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Game theoretical analysis of Kidney Exchange Programs
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Abstract
The goal of a kidney exchange program (KEP) is to maximize number of transplants within a pool of incompatible patient-donor pairs by exchanging donors. A KEP can be modelled as a maximum matching problem in a graph. A KEP between incompatible patient-donor from pools of several hospitals, regions or countries has the potential to increase the number of transplants. These entities aim is to maximize the transplant benefit for their patients, which can lead to strategic behaviours. Recently, this was formulated as a non-cooperative two-player game and the game solutions (equilibria) were characterized when the entities objective function is the number of their patients receiving a kidney. In this paper, we generalize these results for N-players and discuss the impact in the game solutions when transplant information quality is introduced. Furthermore, the game theory model is analyzed through computational experiments on instances generated through the Canada Kidney Paired Donation Program. These experiments highlighting the importance of using the concept of Nash equilibrium, as well as, the anticipation of the necessity to further research for supporting police makers once measures on transplant quality are available.
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The complex multi-criteria optimisation problems arising in Kidney Exchange Programmes have received considerable attention both in practice and in the scientific literature. Whereas theoretical advancements are well reviewed and synthesised, this is not the case for practice. We present a synthesis of models and methods applied in present European Kidney Exchange Programmes, which is based on detailed descriptions we created for this purpose. Most descriptions address national programmes, yet we also present findings on emerging cross-national programmes. The synthesis provides a systematic and detailed description of the models and methods the programmes use, revealing important commonalities as well as considerable variation among them. Rather than distilling a single best practice from these results, we find that the variation in models and methods arises because of variation in country characteristics, policies, and ethics. The synthesised state of the art may benefit future national and cross-national initiatives and direct future theoretical contributions within and across the boundaries of the Operations Research discipline.
Advances in medical technology have made kidney transplants over the blood group barrier feasible. This paper investigates how such technology should be implemented when designing pairwise kidney exchange programs. The possibility to receive a kidney transplant from a blood group incompatible donor motivates an extension of the preference domain, allowing patients to distinguish between compatible donors and half-compatible donors (i.e., blood group incompatible donors that only become compatible after undergoing an immunosuppressive treatment). It is demonstrated that the number of transplants can be substantially increased by providing an incentive for patients with half-compatible donors to participate in kidney exchange programs. The results also suggest that the technology is beneficial for patient groups that are traditionally disadvantaged in kidney exchange programs (e.g., blood group O patients). The positive effect of allowing transplants over the blood group barrier is larger than the corresponding effects of including altruistic patient-donor pairs or of allowing three-way exchanges in addition to pairwise exchanges.
Background:
Considerable differences exist among the living donor Kidney Exchange Programmes (KEPs) that are in use and being built in Europe, contributing to a variation in the number of living donor transplants (Newsletter Transplant; International figures on donation and transplantation 2016). Efforts of European KEPs to exchange (best) practices and share approaches to address challenges have, however, been limited.
Methods:
Experts from 23 European countries, collaborating on the European Network for Collaboration on Kidney Exchange Programmes Cooperation on Science and Technology Action, developed a questionnaire to collect detailed information on the functioning of all existing KEPs in Europe, as well as their opportunities and challenges. Following a comparative analysis, results were synthesized and interpreted by the same experts.
Results:
The practices, opportunities and challenges reported by 17 European countries reveal that some of the 10 operating programs are mature, whereas others are in earlier stages of development. Over 1300 transplants were performed through existing KEPs up to the end of 2016, providing approximately 8% of their countries' living kidney donations in 2015. All countries report challenges to either initiating KEPs or increasing volumes. Some challenges are shared, whereas others differ because of differences in context (eg, country size, effectiveness of deceased donor program) and ethical and legal considerations (eg, regarding living donation as such, nonrelated donors, and altruistic donation). Transnational initiatives have started in Central Europe, Scandinavia, and Southern Europe.
Conclusions:
Exchange of best practices and shared advancement of national programs to address existing challenges, aided by transnational exchanges, may substantially improve access to the most (cost) effective treatment for the increasing number of patients suffering from kidney disease.
Relatives and friends of an end-stage renal disease patient who offer to donate a kidney are often found to be incompatible with their intended recipients. Kidney paired donation matches one patient and his incompatible donor with another patient and donor in the same situation for an organ exchange. Let patient- donor pairs be the vertices of an undirected graph G, with an edge connecting any two reciprocally compatible vertices. A matching in G is a feasible set of paired donations. We describe various optimization problems on kidney paired donation graphs G and the merits of each in clinical transplantation. Because some matches are geographically undesirable, and the expected lifespan of a transplanted kidney depends on the immunologic concordance of donor and recipient, we weight the edges of G and seek a maximum edge-weight matching. Unfortunately, such matchings might not have the maximum cardinality; there is a risk of an unpredictable trade-off between the quality and quantity of paired donations. We propose an edge-weighting of G which guarantees that every matching with maximum weight also has maximum cardinality, and also maximizes the number of transplants for an exceptional subset of recipients, while reducing travel and favoring immunologic concordance.
