122 The UMAP Journal 28. 2(2007) Patiente Receive New Receive New ranns Phase 2 Organs Priority I Matc Plnse 3 Last Priority Match tystem Death Simulate death Fi 2. Flowchart for simulati Summary of Assumptions o Arrivals in the waiting queue, both of cadaver donors and needy patients, independent and randomly distributed The generic U.S. transplant network can be simulated as a rooted tree o Death rate can be approximated as a linear function of time on the waiting Other Countries/ Transplantation Policies We researched the policies of other countries, such as China, Australia he United Kingdom; they differ little from the U.S. policy. China uses from executed prisoners, which we do not believe to be ethical. We decided that the policies of Eurotransplant have the best groundwork: People analyze their policy each year, tweaking the waiting-time point system. The Eurotransplant policy does not emphasize regions as much, with the maximum number of points for distance being 300. In contrast, the number
122 The UMAP ]ournal 28.2 (2007) Figure 2. Flowchart for simulation. Summary of Assumptions "o Arrivals in the waiting queue, both of cadaver donors and needy patients, are independent and randomly distributed " The generic U.S. transplant network can be simulated as a rooted tree "o Death rate can be approximated as a linear function of time on the waiting list. Other Countries' Transplantation Policies We researched the policies of other countries, such as China, Australia, and the United Kingdom; they differ little from the U.S. policy. China uses organs from executed prisoners, which we do not believe to be ethical. We decided that the policies of Eurotransplant have the.best groundwork: People analyze their policy each year, tweaking the waiting-time point system. The Eurotransplant policy does not emphasize regions as much, with the maximum number of points for distance being 300. In contrast, the number
Optimizing the Effectiveness of Organ Allocation 123 of points received for zero HLA mismatch is 400. The Eurotransplant policy also has greater emphasis on providing young children with a kidney match, giving children younger than 6 years an additional 1095 waiting-time points We implemented the Eurotransplant policy in our model to see if thatpolie could also benefit the u.s. but we found little difference. Utilizing Kidney Exchanges e A promising approach for kidney paired exchange is to run the maximal Home ing algorithm over the graph defined by the set of possible exchanges rer, this approach takes away from the autonomy of patients, because it requires them to wait for enough possible pairs to show up before performing the matching, and sometimes it may require them to take a less than perfect We sought to improve this supposedly"optimal solution"by implementing t paired donation in our model. According to each patient's phenotypes, we calculate the expected blood types of the persons parents and siblings, and make that the persons contri bution to the"donor pool. "In other words, the person brings to the transplant network an expected number r of potential donors. We then make the patient perform list paired donation with the topmost person in the current queue who is compatible in blood type to the donor accompanying the new patient. Ac- cording to our research, kidneys from live donors are about 21% better than cadaver kidneys in terms of success rate. Thus, it is in the cadaver- list persons best interest to undergo this exchange We find that for any value of r from 0. 2 to 2, list paired donation drastically decreases the length of the waitlist, by factors as large as 3, and makes the queue ize stabilize( figure 3) Patient Choices What should a patient do when presented with the opportunity for a kid ney? The decision is not clear-cut; for instance, if the patient is offered a poorly matched kidney now, but a well-matched kidney is likely to arrive in areason- able time, the patient should perhaps wait. We examine the this tradeoff. We assume that a patient who has already received a kidney transplant may not receive another in the future while this is not always true, it suffices for the purposes of our model, since we posit a choice between accepting a "lesser"kidney today and a better kidney later. (When a patient receives a second kidney transplant after the first organs failure, there is no reason to expect a better organ, since the patient cannot immediately return to the top of e cadaver kidney queue, and live donors are likely to be more reluctant previous failure
