Analysis of Kidney Transplant System 139 Analysis of Kidney Transplant System Using markov process Models Jeffrey Y. Tang Yue Yang Jingyuan Wu Princeton universit Advisor: Ramin Takloo-Bighash Summary Abstract: We use Markov processes to develop a mathematical model for the U.S. kidney transplant system. We use both mathematical models and computer simulations to analyze the effect of certain parameters on transplant waitlist size and investigate the effects of policy changes on the model's behavior. waitlist members and insufficient deceased donor and living donor ing of new transplants available. Possible policy changes to improve the situation include presumed nsent, tightening qualifications for joining the waitlist, and relaxing the re- quirements for accepting deceased donors Ve also evaluate alternative models from other countries that would reduce the waitlist, and examine the benefits and costs of these models compared with the current U.S. model. We analyze kidney paired exchange along with generic n- cycle kidney exchange, and use our original U.S. model to evaluate the benefits of incorporating kidney exchange. We develop a model explaining the decisions that potential recipients face con- cerning organ transplant, then expand this consumer decision theory model to explain the decisions that potential organ donors face when deciding whether to We finally consider an extreme policy change-the marketing of kidneys for kid- ney transplants-as a method of increasing the live-donor pool to reduce waitlist The UMAP Journal28(2)(2007)139-158. @ Copyright 2007by COMAP, Inc. Allrights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use anted without fee provided that copies are not made or distributed for profit or commercial ntage and that copies bear this notice Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
Analysis of Kidney Transplant System Analysis of Kidney Transplant System Using Markov Process Models Jeffrey Y. Tang Yue Yang Jingyuan Wu Princeton University Princeton, NJ Advisor: Ramin Takloo-Bighash Summary Abstract: We use Markov processes to develop a mathematical model for the U.S. kidney transplant system. We use both mathematical models and computer simulations to analyze the effect of certain parameters on transplant waitlist size and investigate the effects of policy changes on the model's behavior. Our results show that the waitlist size is increasing due to the flooding of new waitlist members and insufficient deceased donor and living donor transplants available. Possible policy changes to improve the situation include presumed consent, tightening qualifications for joining the waitlist, and relaxing the requirements for accepting deceased donors. We also evaluate alternative models from other countries that would reduce the waitlist, and examine the benefits and costs of these models compared with the current U.S. model. We analyze kidney paired exchange along with generic ncycle kidney exchange, and use our original U.S. model to evaluate the benefits of incorporating kidney exchange. We develop a model explaining the decisions that potential recipients face conceming organ transplant, then expand this consumer decision theory model to explain the decisions that potential organ donors face when deciding whether to donate a kidney. We finally consider an extreme policy change-the marketing of kidneys for kidney transplants-as a method of increasing the live-donor pool to reduce waitlist size. The UMAPJournal28 (2) (2007) 139-158. @Copyright2007by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAR 139
140 The UMAP Journal 28.2(2007) Introduction The American organ transplant system is in trouble: Waitlist size is in- creasing;as of February 2007, 94,000 candidates were waiting for a transplant, among them 68,000 waiting for kidneys. We create a mathematical model using a Markov process to examine the effects of parameters on waitlist size and to investigate theeffects of policy changes. Possible policy changes toimprove the situation include assuming that all people are organ donors unless specifically pecified(presumed consent), tightening qualifications for joining the waitlist, and relaxing the requirements for accepting deceased donors. We evaluate alternative models from other countries that could reduce the waitlist, and examine the benefits and costs of these models compared with the current U.S. model. We analyze the Korean kidney paired exchange along with the generic n-cycle kidney exchange, and use our original U.S. model to evaluate the benefits of incorporating the kidney exchange. The Korean model increases the incoming