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