53 M B 50-64 A PRS, XYZ

D3' M B 18-34 B TUW, XYZ MER- ABO D53 F B 50-64 O TUW, XYZ 57% SIBLINGS PRA R4 F H 35-49 O ABC, DEF 82% FRIENDS PRA R54 M W 50-64 O JKL, LMN

D4 F W 35-49 O ABC, DEF D54 F W 35-49 A JKL, LMN 26% SIBLINGS ABO

R5 F H 50-64 O ABC, XYZ R55 M W >65 A JKL, LMN

D5 F H 35-49 A ABC, XYZ 0% SIBLINGS ABO D55 F A 50-64 AB DEF, PRS 0% MERRIAGE ABO

R6 F W 18-34 O PRS, XYZ R56 M W 35-49 O ABC, PRS

D6' M W 35-49 O ABC, DEF 60% MER- POS XE D56 F A 50-64 A ABC, TUW 24% MERRIAGE ABO

R7 F W 35-49 O DEF, XYZ R57 M W >65 O DEF, PRS

D7 F B 50-64 A ABC, PRS 0% FRIENDS ABO D57 F H >65 O ABC, PRS 54% MERRIAGE POS XE

R8 F B 18-34 O XYZ, JKL R58 M B 18-34 O DEF, TUW

D8 F W 18-34 AB ABC, PRS 0% FRIENDS ABO D58 F B 18-34 A DEF, XYZ 0% SIBLINGS ABO

R9 F B 35-49 O ABC, PRS R59 M B 35-49 O DEF, PRS

D9' M W 50-64 A XYZ, TUW 90% MER- PRA D59 F W 35-49 A ABC, XYZ 0% MERRIAGE ABO R10 F B 50-64 AB GHI, DEF SIBLINGS R60 M B 35-49 O ABC, DEF

D10 M B 35-49 AB GHI, DEF 90% PRA D60 F W 50-64 O JKL, GHI 49% MERRIAGE HLA

R11 F B 50-64 O ABC, JKL R61 M B 50-64 O LMN, TUW

D11 F B 35-49 A ABC, JKL 0% SIBLINGS ABO D61 F W 50-64 A PRS, XYZ 0% MERRIAGE ABO

R12 F B 50-64 A ABC, GHI R62 M H 35-49 O GHI, XYZ

D12 F B 50-64 O DEF, XYZ SIBLINGS HLA D62 F H 50-64 A GHI, TUW 0% SIBLINGS ABO Roth, A. E., Sönmez, T., Ünver, M. U., Delmonico, F. L., & Saidman, S. L. (2006). Utilizing List Exchange and