rate of live donors, which is preferable because live- donor transplants lead to higher life expectancy. However, this policy alone cannot reverse the trend in waitlist size. We also develop a model explaining the decisions that potential recipients face concerning organ transplant. We expand this consumer decision theory model to explain the decisions that potential organ donors face when decid ing whether or not to donate a kidney. Finally, we consider an extreme polic change-the marketing of kidneys for kidney transplants as a method of in- creasing the live donor pool to reduce waitlist size. We consider two economic models: one in which the government buys organs from willing donors and offsets the price via a tax, and one in which private firms are allowed to buy or- gans from donors and offer transplants to consumers at the market-equilibrium Pr Task 1: The U.S. Kidney Transplant System Background: Kidney Transplants Blood Type: Recipient and donor must have compatible blood types(Ta- ble 1). o HLA: Recipient and donor must have few mismatches in the HLA antigen locus. Because of diverse allelic variation, perfect matches are rare, Mi matches can cause rejection of the organ o PRA: PRA is a blood test that measures rejection to human antibodies in the body. The value is between 0 and 99, and its numerical value indicates the percent of the U.S. population that the blood,'s antibodies reacts with. High PRA patients have lower success rates among potential donors[U so it is more difficult to locate donate matches for them table 2)
140 The UMAP Journal 28.2 (2007) Introduction The American organ transplant system is in trouble:, Waitlist size is increasing; as of February 2007, 94,000 candidates were waiting for a transplant, among them 68,000 waiting for kidneys. We create a mathematical model using a Markov process to examine the effects of parameters on waitlist size and to investigate the effects of policy changes. Possible policy changes to improve the situation include assfiming that all people are organ donors unless specifically specified (presumed consent), tightening qualifications for joining the waitlist, and relaxing the requirements for accepting deceased donors. We evaluate alternative models from other countries that could reduce the waitlist, and examine the benefits and costs of these models compared with the current U.S. model. We analyze the Korean kidney paired exchange along with the generic n-cycle kidney exchange, and use our original U.S. model to evaluate the benefits of incorporating the kidney exchange. The Korean model increases the incoming rate of live donors, which is preferable because livedonor transplants lead to higher life expectancy. However, this policy alone cannot reverse the trend in waitlist size. We also develop a model explaining the decisions that potential recipients face concerning organ transplant. We expand this consumer decision theory model to explain'the decisions that potential organ donors face when deciding whether or not to donate a kidney. Finally, we consider an extreme policy change-the marketing of kidneys for kidney transplants as a method of increasing the live donor pool to reduce waitlist size. We consider two economic models: one in which the government buys organs from willing donors and offsets the price via a tax, and one in which private firms are allowed to buy organs from donors and offer transplants to consumers at the market-equilibrium price. Task 1: The U.S. Kidney Transplant System Background: Kidney Transplants "* Blood Type: Recipient and donor must have compatible blood types (Table 1). "* HLA: Recipient and donor must have few mismatches in the HLA antigen locus. Because of diverse allelic variation, perfect matches are rare. Mismatches can cause rejection of the organ. "* PRA: PRA is a blood test that measures rejection to human antibodies in the body. The value is between 0 and 99, and its numerical value indicates the percent of the U.S. population that the blood's antibodies reacts with. High PRA patients have lower success rates among potential donors[U so it is more difficult to locate donate matches for them (Table 2)