An accurate model of patient-specific kidney graft survival distributions can help to improve shared-decision making in the treatment and care of patients. In this paper, we propose a deep learning method that directly models the survival function instead of estimating the hazard function to predict survival times for graft patients based on the principle of multi-task learning. By learning to jointly predict the time of the event, and its rank in the cox partial log likelihood framework, our deep learning approach outperforms, in terms of survival time prediction quality and concordance index, other common methods for survival analysis, including the Cox Proportional Hazards model and a network trained on the cox partial log-likelihood.
Kidney paired donation (KPD) is an efficient strategy to circumvent major immunological barriers in patients who have a willing and medically able, but incompatible living donor. In recent years, several countries have developed successful KPD programs, including highly active multicenter alliances in the United States, and even international cross-border kidney shipment has been documented (1-3). In Europe, several countries are running independent KPD programs, with the Dutch program being the most prominent (4). This article is protected by copyright. All rights reserved.
We revisit the problem of designing optimal, individually rational matching mechanisms (in a general sense, allowing for cycles in directed graphs), where each player --- who is associated with a subset of vertices --- matches as many of his own vertices when he opts into the matching mechanism as when he opts out. We offer a new perspective on this problem by considering an arbitrary graph, but assuming that vertices are associated with players at random. Our main result asserts that, under certain conditions, any fixed optimal matching is likely to be individually rational up to lower-order terms. We also show that a simple and practical mechanism is (fully) individually rational, and likely to be optimal up to lower-order terms. We discuss the implications of our results for market design in general, and kidney exchange in particular.
Choosing between multiple living kidney donors, or evaluating offers in kidney paired donation, can be challenging because no metric currently exists for living donor quality. Furthermore, some deceased donor kidneys can result in better outcomes than some living donor kidneys, yet there is no way to compare them on the same scale. To better inform clinical decision-making, we created a living kidney donor profile index (LKDPI) on the same scale as the deceased donor KDPI, using Cox regression and adjusting for recipient characteristics. Donor age over 50 (HR per 10y = 1.151.241.33), elevated body mass index (BMI) (HR per 10 units = 1.011.091.16), African-American race (HR= 1.151.251.37), cigarette use (HR=1.091.161.23), as well as ABO incompatibility (HR= 1.031.271.58), HLA B (HR= 1.031.081.14) mismatches, and DR (HR= 1.041.091.15) mismatches were associated with greater risk of graft loss after living donor transplantation (all p<0.05). Median (IQR) LKDPI score was 13 (1-27); 24.2% of donors had LKDPI<0 (less risk than any DD kidney), and 4.4% of donors had LKDPI>50 (more risk than the median DD kidney). The LKDPI is a useful tool for comparing living donor kidneys to each other and to deceased donor kidneys. This article is protected by copyright. All rights reserved.
There are growing numbers of patients with end-stage renal disease globally at an unexpected rate. Today, the most serious challenge in transplantation is organ shortage; hence, using deceased donor is increasingly encouraged.
The aim of the study was to investigate the differences in survival rates between kidney transplant recipients with deceased donor and living donor.
In a retrospective cohort study, 218 patients who had undergone kidney transplantation in our institute from April 2008 to September 2010 were recruited. Demographics and post-transplantation follow-up data including immunosuppression regimens, rejection episodes, and survival rates were evaluated. The patients were assigned to two groups according to the donor kidney transplantation: group I, living donor kidney transplants; and group II, deceased donor kidney transplants.
Although there were no significant differences in one-year survival rates of patient and graft between study groups, three-years survival rates of patient and graft were significantly longer in living donor kidney transplants in comparison with the deceased donor kidney recipients (P = 0.006 and P = 0.004, respectively). In Cox-regression model after adjusting for other confounding factors such as age, sex, diabetes mellitus, and first- or second-time transplantation, overall patient and graft survivals were also significantly shorter in deceased kidney transplantation than those who received kidney from a living donor (HR, 3.5; 95% CI, 1.2-10.4; and P = 0.02 for patient survival; and HR, 5.4; 95% CI, 1.5-19.5; and P = 0.009 for graft survival).
We found acceptable short-term survival in both groups; however, living donor recipients continue to have better long-term patient and graft survival rates.
