Optimizing the Kidney Exchange Problem with a Budget for Simultaneous Crossmatch Tests

Abstract

Many people in the world suffer from end stage renal disease, which has transplantation as the most effective form of treatment. However, kidneys obtained from deceased donors are not nearly enough to meet demand and willing living donors may display incompatibilities with their intended recipient. Kidney Exchange Programs have emerged as an attempt to answer the transplant shortage and bypass these incompatibility issues between donor-patient pairs. The process of selecting the pairs participating in the transplantation plan requires optimization models, one of which is presented in the current paper. We focus on maximizing the expected number of transplants taking into account that only a given number of actual incompatibility (crossmatch) tests can be made. We present an integer programming model to address this problem when recourse is feasible and we compare computationally its outcomes with other two approaches in 150 instances.

Appendix A Enumerating Configurations and Calculating Their Expected Values

The concept of configuration that we used is based on the definition in [13]. Let us then consider a cycle of length 3. This cycle may contain zero, one, two or three backarcs. In the first case, it has one possible configuration (itself). In the second case, the backarc can be associated to three different arcs (Fig. 6), but all these graphs are isomorphic, belonging to the same equivalence class. In the third case, we also have three different combinations of each two backarcs, but again, they are isomorphic to one another (Fig. 7). In the last case, there is one possible configuration of three backarcs. This gives us four equivalence classes total, which can be observed in Fig. 8.

In order to use the formulas for the expected number of transplants in [15], we must then rearrange each configuration into what we may call “standard form”. This means we will identify which equivalence class this configuration belongs to and consider its representative, calculating the expected number of transplants for this object. Considering the representative of the class each time means the disposition of the arcs in the configuration is always the same, which allows us to use the mentioned formulas regardless of the position of the backarcs (as we can always reduce them to the same four cases).

Fig. 6.

The three possible configurations of a 3-cycle with one backarc (one equivalence class) and the representative on the left.

Fig. 7.

The three possible configurations of a 3-cycle with two backarcs (one equivalence class) and the representative on the left.

Fig. 8.

All four possible configurations (representatives of each equivalence class) of a 3-cycle.