Analysis of Kidney Transplant System 141 Table 1. Compatible blood types [American National Red Cross 2006 Recipient blood type Donor red blood cells must be AB→ B- B+ o一 In].Spopul%3%跳第%3 Relationship between PRa and transplant waiting time [University of Maryland, 2007 Peak PRa Proportion of. Median waiting time waiting list to transplant(days) 2079 21% 80 19% Explanation of model The Organ Procurement and Transplantation Network's(oPtN)priority system for assigning and allocating kidneys is used as the core model for the current U.S. transplantation system [Organ Procurement.. 2006]. The OPtN kidney network is divided into three levels: the local level, the regional level, and the national level. There are 270 individual transplant centers distributed throughout the U.S.[ Dept of Health and Human Services 20071, organized into The priority system for allocation of deceased-donor kidneys to candidates on the waitlist takes into account proximity of recipient to donor, recipient wait time, and match to donor, with location carrying greater weight, according to a point system[Organ Procurement.. 2006 Wait time points A candidate receives one point foreach year on the waiting t.A candidate also an additional fraction of a point based on rank on the list: With n candidates on the list, the rth-longest-waiting candidate gets 1-(r-1)/n points. So, for example, the longest-waiting candidate(r=1) gets one additional point, the newest arrival on the list(r= n) gets 1/n Age points The young receive preferential treatment because their expected
Analysis of Kidney Transplant System Table 1. Compatible blood types [American National Red Cross 2006]. Recipient blood type Donor red blood cells must be: AB+ 0- 0+ A- A+ B- 3+ AB- AB+ AB- 0- A- B- ABA+ 0- 0+ A- A+ "A- 0- AB+ 0- 0+ B- B+ B- 0- B- 0+ 0- 0+ 0- 0- In U.S. population: 7% 38% 6% 34% 2% 9% 1% 3% Table 2. Relationship between PRA and transplant waiting time [University of Maryland ... 2007]. Peak PRtA Proportion of Median waiting time waiting list to transplant (days) 0-19 60% 490 20-79 21% 1,042 80+ 19% 2,322 Explanation of Model - The Organ Procurement and Transplantation Network's (OPTN) priority system for assigning and allocating kidneys is used as the core model for the current U.S. transplantation system [Organ Procurement ... 2006]. The OPTN kidney network is divided into three levels: the local level, the regional level, and the national level. There are 270 individual transplant centers distributed throughout the U.S. [Dept. of.Health and Human Services 2007], organized into 11 geographic regions. The priority system for allocation of deceased-donor kidneys to candidates on the wraitlist takes into account proximity of recipient to donor, recipient wait time, and match to donor, with location carrying greater weight, according to a point system [Organ Procurement... 2006]: "* Wait time points A candidate receives one point for each year on the waiting list. A candidate also an additional fraction of a point based on rank on the list: With n candidates on the list, the rth-longest-waiting candidate gets 1 - (r - 1)/n points. So, foi example, the longest-waiting candidate (r = 1) gets one additional point, the newest arrival on the list (r = n) gets 1/n additional points. "* Age points Theyoung receive preferential treatmentbecause their expected 141
142 The UMAP Journal 28.2(2007) lifetime with the transplant is greater. Children below 11 years of age get 4 additional points, and those between 11 and 18 get 3 additional points. o hla mismatch points Because there are two chromosomes, the possible number m of mismatches in the donor-recipient( DR) locus of the hla se quence is 0, 1, or 2. A candidate-donor pair gets 2-m points Model Setup Wemodel theentry and exit of candidates from the waitlistwitha continuous- time Markov birth/ death process [Ross 2002]. It accommodates reduction of the waitlist size(arrivals of living donors and deceased donors and deaths and recoveries of waitlist candidate)and waitlist additions. o In 2006, 29, 824 patients were added to the kidney transplant waitlist, while 5,914 transplants had living donors, so 5914/(29824+ 5914)N% of in- coming patient cases have a willing compatible living donor. o The procedure for allocating deceased-donor kidneysis [Organ Procurement .2006,353-16f: First, match the donor blood type with compatible recipient blood types. exceptions are, Type O donors must be donated to type O recipients first, and Type B donors must be donated to type B recipients first Perfect matches(same blood type and no HLA mismatch)receive first If a kidney with blood type O or B has no perfect-matching candidates in the above procedure, then the pool is reopened for all candidates In the 17% of cases of no a perfect match with any recipient [wikipedia 2007, then sort by PRa value(higher priority to high PRA; high PRA long time), then atibility, which likely means being on the waitlist for a means low c by regional location of the kidney, then by points in the oint allocation system Summary of Markov process Let N, be a random variable indicating the number of people in the waitlist at time t. The properties of N can be generalized in Figure 1, where o Each arrow represents a possible event at the current state(N). o The rate at which each event occurs is exponentially distributed