Barter exchange markets are markets in which agents seek to directly trade their goods with each other. Exchanges occur in cycles or in chains in which each agent gives a good to the next agent. Kidney exchange is an important type of barter exchange market that allows incompatible patient–donor pairs to exchange kidneys so the involved patients can receive a transplant. The clearing problem is to find an allocation of donors to patients that is optimal with respect to multiple criteria. To achieve the best possible score on all criteria, long cycles and chains are often needed, particularly when there are many hard-to-match patients. In this paper we show why this may pose difficulties for existing approaches to the optimization of kidney exchanges. We then present a generic iterative branch-and-price algorithm that can deal effectively with multiple criteria, and we show how the pricing problem may be solved in polynomial time for a general class of criteria. Our algorithm is effective even for large, realistic patient–donor pools. Our approach and its effects are demonstrated by using simulations with kidney exchange data from the Netherlands and the United States.
As multi‐hospital kidney exchange has grown, the set of players has grown from patients and surgeons to include hospitals. Hospitals can choose to enroll only their hard‐to‐match patient–donor pairs, while conducting easily arranged exchanges internally. This behavior has already been observed.
We show that as the population of hospitals and patients grows, the cost of making it individually rational for hospitals to participate fully becomes low in almost every large exchange pool (although the worst‐case cost is very high), while the cost of failing to guarantee individual rationality is high—in lost transplants. We identify a mechanism that gives hospitals incentives to reveal all patient–donor pairs. We observe that if such a mechanism were to be implemented and hospitals enrolled all their pairs, the resulting patient pools would allow efficient matchings that could be implemented with two‐ and three‐way exchanges.
In kidney exchanges, hospitals share patient lists and receive transplantations. A kidney-paired donation (KPD) mechanism needs to promote full sharing of information about donor-patient pairs, and identify a Pareto efficient outcome that also satisfies participation constraints of hospitals. We introduce a random graph model of the KPD exchange and then fully characterize the structure of the efficient outcome and the expected number of transplantations that can be performed. Random graph theory allows early experimental results to be explained analytically, and enables the study of participation incentives in a methodological way. We derive a square-root law between the welfare gains from sharing patient-donor pairs in a central pool and the individual sizes of hospitals, illustrating the urgent need for the nationwide expansion of such programs. Finally, we establish through theoretical and computational analysis that enforcing simple individual rationality constraints on the outcome can mitigate the negative impact of strategic behavior by hospitals.
This paper is devoted to bilevel optimization, a branch of mathematical program- ming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution ap- proaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints).
The profound organ shortage has resulted in longer waiting times and increased mortality for those awaiting kidney transplantation. Consequently, patients are turning to older living donors. It is unclear if an upper age limit for donation should exist, both in terms of recipient and donor outcomes.
In the United States, 219 healthy adults aged ≥70 have donated kidneys at 80 of 279 transplant centers. Competing risks models with matched controls were used to study the independent association between older donor age and allograft survival, accounting for the competing risk of recipient mortality as well as other transplant factors.
Among recipients of older live donor allografts, graft loss was significantly higher than matched 50-to 59-year-old live donor allografts (subhazard ratio [SHR] 1.62, 95% confidence interval [CI] 1.16 to 2.28, P = 0.005) but similar to matched nonextended criteria 50-to 59-year-old deceased donor allografts (SHR 1.19, 95% CI 0.87 to 1.63, P = 0.3). Mortality among living kidney donors aged ≥70 was no higher than healthy matched controls drawn from the NHANES-III cohort; in fact, mortality was lower, probably reflecting higher selectivity among older live donors than could be captured in National Health and Nutrition Examination Survey III (NHANES-III; HR 0.37, 95% CI 0.21 to 0.65, P < 0.001).
These findings support living donation among older adults but highlight the advantages of finding a younger donor, particularly for younger recipients.
Renal transplantation has dramatically improved the survival rate of hemodialysis patients. However, with a growing proportion of marginal organs and improved immunosuppression, it is necessary to verify that the established allocation system, mostly based on human leukocyte antigen matching, still meets today's needs. The authors turn to machine-learning techniques to predict, from donor-recipient data, the estimated glomerular filtration rate (eGFR) of the recipient 1 year after transplantation.
The patient's eGFR was predicted using donor-recipient characteristics available at the time of transplantation. Donors' data were obtained from Eurotransplant's database, while recipients' details were retrieved from Charité Campus Virchow-Klinikum's database. A total of 707 renal transplantations from cadaveric donors were included.
Two separate datasets were created, taking features with <10% missing values for one and <50% missing values for the other. Four established regressors were run on both datasets, with and without feature selection.
The authors obtained a Pearson correlation coefficient between predicted and real eGFR (COR) of 0.48. The best model for the dataset was a Gaussian support vector machine with recursive feature elimination on the more inclusive dataset. All results are available at http://transplant.molgen.mpg.de/.
For now, missing values in the data must be predicted and filled in. The performance is not as high as hoped, but the dataset seems to be the main cause.
Predicting the outcome is possible with the dataset at hand (COR=0.48). Valuable features include age and creatinine levels of the donor, as well as sex and weight of the recipient.
