《数学建模》美赛优秀论文：2007 C O Optimizing the Effectiveness of Organ Allocation

Optimizing the effectiveness of Organ Allocation 117 Optimizing the Effectiveness of Organ Allocation Matthew rognlie eng Amy Wen Duke University Durham, Nc Advisor: David Kraines Introduction The first successful organ transplant occurred in 1954, a kidney transplan etween twin brothers in Boston [Woodford 2004]. Since then, although the number of transplants per year has steadily risen, the number of organ donors has not kept up with demand [Childress and Liverman 2006](Figure 1) To ensure equitable distribution of available organs, Congress passed the National Organ Transplant Act in 1984. The act established the Organ Pro- curement and Transplantation Network(OPTN), a regionalized network for organ distribution [Conover and Zeitler 2006]. In 2000, the U.S. Department of Health and Human Services(HHS)implemented a additional policy called the Final Rule, which ensured that states could not interfere with OptN policies that require organ sharing across state lines [Organ Procurement.. 1999] The organ matching process involves many factors, whose relative im tance depends on the type of organ involved. These include compatibility,re- gion, age, urgency of patient, and waitlist time [Organ Procurement.. 2006 though most countries use the same basic matching processes, systems var in their emphasis on particular parameters [Transplantation Society..2002; UK Transplant 2007; Doxiadis et al. 2004; De Meester et al. 1998]1 In 2006, kidneys comprised 59% of all organs transplanted [Organ Procure- ment..2007]. In determining compatibility in kidney transplants, doctor look a ABo blood type: The aBo blood type indicates the presence of two types of antigens, A and B, present in the patient's body. Antigens are foreign molecules or substances that triggeranimmuneresponse. People with blood The UMAP Journal28(2)(2007)117-138. Copyright 2007 by COMAP, Inc. Allrights reserved Permission to make digital or hard copies of part or all of this work for personal or classroom use ad wanted without fee provided that copies are not made or distributed for profit or commercial

Optimizing the Effectiveness of Organ Allocation Optimizing the Effectiveness of Organ Allocation Matthew Rogniie Peng Shi Amy Wen Duke University Durham, NC Advisor: David Kraines Introduction The first successful organ transplant occurred in 1954,a kidnej, transplant between twin brothers in Boston [Woodford 2004]. Since then, although the number of transplants per year has steadily risen, the number of organ donors has not kept up with demand [Childress and Liverman 2006] (Figure 1). To ensure equitable distribution of available organs, Congress passed the National Organ Transplant Act in 1984. The act established the Organ Pro-' curement and Transplantation Network (OPTN), a regionalized network for organ distribution [Conover and Zeitler 2006]. In 2000, the U.S. Department of Health and Human Services (HHS) implemented a additional policy called the Final Rule, which ensured that states could not interfere with OPTN policies that require organ sharing across state lines [Organ Procurement... 1999]. The organ matching .process involves many factors, whose relative impor￾tance depends on the type of organ involved. These include compatibility, re￾gion, age, urgency of patient, and waitlist time [Organ Procurement... 2006]. Although most countries use the same basic matching processes, systems vary in their emphasis on particular parameters [Transplantation Society ... 2002; UK Transplant 2007; Doxiadis et al. 2004; De Meester et al. 1998]. In 2006, kidneys comprised 59% of all organs transplanted [Organ Procure￾ment ... 2007]. In determining compatibility in kidney transplants, doctors look at: * ABO blood type: The ABO blood type indicates the presence of two types of antigens, A and B, present in the patient's body. Antigens are foreign molecules or substances that trigger an immune response. People withblood The UMAPJournal28(2) (2007) 117-138. @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 117

118 The UMAP Journal 28.2(2007) supply and Demand of Organs vs Year 品品2o苏s 941995199619971998 99200020012002200320042005 Figure 1. Number of transplants and cadaveric organ donors. Source: OPTN Annual Report 2005 [U.S. Organ Procurement .. 2005] type a have antigen a in their body, people with blood type B have antigen B, people with blood type AB have both, and people with blood type Ohave neither. a person with blood type AB can receive an organ from anyone, a person with blood type A or B can rece ive an organ from a person of blood type o or the same blood type, and a person with blood type O can receive an organ only from someone with typo blood Human Leukocyte Antigens(HLA): HLA indicates a persons tissue type, whose most important components are the A, B, and DR antigens, Each antigen consists of two alleles, and matching all six components results in a significantly increased success rate for kidney transplants. Patients with mismatched components, however, can still survive for many years [U.S Organ Procurement.. 2005]. o Panel Reactive Antibody(PRA): PRA is a blood test, measuring the per entage of the U.S. population that blood samples are likely to react with. It tests for the presence of antibodies, proteins that bind to foreign molecules [University of Maryland. 2004a]. Blood can become more sensitive due to previous transplants, blood transfusions, or pregnancies [Duquesnoy 2005]