142 The UMAP Journal 28.2(2007) lifetime with the transplant is greater. Children below 11 years of age get 4 additional points, and those between 11 and 18 get 3 additional points. e HLA mismatch points Because there are two chromosomes, the possible number m of mismatches in the donor-recipient (DR) locus of the I-ILA sequence is 0, 1, or 2. A candidate-donor pair gets 2 - m points. Model Setup We model the entry and exit of candidates from the waitlist with a continuoustime Markov birth/death process [Ross 2002]. It accommodates reduction of the waitlist size (arrivals of living donors and deceased donors and deaths and recoveries of waitlist candidate) and waitlist additions. "* In 2006, 29,824 patients were added to the kidney transplant waitlist, while 5,914 transplants had living donors, so 5914/(29824 + 5914) ;-, 17% of incoming patient cases have a willing compatible living donor. "* The procedure for allocating deceased-donor kidneys is [Organ Procurement ... 2006, 3.5, 3-16ff]: - First, match the donor blood type with compatible recipientblood types. The only exceptions are: * Type 0 donors must be donated to type 0 recipients first, and: * Type B donors must be donated to type B recipients first. - Perfect matches (same blood type and no I-ILA mismatch) receive first priority. - If a kidney with blood type 0 or B has no perfect-matching candidates in the above procedure, then the pool is reopened for all candidates. - In the 17% of cases of no a perfect match with any recipient [Wikipedia 20071, then sort by PRA value (higher priority to high PRA; high PRA means low compatibility, which likely means being on the waitlist for a long time), then by regional location of the kidney, then by points in the point allocation system. Summary of Markov Process Let Nt be a random variable indicating the number of people in the waitlist at time t. The properties of Nt can be generalized in Figure 1, where "* Each arrow represents a possible event at the current state (N). "* The rate at which each event occurs is exponentially distributed
Analysis of Kidney Transplant System 143 Deceased donor transplant ewwaitlist arriva Waist Death avallable Condition better. Figure 1. Markov process model of waitlist. After the event occurs, by memorylessness of the exponential distribution, the time is reset to zero, as if nothing has happened Wait time is assumed zero for compatible live donor transplants Because there are so many local centers(270), we simplify our model to consider the region (of which there are 11) as the lowest level of waitlist Candidates who become medically unfit surgery are removed from the wait list and in our model are classified as deaths Candidates whose conditions improve enough are removed from the wait list. Both these people and those recovering from surgery have exponential remaining lifetime with mean 15 years Weuse the parametervaluesin Table 3, which come from the OPTN database using values from 2006. Table 3. Means of exponential distributions Symbol Mean 817d λ3=A1+λ2 coming patients with living donors available 16.2d total incoming patients(independants 817+162=979d rivals of de 26.9 d [Norman 2005 waitlist deaths 270d waitlist condition improves per day A4=A1+42+u, waitlist departures(independent RVs) 269+27.0+24=563d time of life after surgery 0, if candidate dies [European Medical Tourism 20071 15y with transplant
Analysis of Kidney Transplant System 143 beceased donor transpolant New-waitlist arrival Incohindoatients (W With living donors Conditi on Better. " (4) -E)p (tti Figure 1. Markov process model of waitlist. "* After the event occurs, by memorylessness of the exponential distribution, the time is reset to zero, as if nothing has happened. "* Wait time is assumed zero for compatible live donor transplants. "• Because there are so many local centers (270), we simplify our model to consider the region (of which there are 11) as the lowest level of waitlist candidates. "* Candidates who become medically unfit surgery are removed from the wait list and in our model are classified as deaths. "* Candidates whose conditions improve enough are removed from the waitlist. Both these people and those recovering from surgery have exponential remaining lifetime with mean 15 years. * We use the parameter values in Table 3, which come from the OPTN database using values from 2006. Table 3. Means of exponential distributions. Symbol Rate Mean A1 new waitlist arrivals 81.7 d A2 incoming patients with living donors available 16.2 d A3 A1 + A2 total incoming patients (independent RVs) 81.7 + 16.2 = 97.9 d Al arrivals of deceased donor transplants 26.9 d [Norman 2005] 112 waitlist deaths 27.0 d A3 waitlist condition improves per day 2.4 d 114 Al + J2 + 13p waitlist departures (independent RVs) 26.9 + 27.0+ 2.4 = 56.3 d TAB time of life after surgery 0, if candidate dies; [European Medical Tourism 2007] 15 y with transplant