We show that kidney exchange markets suffer from market failures whose remedy could increase transplants by 30 to 63 percent. First, we document that the market is fragmented and inefficient; most transplants are arranged by hospitals instead of national platforms. Second, we propose a model to show two sources of inefficiency: hospitals only partly internalize their patients’ benefits from exchange, and current platforms suboptimally reward hospitals for submitting patients and donors. Third, we calibrate a production function and show that individual hospitals operate below efficient scale. Eliminating this inefficiency requires either a mandate or a combination of new mechanisms and reimbursement reforms. (JEL D24, D47, I11)
The efficient allocation of limited resources is a classical problem in economics and computer science. In kidney exchanges, a central market maker allocates living kidney donors to patients in need of an organ. Patients and donors in kidney exchanges are prioritized using ad-hoc weights decided on by committee and then fed into an allocation algorithm that determines who get what---and who does not. In this paper, we provide an end-to-end methodology for estimating weights of individual participant profiles in a kidney exchange. We first elicit from human subjects a list of patient attributes they consider acceptable for the purpose of prioritizing patients (e.g., medical characteristics, lifestyle choices, and so on). Then, we ask subjects comparison queries between patient profiles and estimate weights in a principled way from their responses. We show how to use these weights in kidney exchange market clearing algorithms. We then evaluate the impact of the weights in simulations and find that the precise numerical values of the weights we computed matter little, other than the ordering of profiles that they imply. However, compared to not prioritizing patients at all, there is a significant effect, with certain classes of patients being (de)prioritized based on the human-elicited value judgments.
A kidney exchange is an organized barter market where patients in need of a kidney swap willing but incompatible donors. Determining an optimal set of exchanges is theoretically and empirically hard. Traditionally, exchanges took place in cycles, with each participating patient-donor pair both giving and receiving a kidney. The recent introduction of chains, where a donor without a paired patient triggers a sequence of donations without requiring a kidney in return, increased the efficacy of fielded kidney exchanges---while also dramatically raising the empirical computational hardness of clearing the market in practice. While chains can be quite long, unbounded-length chains are not desirable: planned donations can fail before transplant for a variety of reasons, and the failure of a single donation causes the rest of that chain to fail, so parallel shorter chains are better in practice.
In this paper, we address the tractable clearing of kidney exchanges with short cycles and chains that are long but bounded. This corresponds to the practice at most modern fielded kidney exchanges. We introduce three new integer programming formulations, two of which are compact. Furthermore, one of these models has a linear programming relaxation that is exactly as tight as the previous tightest formulation (which was not compact) for instances in which each donor has a paired patient. On real data from the UNOS nationwide exchange in the United States and the NLDKSS nationwide exchange in the United Kingdom, as well as on generated realistic large-scale data, we show that our new models are competitive with all existing solvers---in many cases outperforming all other solvers by orders of magnitude.
Finally, we note that our position-indexed chain-edge formulation can be modified in a straightforward way to take post-match edge failure into account, under the restriction that edges have equal probabilities of failure. Post-match edge failure is a primary source of inefficiency in presently-fielded kidney exchanges. We show how to implement such failure-aware matching in our model, and also extend the state-of-the-art general branch-and-price-based non-compact formulation for the failure-aware problem to run its pricing problem in polynomial time.
Kidney exchange programs have been set in several countries within national, regional or hospital frameworks, to increase the possibility of kidney patients being transplanted. For the case of hospital programs, it has been claimed that hospitals would benefit if they collaborated with each other, sharing their internal pools and allowing transplants involving patients of different hospitals. This claim led to the study of multi-hospital exchange markets. We propose a novel direction in this setting by modeling the exchange market as an integer programming game. The analysis of the strategic behavior of the entities participating in the kidney exchange game allowed us to prove that the most rational game outcome maximizes the social welfare and that it can be computed in polynomial time. © 2016 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society
Kidney exchange provides a life-saving alternative to long waiting lists for patients in need of a new kidney. Fielded exchanges typically match under utilitarian or near-utilitarian rules; this approach marginalizes certain classes of patients. In this paper, we focus on improving access to kidneys for highly-sensitized, or hard-to-match, patients. Toward this end, we formally adapt a recently introduced measure of the tradeoff between fairness and efficiency - the price of fairness - to the standard kidney exchange model. We show that the price of fairness in the standard theoretical model is small. We then introduce two natural definitions of fairness and empirically explore the tradeoff between matching more hard-to-match patients and the overall utility of a utilitarian matching, on real data from the UNOS nationwide kidney exchange and simulated data from each of the standard kidney exchange distributions. Copyright © 2014, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved.
