Optimal control of a paired-kidney exchange program

Organ exchanges are expected to increase the utilization of living donors and to alleviate the critical shortage of organs for transplantation. The typical arrangement involves a direct exchange between two blood-type incompatible donor-candidate pairs. An alternate possibility is an indirect exchange between one such pair and the highest priority candidate on the regular waiting list for cadaveric organs. This paper focuses on the mix of direct and indirect exchanges that maximizes the expected total discounted quality-adjusted life years (QALY) of the candidates in the participating pairs. Direct exchanges are preferable because the candidate receives a living-donor organ instead of the inferior cadaveric organ an indirect exchange provides. However, the latter involves a shorter wait. To capture this tradeoff, we develop a double-ended queueing model for an exchange system with two types of donor-candidate pairs, and obtain an optimal dynamic exchange policy by invoking a Brownian approximation. The policy takes the form of a two-sided regulator in which new pairs will join the exchange system to wait for a direct exchange if and only if the process modeling the exchange system is within the regulator's two barriers. In all other circumstances, new pairs will participate in an indirect exchange. Expressions for the optimal barriers are obtained under a variety of assumptions about the objective function, including one of complete candidate autonomy. The analysis identifies three design principles that will amplify the likelihood of an exchange program's success. First, exchange programs must involve the coordinated activities of multiple local transplant centers to enjoy the substantial benefits of resource pooling. Second, participant wait must be controlled through indirect exchanges. Third, the program must respect participants' autonomy and weigh that autonomy against the broader goal of maximizing their overall welfare.