144 The UMaP Journal 28. 2 (2007) Analysis of Model Our two variables to indicate strength of model strategy are the number of people in the waitlist(or the number of people who get transplants)and optimizing the matches so as to maximize lifetime after receiving a transplant. Efficient Allocation of Kidney Transplants We build a new model to take into account the effects of both distance and optimal match. A kidney arriving at a center can be given to the best matching candidate at that center, the best in the region, or the best in the countr Of 10,000 candidate recipients, on average 37 are from the center, 873 are from the region outside the center, and 9,090 are from the nation outside the region. Using a uniform distribution on (0, 1), we randomly assign scores to each of the 10,000, rank them by score, and take the highest rank at each level. We iterate this process 10,000 times and find the average rank of the toy candidate in each area(Table 4). Table 4 Average quality of top candidate in each area. Probability that top candidate Average rank(from botton is in this group Center 037% 97397 Region outside center I-270 8.72% Nation outside region 1 90.90 Transportation of the kidney can lead to damage, because of time delay in transplanting. Thus, we posit a damage function f that depends on the location of the recipient: lower in the center, slightly higher in the region but outside the center, and even higher in the country but outside the f(local)< f(regional)< f(national) Let us assume that when a kidney arrives in a center, it goes to the center, the region, or outside the region with probabilities a1, a2, and a3. Let G be the weighted score for the kidney, with G= a1 (1-f(local ) scorelocal+a2(1-f(regional)scorereglonal +a3(1-f(national).scorenat and expected value E(G)=a1(1-f(oca)9739.7+a2:(1- f(regional)97 (1-f(national) Optimizing G as a function of the ai is a linear programming problem, but we cannot solve it without assessing the damage function for different regions
144 The UMAP Journal 28.2 (2007) Analysis of Model Our two variables to indicate strength of model strategy are the number of people in the waitlist (or the number of people who get transplants) and optimizing the matches so as to maximize lifetime after receiving a transplant. Efficient Allocation of Kidney Transplants We build a new model to take into account the effects of both distance and optimal match. A kidney arriving at a center can be given to the best matching candidate at that center, the best in the region, or the best in the country. Of 10,000 candidate recipients, on average 37 are from the center, 873 are from the region outside the center, and 9,090 are from the nation outside the region. Using a uniform distribution on (0,1), we randomly assign scores to each of the 10,000, rank them by score, and take the highest rank at each level. We iterate this process 10,000 times and find the average rank of the top candidate in each area (Table 4). Table 4. Average quality of top candidate in each area. Probability that top candidate Average rank (from bottom) is in this group of top candidate among 10,000 Center = 0.37% 9739.7 Region outside center 11 o 8.72% 9989.7 Nation outside region 1 90.90% 9999.9 Transportation of the kidney can lead to damage, because of time delay in transplanting. Thus, we posit a damage function f that depends on the location of the recipient: lower in the center, slightly higher in the region but outside the center, and even higher in the country but outside the region, i.e., f(local) < f(regional) < f(national). Let us assume that when a kidney arrives in a center, it goes to the center, the region, or outside the region with probabilities al, a2, and a3. Let G be the weighted score for the kidney, with G = a1 • (1 - f(local)) , scorelo,.l + a2 . (I - f(regional)) • scoreregional + a3" (1 - f (national)) , scorena,tonal. (1) and expected value E(G) = al' (1 - f(local)). 9739.7 + a2. (1 - f(regional)). 9989.7 + a3 . (1 - f(national)). 9999.9. (2) Optimizing G as a function of the ai is a linear programming problem, but we cannot solve it without assessing the damage function for different regions