As kidney exchange programs are growing, manipulation by hospitals becomes more of an issue. Assuming that hospitals wish to maximize the number of their own patients who receive a kidney, they may have an incentive to withhold some of their incompatible donor–patient pairs and match them internally, thus harming social welfare. We study mechanisms for two-way exchanges that are strategyproof, i.e., make it a dominant strategy for hospitals to report all their incompatible pairs. We establish lower bounds on the welfare loss of strategyproof mechanisms, both deterministic and randomized, and propose a randomized mechanism that guarantees at least half of the maximum social welfare in the worst case. Simulations using realistic distributions for blood types and other parameters suggest that in practice our mechanism performs much closer to optimal.
Kidney exchanges enable transplants when a pair of a patient and an incompatible donor is matched with other similar pairs. In multi-hospital kidney exchanges pairs are pooled from multiple hospitals, and each hospital is able to decide which pairs to report and which to hide and match locally. Modeling the problem as a maximum matching on a random graph, we first establish that the expected benefit from pooling scales as the square-root of the number of pairs in each hospital. We design the xCM mechanism, which achieves efficiency and incentivizes hospitals of moderate-to-large size to fully report their pairs. Reciprocal pairs are crucial in the design, with the probabilistic uniform rule used to ensure incentive alignment. By grouping certain pair types into so-called virtual-reciprocal pairs, xCM extends to handle 3-cycles. We validate the performance of xCM in simulation, demonstrating its efficiency and incentive advantages over the Bonus mechanism (Ashlagi and Roth, 2013).
In recent years several countries have set up policies that allow exchange of kidneys between two or more incompatible patient-donor pairs. These policies lead to what is commonly known as kidney exchange programs. The underlying optimization problems can be formulated as integer programming models. Previously proposed models for kidney exchange programs have exponential numbers of constraints or variables, which makes them fairly difficult to solve when the problem size is large. In this work we propose two compact formulations for the problem, explain how these formulations can be adapted to address some problem variants, and provide results on the dominance of some models over others. Finally we present a systematic comparison between our models and two previously proposed ones via thorough computational analysis. Results show that compact formulations have advantages over non-compact ones when the problem size is large.
We study the computational problem of identifying optimal sets of kidney exchanges in the UK. We show how to expand an integer programming-based formulation [1,19] in order to model the criteria that constitute the UK definition of optimality. The software arising from this work has been used by the National Health Service Blood and Transplant to find optimal sets of kidney exchanges for their National Living Donor Kidney Sharing Schemes since July 2008. We report on the characteristics of the solutions that have been obtained in matching runs of the scheme since this time. We then present empirical results arising from the real datasets that stem from these matching runs, with the aim of establishing the extent to which the particular optimality criteria that are present in the UK influence the structure of the solutions that are ultimately computed. A key observation is that allowing 4-way exchanges would be likely to lead to a significant number of additional transplants.
End-stage renal disease (ESRD) is the ninth-leading cause of death in the U.S. Transplantation is the most viable renal replacement therapy for ESRD patients, but there is a severe disparity between the demand for kidneys for transplantation and the supply. This shortage is further complicated by incompatibilities in blood-type and antigen matching between patient-donor pairs. Paired kidney exchange (PKE), a cross-exchange of kidneys among incompatible patient-donor pairs, overcomes many difficulties in matching patients with incompatible donors. In a typical PKE, transplantation surgeries take place simultaneously so that no donor may renege after her intended recipient receives the kidney. Therefore, in a PKE, the occurrence of a transplantation requires compatibility among the pairs' willingnesses to exchange. We consider an arbitrary number of autonomous patients with probabilistically evolving health statuses in a prearranged PKE, and model their transplant timing decisions as a discrete-time non-zero-sum noncooperative stochastic game. We explore necessary and sufficient conditions for patients' decisions to be a stationary-perfect equilibrium, and formulate a mixed-integer linear programming representation of equilibrium constraints, which provides a characterization of the socially optimal stationary-perfect equilibria. We carefully calibrate our model using a large scale nationally representative clinical data, and empirically confirm that randomized strategies, which are less consistent with clinical practice and rationality of the patients, do not yield a significant social welfare gain over pure strategies. We also quantify the social welfare loss due to patient autonomy and demonstrate that maximizing the number of transplants may be undesirable. Our results highlight the importance of the timing of an exchange and the disease severity on matching patient-donor pairs.
Most transplanted kidneys are from cadavers, but there are also many transplants from live donors. Recently, there have started
to be kidney exchanges involving two donor-patient pairs such that each donor cannot give a kidney to the intended recipient because of immunological
incompatibility, but each patient can receive a kidney from the other donor. Exchanges are also made in which a donor-patient
pair makes a donation to someone waiting for a cadaver kidney, in return for the patient in the pair receiving high priority
for a compatible cadaver kidney when one becomes available. There are stringent legal/ethical constraints on how exchanges
can be conducted. We explore how larger scale exchanges of these kinds can be arranged efficiently and incentive compatibly,
within existing constraints. The problem resembles some of the “housing” problems studied in the mechanism design literature
for indivisible goods, with the novel feature that while live donor kidneys can be assigned simultaneously, cadaver kidneys
cannot. In addition to studying the theoretical properties of the proposed kidney exchange, we present simulation results
suggesting that the welfare gains from larger scale exchange would be substantial, both in increased number of feasible live
donation transplants, and in improved match quality of transplanted kidneys.