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44

R13 F B 50-64 A DEF, GHI R63 M H 35-49 O GHI, JKL

D13 F H >65 B DEF, XYZ 0% FRIENDS ABO D63 F H 18-34 O ABC, PRS SIBLINGS HLA

R14 F W 35-49 O ABC, DEF R64 M H 35-49 O JKL, LMN

D14' F W 50-64 O PRS, XYZ 20% SIBLINGS HLA D64 F B 18-34 O TUW, PRS 42% MERRIAGE HLA

R15 F W 50-64 O TUW, PRS R65 M W 50-64 O ABC, JKL

D15 F W 50-64 A TUW, PRS 0% SIBLINGS ABO D65 M W 35-49 B ABC, XYZ 0% SIBLINGS ABO

R16 F W 35-49 B DEF, PRS R66 M W 50-64 O DEF, GHI

D16 M W 18-34 A GHI, PRS 60% SIBLINGS ABO D66 M W 50-64 B JKL, LMN 0% SIBLINGS ABO

R17 F W 50-64 B ABC, XYZ R67 M W >65 O ABC, TUW

D17 M W 18-34 AB ABC, PRS 26% SON ABO D67 M W 18-34 O ABC, PRS 92% SON PRA

R18 F W 50-64 O JKL, PRS R68 M W >65 O ABC, DEF

D18 M W 18-34 A JKL, XYZ 43% SON ABO D68 M W 50-64 B GHI, JKL 82% SIBLINGS ABO

R19 F W 35-49 AB ABC, DEF R69 M W >65 O GHI, XYZ

D19 M W 18-34 O GHI, JKL 0% SIBLINGS HLA D69 M W 35-49 A GHI, PRS 0% SON ABO

R20 F W 18-34 A ABC, DEF R70 M W 50-64 O JKL, LMN

D20 M W 18-34 A ABC, PRS 80% SIBLINGS PRA D70 M W 50-64 A JKL, LMN 87% SIBLINGS ABO

R21 F W 35-49 O PRS, TUW R71 M W 50-64 O ABC, TUW

D21 M W 35-49 A PRS, TUW 0% SIBLINGS ABO D71 M W 35-49 A PRS, TUW 0% SIBLINGS ABO

R22 F W 35-49 A PRS, ABC R72 M B 50-64 O DEF, GHI

D22 M W 35-49 O PRS, TUW 56% SIBLINGS PRA D72 M W 35-49 B JKL. XYZ 0% FRIENDS ABO

R23 F B 50-64 A GHI, JKL R73 M B 35-49 O ABC, TUW

D23 M B 50-64 O ABC, DEF 0% SIBLINGS HLA D73 M W 35-49 AB ABC, PRS 0% FRIENDS ABO

R24 F B 35-49 O GHI, XYZ R74 M B 35-49 O PRS, XYZ

D24 M W 50-64 A DEF, XYZ 47% MERRIAGE ABO D74 M W 35-49 B JKL, DEF 0% FRIENDS ABO

R25 F B 50-64 O GHI, XYZ R75 M B >65 O GHI, PRS

D25 M B 35-49 A DEF, XYZ 0% SIBLINGS ABO D75 M B 50-64 A GHI, XYZ 0% SIBLINGS ABO

R26 F W 50-64 A ABC, XYZ R76 M W 18-34 A ABC, JKL

D26 M W 50-64 O ABC, XYZ 90% SIBLINGS PRA D76 M W 35-49 O PRS, TUW 0% SIBLINGS HLA

R27 F B 50-64 A ABC, DEF R77 M B 50-64 A ABC, DEF

D27 M W 50-64 O DEF, TUW 83%

MER-RIAGE POS XE D77 M B 35-49 O XYZ, GHI 0% SIBLINGS HLA

R28 F B 35-49 A DEF, TUW R78 M B 50-64 O ABC, XYZ

D28 M W 50-64 B PRS, TUW 0%

MER-RIAGE ABO D78 M B 35-49 A ABC, LMN 0% SIBLINGS ABO

R29 F B 18-34 O ABC DEF R79 M B 35-49 A ABC, DEF

D29 M W 50-64 A GHI, JKL 0%

MER-RIAGE ABO D79 M B 18-34 AB ABC, DEF 44% SIBLINGS ABO

R30 F B 35-49 B ABC, XYZ F80 M B 35-49 A DEF, XYZ

D30 M W 35-49 A JKL, TUW 0%

MER-RIAGE ABO D80 M B 18-34 O GHI, PRS 0% SIBLINGS HLA