Analysis of Kidney Transplant System 145 Minimizing the Waitlist There are some who argue that the wait time assignment is too lax and leads to unfair waitlists. In the current system, urgency is specifically stated as have no effect on the points used to determine who receives a transplant Organ Procurement.. 2006]. A patient is permitted to join the waitlist (in nore than one region, even) when kidney filtration rate falls below a particular value or when dialysis begins. Getting on the waitlist as early as possible helps pad"the points for waiting time. a patient not yet on dialysis can afford to wait longer yet may receive a kidney sooner than others joining later who have more urgent need. Urgency has no effect on a patient's rank for receiving a Iney. a possible solution is to tighten the conditions for joining the waitlist, so thatthat a patient's wait time begins at dialysis. This policy would slow the rate of growth of the waitlist, at the expense of more waitlisted patients dying A strategy to increase the rate of deceased-donor arrival, already policy in Illinois, is to presume that everyone desires to be an organ donor unless they ecifically opt out. Figure 2 shows the field space of combinations from rates for these two Net Waitlist Arrival Rate /yr 己 5045 131 igure 2. Net waitlist arrival rate per Using both strategies could make net waitlist arrival rate negative, for ex- mple, if waitlist arrivals can be decreased by 25% and donor size by 175 Model Strengths The Markov process, with exponentially distributed entry/exit times, makes calculations simple
Analysis of Kidney Transplant System 145 Minimizing the Waitlist There are some who argue that the wait time assignment is too lax and leads to unfair waitlists. In the current system, urgency is specifically stated as have no effect on the points used to determine who receives a transplant [Organ Procurement ... 2006]. A patient is permitted to join the waitlist (in more than one region, even) when kidney filtration rate falls below a particular value or when dialysis begins. Getting on the waitlist as early as possible helps "pad" the points for waiting time. A patient not yet on dialysis can afford to wait longer yet may receive a kidney sooner than others joining later who have more urgent need. Urgency has no effect on a patient's rank for receiving a kidney. A possible solution is to tighten the conditions for joining the waitlist, so that that a patient's wait time begins at dialysis. This policy would slow the rate of growth of the waitlist, at the expense of more waitlisted patients dying. A strategy to increase the rate of deceased-donor arrival, already policy in Illinois, is to presume that everyone desires to be an organ donor unless they specifically opt out. Figure 2 shows the field space of combinations from rates for these two policies. C Net Waitlist Arrival Rate / yr 7291 m . S~5045 2975 3131 -1050 -2100 S5 10 is 20 25 30 Increase in deceased donors due to presumed consent Figure 2. Net waitlist arrival rite per year. Using both strategies could make net waitlist arrival rate negative, for example, if waitlist arrivals can be decreased by 25% and donor size by 17%. Model Strengths 9 The Markov process, with exponentially distributed entry/exit times, makes calculations simple
146 The UMAP Journal 28. 2 (2007) o Minimizing the waitlist depends on only two variables The modelincorporates HLA values, PRA distributions, no-mismatch prob. bilities, region distribution, and blood-type distribution and compatibility o The model is compatible with alternative strategies, such as a paired ex- stem Model Weaknesses o Remaining lifetime after surgery should be adjusted, since an exponential distribution for remaining lifetime is appropriate only until a certain age o The model cannot account for patients'. We assume that all patients offered kidney take it if the hla value is reasonable, which may not be the case. o The model does not make distinctions for race and socioeconomic status. Different races have differing wait times [Norman 2005, 457] o We assume independence of random variables, so thatincreasing or decreas- ing parameters will not affect other parameters o Ouremphasis on waitlistsizeneglects waitlist time; anotherapproach would be to try to minimize waitlist wait time Tasks 2 and 3: Kidney Paired Exchange Background As noted at the University of Chicago Hospitals, "In 10 to 20 percent of cases at the Hospitals, Patients who need a kidney transplant have family or friends who agree to donate, but the willing donor is found to be biologically unsuited for that specific recipient"[Physicians propose.. 