The growth of kidney exchange presents new challenges for the design of kidney exchange clearinghouses. The players now include directors of transplant centers, who see sets of patient-donor pairs, and can choose to reveal only difficult-to-match pairs to the clearinghouse, while withholding easy-to-match pairs to transplant locally. This reduces the number of transplants. We discuss how the incentives for hospitals to enroll all pairs in kidney exchange can be achieved, and how the concentration of hard to match pairs increases the importance of long, non-simultaneous nondirected donor chains.
A general procedure is presented for computing the best, 2nd best,…, Kth best solutions to a given discrete optimization problem. If the number of computational steps required to find an optimal solution to a problem with n(0, 1) variables is c(n), then the amount of computation required to obtain the if best solutions is O(K nc (n)).
The procedure specializes to published procedures of Murty and of Yen for the assignment problem and the shortest path problem, respectively. A method is presented for reducing the required amount of storage by a factor of n, compared with the algorithms of Murty and of Yen. It is shown how the K shortest (loopless) paths in an n-node network with positive and negative arcs can be computed with an amount of computation which is O(Kn ³ ). This represents an improvement by a factor of n, compared with Yen's algorithm.
We give efficient algorithms to sample uniformly, and count approximately, the solutions to a zero-one knapsack problem. The algorithm is based on using dynamic programming to provide a deterministic relative approximation. Then "dart throwing" techniques are used to give arbitrary approximation ratios. We also indicate how further improvements can be obtained using randomized rounding. We extend the approach to several related problems: the m-constraint zero-one knapsack, the general integer knapsack (including its m-constraint version) and contingency tables with constantly many rows.
As multi-hospital kidney exchange clearinghouses have grown, the set of players has grown from patients and surgeons to include hospitals. Hospitals have the option of enrolling only their hard-to-match patient-donor pairs, while conducting easily arranged exchanges internally. This behavior has already started to be observed.
We show that the cost of making it individually rational for hospitals to participate fully is low in almost every large exchange pool (although the worst-case cost is very high), while the cost of failing to guarantee individually rational allocations could be large, in terms of lost transplants. We also identify an incentive compatible mechanism.
In barter-exchange markets, agents seek to swap their items with one another, in order to improve their own utilities. These swaps consist of cycles of agents, with each agent receiving the item of the next agent in the cycle. We fo- cus mainly on the upcoming national kidney-exchange mar- ket, where patients with kidney disease can obtain compat- ible donors by swapping their own willing but incompatible donors. With over 70,000 patients already waiting for a cadaver kidney in the US, this market is seen as the only ethical way to significantly reduce the 4,000 deaths per year attributed to kidney disease. The clearing problem involves finding a social welfare max- imizing exchange when the maximum length of a cycle is fixed. Long cycles are forbidden, since, for incentive reasons, all transplants in a cycle must be performed simultaneously. Also, in barter-exchanges generally, more agents are affected if one drops out of a longer cycle. We prove that the clearing problem with this cycle-length constraint is NP-hard. Solv- ing it exactly is one of the main challenges in establishing a national kidney exchange. We present the first algorithm capable of clearing these markets on a nationwide scale. The key is incremental prob- lem formulation. We adapt two paradigms for the task: constraint generation and column generation. For each, we develop techniques that dramatically improve both runtime and memory usage. We conclude that column generation scales drastically better than constraint generation. Our al- gorithm also supports several generalizations, as demanded by real-world kidney exchanges. Our algorithm replaced CPLEX as the clearing algorithm of the Alliance for Paired Donation, one of the leading kid- ney exchanges. The match runs are conducted every two weeks and transplants based on our optimizations have al- ready been conducted.
We consider a generalization of the assignment problem in which an integer k is given and one wants to assign k rows to k columns so that the sum of the corresponding costs is a minimum. The problem can be seen as a 2-matroid intersection, hence is solvable in polynomial time; immediate algorithms for it can be obtained from transformation to min-cost flow or from classical shortest augmenting path techniques. We introduce original preprocessing techniques for finding optimal solutions in which g⩽k rows are assigned, for determining rows and columns which must be assigned in an optimal solution and for reducing the cost matrix. A specialized primal algorithm is finally presented. The average computational efficiency of the different approaches is evaluated through computational experiments.
An artificial neural networks (ANNs) model was developed to predict 5-year graft survival of living-donor kidney transplants. Predictions from the validated ANNs were compared with Cox regression-based nomogram.