R31 F H 50-64 A ABC, GHI R81 M W >65 O DEF, XYZ

D31 M H 18-34 O DEF, TUW 32% SON HLA D81 M W 35-49 O DEF, XYZ 57% SON POS XE

R32 F H 50-64 O DEF, GHI F82 M H >65 O GHI, XYZ

D32 M H 35-49 A DEF, XYZ 0% SIBLINGS ABO D82 M H 50-64 A GHI, TUW 0% SIBLINGS ABO

R33 F H 50-64 O ABC, DEF F83 M B 50-64 A JKL, PRS

D33' F H 50-64 O PRS, TUW 0% SIBLINGS HLA D83 M B 50-64 O JKL, TUW 36% SIBLINGS PRA

R34 F W >65 A DEF, JKL F84 M B >65 A LMN, JKL

D34 M W 50-64 O DEF, JKL 56% SIBLINGS PLA D84 M B 50-64 B LMN, TUW SIBLINGS ABO

R35 F W 50-64 O DEF, XYZ F85 M B 50-64 A ABC, DEF

D35 M A 35-49 A ABC, XYZ 0%

MER-RIAGE ABO D85 M B 35-49 O DEF, JKL 42% SIBLINGS PRA

R36 F H >65 A GHI, JKL R86 M B 50-64 B ABC, DEF

46

R37 F A >65 AB JKL, XYZ R87 M H 18-34 O PRS, TUW

D37 M W 50-64 O ABC, PRS 68% MERRIAGE HLA D87 M W 18-34 A ABC, JKL 85%

MER-RIAGE ABO

R38 F W >65 O ABC, XYZ R88 M H >65 O DEF, GHI

D38 M W 18-34 A ABC, PRS 0% SON ABO D88 M W >65 A LMN, XYZ 0% FRIENDS ABO

R39 F A >65 O DEF, PRS R89 M H 50-64 A ABC, XYZ

D39 M W 50-64 A TUW, XYZ 0% MERRIAGE ABO D89' M W 50-64 O DEF, PRS 0% FRIENDS HLA

R40 F A >65 A DEF, TUW R90 M B 50-64 B GHI, JKL

D40 M W 50-64 O DEF, XYZ 84% MERRIAGE POS XE D90 M B 34-49 O ABC, PRS 81% SIBLINGS HLA

R41 F A >65 B GHI, XYZ R91 M B 18-34 O ABC, TUW

D41 M H 50-64 A ABC, DEF 0% MERRIAGE ABO D91 M H 35-49 A PRS, XYZ 0% FRIENDS ABO

R42 F H 35-49 B DEF, GHI R92 M H 50-64 A GHI, JKL

D42 M W 18-34 A DEF, TUW 58% MERRIAGE ABO D92 M W 50-64 O PRS, TUW 0% FRIENDS HLA

R43 M W 18-34 A ABC, TUW R93 M H >65 A PRS, TUW

D43 F W 18-34 B ABC, TUW 0% SIBLINGS ABO D93 M H 35-49 B DEF, GHI 0% SON ABO

R44 M W 18-34 A DEF, XYZ R94 M H 50-64 A ABC, JKL

D44 F W >65 AB DEF, PRS 0% MOTHER ABO D94 M A 50-64 B GHI, TUW 0% FRIENDS ABO

R45 M W 35-49 O LMN, PRS R95 M H 50-64 B DEF, GHI

D45 F W 35-49 B LMN, TUW 0% SIBLINGS ABO D95 M H 35-49 A DEF, PRS 0% SIBLINGS ABO

R46 M W 35-49 O ABC, DEF R96 M H 50-64 AB TUW, ABC

D46 F W 35-49 A PRS, XYZ 0% SIBLINGS ABO D96 M H 35-49 O XYZ, PRS SIBLINGS HLA

R47 M W 35-49 A DEF, XYZ R97 M A 50-64 O ABC, DEF

D47 F W 35-49 O DEF, PRS SIBLINGS HLA D97 M W 50-64 AB ABC, XYZ 0% FRIENDS ABO

R48 M W 35-49 B ABC, DEF R98 M A 50-64 O TUW, XYZ

D48 F W 35-49 A ABC, DEF 0% SIBLINGS AB0 D98 M W 35-49 A TUW, PRS 0% MERRIAGE ABO

R49 M W 50-64 B GHI, XYZ R99 M A 35-49 O DEF, JKL

D49 F W 50-64 A ABC, XYZ 0% SIBLINGS ABO D99 M A 18-34 A DEF, XYZ 0% SIBLINGS ABO

R50 M W 50-64 A PRS, TUW R100 M A 50-64 B GHI, JKL