1997] of patient/ donor candidates. Each donor is incompatible with the intended patient but compatible with the other patient. Surgery is performed simulta- neously in the same hospital on four people, with two kidney removals and two kidney transplant However, fornot all patient-donor pairs will the ere be a mutually compatible partner pair. In such a case, it is possible for the cycle to expand to n patient- donor pairs, with each donor giving to a compatible stranger patient(Figure 4) Since such an exchange requires at least 2n surgeons at the same hospital, higher-order exchanges are less desirable on logistic ground A kidney paired exchange program does not affect the intrinsic model out- lined for Task 1. The only change when live incompatible pairs get swapped is
146 The UMAP Journal 28.2 (2007) "* Minimizing the waitlist depends on only two variables. "* The model incorporates HLA values, PRA distributions, no-mismatch probabilities, region distribution, and blood-type distribution and compatibility requirements. "* The model is compatible with alternative strategies, such as a paired exchange system. Model Weaknesses "* Remaining lifetime after surgery should'be adjusted, since an exponential distribution for remaining lifetime is appropriate only until a certain age. "o The model cannot account for patients'. We assume that all patients offered a kidney take it if the I-LA value is reasonable, which may not be the case. "* The model does not make distinctions for race and socioeconomic status. Different races have differing wait times [Norman 2005, 457]. "o We assume independence of random variables, so that increasing or decreasing parameters will not affect other parameters. "* Our emphasis on waitlist size neglects waitlist time; another approach would be to try to minimize waitlist wait time. Tasks 2 and 3: Kidney Paired Exchange Background As noted at the University of Chicago Hospitals, "In 10 to 20 percent of cases at the Hospitals, patients who need a kidney transplant have family or friends who agree to donate, but the willing donor is found to be biologically unsuited for that specific recipient" [Physicians propose ... 1997]. In the simple kidney paired exchange system (Figure 3), there are two pairs of patient / donor candidates. Each donor is incompatible with the intended patient but compatible with the other patient. Surgery is performed simultaneously in the same hospital on four people, with two kidney removals and two kidney transplants. However, for not all patient-donor pairs will there be a mutually compatible partner pair. In such a case, it is possible for the cycle to expand to n patientdonor pairs, with each donor giving to a compatible stranger patient (Figure 4). Since such an exchange requires at least 2n surgeons at the same hospital, higher-order exchanges are less desirable on logistic grounds. A kidney paired exchange program does not affect the intrinsic model outlined for Task 1. The only change when live incompatible pairs get swapped is
Analysis of Kidney Transplant System 147 Figure 4. Kidney paired exchange system. s Donot Figure 5. n-way kidney exchange. that the rate of candidates entering the waitlist is reduced. However, for those in the waitlist, the same procedure is still being used Kidney paired exchanges have higher priority than larger cycles, for logisti- cal reasons Recipients must receive a transplant in the region in which they are on the waitlist(this reduces travel time) Exchanges withno mismatch are prioritized over exchanges with mismatch. Waiting time for an exchange is assumed to be 0, as was live donor matches in Task 1. After an exchange, all individuals involved are removed from the pools of donors and recipients We use Region 9 as a sample region to test our model. Region 9 has a waitlist(6058 )similar to the average waitlist per region, and 909 candidates (15%) have willing but incompatible donors. We ran our simulation 100 times and computed averages. Table 5 shows extrapolationof the Analysis In 2006, there were 26,689 kidney transplants nationwide, including only a few kidney paired exchanges. The approximately 9, 656 additional transplants yearly indicated in Table 5 would have been a 36% increase and would have reduce the waitlist correspondingly by 14%, from 69, 983 to 60, 327