Out of 1900 patients with living-donor kidney transplant; 1581 patients were used for training of the ANNs (training group), the remainder 319 patients were used for its validation (testing group). Many variables were correlated with the graft survival by univariate analysis. Significant ones were used for ANNs construction of a predictive model. The same variables were subjected to a multivariate statistics using Cox regression model; their result was the basis of a nomogram construction. The ANNs predictive model and the nomogram were used to predict the graft survival of the testing group. The predicted probability(s) was compared with the actual survival estimates.
The ANNs sensitivity was 88.43% (95% confidence interval [CI] 86.4-90.3), specificity was 73.26% (95% CI 70-76.3), and predictive accuracy was 88% (95% CI 87-90) in the testing group, whereas nomogram sensitivity was 61.84% (95% CI 50-72.8) with 74.9% (95% CI 69-80.2) specificity and predictive accuracy was 72% (95% CI 67-77). The positive predictive value of graft survival was 82.1% and 43.5% for the ANNs and Cox regression-based nomogram, respectively, and the negative predictive value was 82% and 86.3% for the ANNs and Cox regression-based nomogram, respectively. Predictions by both models fitted well with the observed findings.
These results suggest that ANNs was more accurate and sensitive than Cox regression-based nomogram in predicting 5-year graft survival.
The shortage of cadaveric donor kidneys for transplantation has forced a re-evaluation of the limits on donor age acceptability. However, as more kidneys from older donors have been transplanted, a significantly lower graft survival has been noted among their recipients. The impact of utilizing older donor kidneys and the relative importance of donor age with respect to other factors has not been clarified. A total of 43,172 cadaver donor transplants reported to the UNOS Scientific Renal Transplant Registry between 1987 and 1995 were the subjects of this study. Cox regression analysis was utilized to assess the joint effects on graft survival of donor age and HLA mismatch, recipient sex, race, age, original disease, donor death cause, cold ischemia time, and transplant year. Increased first day anuria, dialysis requirement, and discharge serum creatinine were noted with increasing donor age. Moreover, long-term graft and patient survival diminished as donor age increased. The 5-yr graft survival of zero HLA-A,B,DR mismatched kidneys fell steadily from 81% when the donor was aged 21-30 to 39% when the donor was over age 60. The reported causes of kidney transplant failure were remarkably similar for old and young donors. The best transplant results were obtained with zero HLA-A,B,DR mismatched transplants from young donors and the worst with older donor kidneys, regardless of HLA compatibility. We calculated that up to 21% of kidney failures resulted from insufficient renal mass due to age and were incorrectly attributed to chronic rejection.
Recent dramatic improvements in the results of kidney transplantation have markedly intensified the demand for transplantable kidneys. Waiting lists for renal transplantation have grown rapidly in most transplantation centers throughout the world, with no evidence of a parallel increase in the actual number of available renal allografts. This situation has created a demand for kidneys that cannot be met by currently available sources of supply. This is a consequence of the recent applications to the bedside of the results of almost 25 years of exacting research efforts, whose success has included (1) a further identification of the highly polymorphic components of the major histocompatibility complex in man, HLA, located on the sixth human chromosome; (2) reports of the highly beneficial effects of blood transfusion in transplantation, and especially of the utilization of donor-specific transfusions; (3) selective modulation of specific subpopulations of immunologically competent cells, following the discovery of Borel in 1972 of the properties of a metabolite isolated from Tolypocladium Inflatum Gans, which culminated in the beginning of a new generation of immunosuppressive agents. Cyclosporine has dramatically improved the overall success of renal transplantation; it has a particularly profound effect upon the postoperative morbidity and prevention of complications in such procedures; (4) the utilization of monoclonal antibodies for selective inactivation of specific receptor sites on the surface of immunologically competent cells has, in addition, provided a new approach to the treatment and possibly the prevention of kidney transplant rejection response; (5) another important series of studies has led to the utilization of total lymphoid irradiation in the preoperative preparation of prospective kidney transplant recipients. A number or reports indicate that this technique may be effective in combination with steroids and/or antithymocyte globulin, or with cyclosporine, in producing long-term allograft survival. Pari passu with these developments, the mortality rate of kidney transplantations has decreased to 4% to 7%, and the success rate of living-related donor kidney allografts (HLA-identical) is 95%, with an 80% to 85% survival of parental or sibling-to-sibling HLA-haplo-identical kidneys, and 55% for fully incompatible living-related donor allografts. The success rate of HLA-identical cadaver donor kidney allografts approximates 80% i.e., a result that is close to the data obtained earlier with renal allografts from sibling and/or parental donors. The results reported by Opelz and associates have shown a significant improvement in the results of cadaver kidney transplantation in recipients on cyclosporine, and these results were particularly favorable in the presence of two or more HLA antigen sharing between the donor and the recipient.