Optimizing the Effedtiveness of Organ Allocation 123 of points received for zero HLA mismatch is 400. The Eurotransplant policy also has greater emphasis on providing young dcildren with a kidney match, giving children younger than 6 years an additional 1095 waiting-time points. We implemented the Eurotransplant policy in our model to see if that policy could also benefit the U.S., but we found little difference. Utilizing Kidney Exchanges A promising approach for kidney paired exchange is to run the maximal matching algorithm over the graph defined by the set of possible exchanges. However, this approach takes away from the autonomy of patients, because it requires them to wait for enough possible pairs to show up before performing the matching, and sometimes it may require them to take a less than perfect matching. We sought to improve this supposedly "optimal solution" by implementing list paired donation in our model. According to each patient's phenotypes, we calculate the expected blood types of the person's parents and siblings, and make that the person's contribution to the "donor pool." In other words, the person brings to the transplant network an expected number r of potential donors. We then make the patient perform list paired donation with the topmost person in the current queue who is compatible in blood type to the donor accompanying the new patient. According to our research, kidneys from live donors are about 21% better than cadaver kidneys in terms of success rate. Thus, it is in the cadaver-list person's best interest to undergo this exchange. We find that for any value of r from 0.2 to 2, list paired donation drastically decreases the length of the waitlist, by factors as large as 3, and makes the queue size stabilize (Figure 3). Patient Choices What should a patient do when presented with the oppor'tunity for a kidney? Tfie decision is not clear-cut; for instance, if the patient is offered a poorly matched kidney now, but a well-matched kidney is likely to arrive in a reasonable time, the patient should perhaps wait. We examine the this tradeoff. We assume that a patient who has already received a kidney transplant may not receive another in the future. While this is not always true, it suffices for the purposes of our model, since we posit a choice between accepting a "lesser" kidney today and. a better kidney later. (When a patient receives a second kidney transplant after the first organ's failure, there is no reason to expect a better organ, since the patient cannot immediately return to the top of the cadaver kidney queue, and live donors are likely to be more reluctant after a previous failure.)
124 The UMAP Journal 28.2(2007) Max Wait Timo(Dond 02 一 Donavon Rate10 Figure 3. Wait time (in days) for various values of donation rate r, with list paired donation, applied over time to the current waitlist. We assume that patients want to maximize expected years of life. Let there be a current transplant available to the patient; we call this the immediate alternative and denote it by Ao. The patient and doctor have some estimate of how this transplant will affect survival; we assume that they have a survival function so(0, t)that describes chance of being alive at timet after the transplant. We further assume that this survival function is continuous and has limit zero at infinity: In other words, the patient is neither strangely prone to die in some infinitesimal instant nor capable of living forever. The patient also has a set of possible future transplants, which we callfil- ture alternatives and write as(Al, A2, .. An). Each future alternative Ai also has a corresponding survival function si(to, t), where to is the starting tim of transplant and t is the current time. We assume that there is a constant robability pi that alternative A will become available at any time. While this is not completely true, we include it to make the problem manageable: More complicated derivations would incorporate outside factors whose complexity would overwhelm our current framework. Finally, if the patient opts for a fu- ture alternative and delays transplant, survival is governed by a default survival ction sd
124 The UMAP Journal 28.2 (2007) "•-0- Donation Rate 2.01 1.5 15 6 7 8 9 10 Time (Years) Figure 3. Wait time (in days) for various values of donation rate r, with list paired donation, applied over time to the current waitlist. We assume that patients want to maximize expected years of life. Let there be a current transplant available to the patient; we call this the immediate alternative and denote it by Ao. The patient and doctor have some estimate of how this transplant will affect survival; we assume that they have a survivalfunction so (0, t) that describes chance of being alive at time t after the transplant. We further assume that this survival function is continuous and has limit zero at infinity: In other words, the patient is neither strangely prone to die in some infinitesimal instant nor capable of living forever. The patient also has a set of possible future transplants, which we callfitture alternatives and write as (A1, A 2,... , .A,,). Each future alternative .A- also has a corresponding survival function si(to, t), where to is the starting time of transplant and t is the current time. We assume that there is a constant probability pi that alternative Ai4 will become available at any time. While this is not completely true, we include it to make the problem manageable: More complicated derivations would incorporate outside factors whose complexity would overwhelm our current framework. Finally, if the patient opts for a future alternative and delays transplant, survival is governed by a default survival function Sd