Analysis of Kidney Transplant System incompab compatIblenepabe Figure 4. Kidney paired exchange system. DanorlI Doncr2 Patient N tsi re PaPatinn IA Patilent 2 Paint ,Patient 3' Donor A Figure 5. n-way kidney exchiange. that the rate of candidates entering the waitlist is reduced. However, -for those in the waitlist, the same procedure is still being used. "* Kidney paired exchanges have higher priority than larger cycles, for logistical reasons. "* Recipients must receive a transplant in the region in which they are on the waitlist (this reduces travel time). "* Exchanges with no mismatch are prioritized over exchanges with mismatch. Waiting time for an exchange is assumed to be 0, as was live donor matches in Task 1. After an exchange, all,individuals involved are removed from the pools of donors and recipients. We use Region 9 as a sample region to test our model. Region 9 has a waitlist (6058) similar to the average waitlist per region, and 909 candidates (15%) have willing but incompatible donors. We ran our simulation 100 times and computed averages. Table 5 shows extrapolation of the results nationwide. Analysis In 2006, there were 26,689 kidney transplants nationwide, including only a few kidney paired exchanges. The approximately 9,656 additional transplants yearly indicated in Table 5 would have been a 36% increase and would have reduce the waitlist correspondingly by 14%, from 69,983 to 60,327. 147
148 The UMAP Journal 28. 2(2007) Table 5 Averaged results of repeated simulations of multiple-pair transplant exchange nationwide (extrapolated from Region 9 data). Kind of match Transplants Percentage 2-way no mismatch 2-way non-perfect 9646 0 3-way non-perfect otal transplants 9656 h willing b 10497 Another option is to consider multiple exchanges for all donor-recipient pairs in a particular center. This minimizes the travel time required for the patients, while improving the computational power of the search algorithm. A center has on average 259 candidates, of whom 39 have willing but incompat- ible donors available. For this sample size, we get on average 25 transplants (65%), compared to 92% under exchange at the regional level. Furthermore, the proportion of high-quality transplants is also smaller. The benefits of a center- only exchange system are personal and psychological: Patients live close to the surgery location, which means better support from both family and familiar physicians Task 4: Patient Choice Theory Suppose a patient is offered a barely compatible kidney from the cadaver queue. There are two options: take the bad-match kidney immediately, or ● wait for a better match from the cadaver queue or We consider two cases: without paired exchange and with it. Model 1: Decision Scenario without Paired Exchange Of transplants with poorly matched kidneys, 50% fail after 5-7 years. So we assume that the lifetime after a poorly matched kidney transplant is expo. nentially distributed with mean 6 years[Norman 2005, 458 ables with mean X by solving P(survive t years)=edes to exponential vari- We translate data of Table 6 on survival probabilitic
148 T77e UMAP Journal 28.2 (2007) Table S. Averaged results of repeated simulations of multiple-pair transplant exchange nationwide (extrapolated from Region 9 data). Kind of match Transplants Percentage 2-way no mismatch 2 2-way non-perfect 9,646 3-way no mismatch 0 3-way non-perfect 8 Total transplants 9,656 92% Candidates with willing but incompatible donor 10,497 Another option is to consider multiple exchanges for all donor-recipient pairs in a particular center. This minimizes the travel time required for the patients, while improving the computational power of the search algorithm. A center has on average 259 candidates, of whom 39 have willing but incompatible donors available. For this sample size, we get on average 25 transplants (65%), compared to 92% under exchange at the regional level. Furthermore, the proportion of high-quality transplants is also smaller. The benefits of a centeronly exchange system are personal and psychological: Patients live close to the surgery location, which means better support from both family and familiar physicians. Task 4: Patient Choice Theory Suppose a patient is offered a barely compatible kidney from the cadaver queue. There are two options: "o take the bad-match kidney immediately, or "* wait for a better match, - from the cadaver queue or - from a paired exchange. We consider two'cases: without paired exchange and with it. Model 1: Decision Scenario without Paired Exchange Of transplants with poorly matched kidneys, 50% fail after 5-7 years. So we assume that the lifetime after a poorly matched kidney transplant is exponentially distributed with mean 6 years [Norman 2005, 458]. We translate data of Table 6 on survival probabilities to exponential variables with mean A by solving P(survive t years) = e-At