To alleviate the organ shortage, the use of more living donors is strongly recommended world wide. A living donor exchange (swap) program was launched in Korea. After the success of a direct swap program between two families, we have developed the swap-around program to expand the donor pool by enrolling many kinds of unrelated donors. Herein, we report our results of a living donor exchange program.
This retrospectively review of 978 recipients of kidney transplants from living donors, included analysis of donor-recipient relationships, mode of donor recruitment, episodes of acute rejection, and 5-year patient/graft survivals.
Transplantation was performed in 101 patients (10.3%) by way of the swap program. The proportion of swap patients among the number of unrelated donor renal transplants has been increasing from 4.2% to 46.6%. The incidence of acute rejection and 5-year patient/graft survival rates were comparable between the groups.
We have achieved some success in reducing the organ shortage with a swap program in addition to our current unrelated living donor programs without jeopardizing graft survival. Potentially exchangeable donors should undergo strict medical evaluation by physicians and social evaluation by social workers and coordinators as a pre-requisite for kidney transplantation. Expanding the swap around program to a regional or national pool could be an option to reduce the organ donor shortage in the future.
To expand the opportunity for paired live donor kidney transplantation, computerized matching algorithms have been designed to identify maximal sets of compatible donor/recipient pairs from a registry of incompatible pairs submitted as candidates for transplantation.
Demographic data of patients who had been evaluated for live donor kidney transplantation but found to be incompatible with their potential donor (because of ABO blood group or positive crossmatch) were submitted for computer analysis and matching. Data included ABO and HLA types of donor and recipient, %PRA and specificity of recipient alloantibody, donor/recipient relationship, and the reason the donor was incompatible. The data set used for the initial simulation included 29 patients with one donor each and 16 patients with multiple donors for a total of 45 patients and 68 donor/patient pairs. In addition, a simulation based on OPTN/SRTR data was used to further assess the practical importance of multiple exchange combinations.
If only exchanges involving two patient-donor pairs were allowed, a maximum of 8 patient-donor pairs in the data set could exchange kidneys. If three-way exchanges were also allowed, a maximum of 11 pairs could exchange kidneys. Simulations with OPTN/SRTR data demonstrate that the increase in the number of potential transplants if three-way exchanges are allowed is robust, and does not depend on the particular patients in our sample.
A computerized matching protocol can be used to identify donor/recipient pairs from a registry of incompatible pairs who can potentially enter into donor exchanges that otherwise would not readily occur.
Predicting the outcome of kidney transplantation is clinically important and computationally challenging. The goal of this project was to develop the models predicting probability of kidney allograft survival at 1, 3, 5, 7, and 10 years. Kidney transplant data from the United States Renal Data System (January 1, 1990, to December 31, 1999, with the follow-up through December 31, 2000) were used (n = 92,844). Independent variables included recipient demographic and anthropometric data, end-stage renal disease course, comorbidity information, donor data, and transplant procedure variables. Tree-based models predicting the probability of the allograft survival were generated using roughly two-thirds of the data (training set), with the remaining one-third left aside to be used for models validation (testing set). The prediction of the probability of graft survival in the independent testing dataset achieved a good correlation with the observed survival (r = 0.94, r = 0.98, r = 0.99, r = 0.93, and r = 0.98) and relatively high areas under the receiving operator characteristic curve (0.63, 0.64, 0.71, 0.82, and 0.90) for 1-, 3-, 5-, 7-, and 10-year survival prediction, respectively. The models predicting the probability of 1-, 3-, 5-, 7-, and 10-year allograft survival have been validated on the independent dataset and demonstrated performance that may suggest implementation in clinical decision support system.
In connection with an earlier paper on the exchange of live donor kidneys (Roth, Snmez, and nver 2004) the authors entered into discussions with New England transplant surgeons and their colleagues in the transplant community, aimed at implementing a Kidney Exchange program. In the course of those discussions it became clear that a likely rst step will be to implement pairwise exchanges, between just two patient-donor pairs, as these are logistically simpler than exchanges involving more than two pairs. Furthermore, the experience of these surgeons suggests to them that patient and surgeon preferences over kidneys should be 0-1, i.e. that patients and surgeons should be indierent among kidneys from healthy donors whose kidneys are compatible with the patient. This is because, in the United States, transplants of compatible live kidneys have about equal graft survival probabilities, regardless of the closeness of tissue types between patient and donor (unless there is a rare perfect match). In the present paper we show that, although the pairwise constraint eliminates some potential exchanges, there is a wide class of constrained-e cient mechanisms that are strategy-proof when patientdonor pairs and surgeons have 0-1 preferences. This class of mechanisms includes deterministic mechanisms that would accomodate the kinds of priority setting that organ banks currently use for the allocation of cadaver organs, as well as stochastic mechanisms that allow considerations of distributive justice to be addressed.
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