118 The UMAP Journal 28.2 (2007) a C7 0 C 0 0 56 Co IL .4 z 3 2 x 10' Supply and Demand of Organs vs. Year 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Year 2005 Figure 1. Number of transplants and cadaveric organ donors. Source: OPTN Annual Report 2005 [U.S. Organ Procurement... 2005]. type A have antigen A in their body, people with blood type B have antigen B, people with blood type AB have both, and people with blood type 0 have neither. A person with blood type AB can receive an organ from anyone, a person with blood type A or B can receive an organ from a person of blood type 0 or the same blood type, and a person with blood type 0 can receive an organ only from someone with type 0 blood. o Human Leukocyte Antigens (HLA): HLA indicates a person's tissue type, whose most important components are the A, B, and DR antigens. Each antigen consists of two alleles, and matching all six components results in a significantly increased success rate for kidney transplants. Patients with mismatched components, however, can still survive for many years [U.S. Organ Procurement ... 2005]. * Panel Reactive Antibody (PRA): PRA is a blood test, measuring the per￾centage of the U.S. population that blood samples are likely to react with. It tests for the presence of antibodies, proteins that bind to foreign molecules [University of Maryland ... 2004a]. Blood can become more sensitive due to previous transplants, blood transfusions, or pregnancies [Duquesnoy 2005]. I I I I I I I I I ci p ci ci l F-if----Wait-lislt- 1---13--. Donors

Optimizing the effectiveness of Organ Allocation 119 Kidney transplants are common partly because kidneys, unlike most other organs, can be safely obtained from live donors. In fact, live-donor kidneys are ore effective than cadaveric kidneys, with longer half-lives and lower rejec- tion rates [Gentry et al. 2005]. Over 75% of living donors in 2004 were related parents, siblings, spouses) to the transplantrecipients[ Childress and Liverman 2006]. However, some people willing to donate to an intended recipient can- not because of blood type or HLA incompatibility, leaving over 30%o of patients without a suitable kidney transplant [Segev et al. 2005]. One solution is kidney paired donation(KPD), which matches two incompatible donor-recipient pairs where the donor of each pair is compatible with the recipient of the other, sat- isfying both parties [Ross et al. Another is list paired donation(LPD),where a recipient receives higher priority on the waitlist if an associated donor gives to another compatible recipient on the waitlist [Gentry et al. 2005] We incorporate all these factors in modeling the various aspects of trans- plantation. First, we focus on the U.S. network and produce a generic model of the processes that impact the number of people on the waitlist, the number of transplants, and the length of wait time. To illustrate our model, we use data specific to kidney transplants, and also examine the policy of Eurotransplant for ideas on improving the current U.S. system. We then construct a model of list paired donation to determine how to maximize the number of exchanges while maintaining compatibility. Finally, we analyze the implications of our model for patient and donor decisions, taking note of important ethical and political issues. Generic U.S. Transplant network Overview We model the generic U.S. transplant network as a rooted tree(growing downward). The rootrepresents theentirenetwork, anditsimmediate children represent theregions. Eachnode represents somekind oforganization, whether an Organ Procurement Center, a state organization, or a interstate region. At nere is a patient wait list, the concatenation of the wait list of the node s children We approximate the network's functioning as a discrete-time process, in which each time step is one day with four phases In phase I, patients are added to the leaf nodes. We approximate the rate of wait list addition by a Poisson process; doing so is valid because we can reasonably assume that the arrivals areindependent, identically distributed, and approximately constant from year to year. Suppose that this number of candidates added to the wait list at time t is addition, then we model tont+ ont =R)=

Optimizing the Effectiveness of Organ Allocation 119 Kidney transplants are common partly because kidneys, unlike most other organs, can be safely obtained from live donors. In fact, live-donor kidneys are more effective than cadaveric kidneys, with longer half-lives and lower rejec￾tion rates [Gentry et al. 2005]. Over 75% of living donors in 2004 were related (parents, siblings, spouses) to the transplant recipients [Childress and Liverman 2006]. However, some people willing to donate to an intended recipient can￾not because of blood type or HLA incompatibility, leaving over 30% of patients without a suitable kidney transplant [Segev et al. 2005]. One solution is kidney paired donation (KPD), which matches' two incompatible donor-recipient pairs where the donor of each pair is compatible with the recipient of the other, sat￾isfying both parties [Ross et al.. Another is list paired donation (LPD), where a recipient receives higher priority on the waitlist if an associated donor gives to another compatible recipient on the waitlist [Gentry et al. 2005]. We incorporate all these factors in modeling the various aspects of trans￾plantation. First, we focus on the U.S. network and produce a generic model of the processes that impact the number of people on the waitlist, the number of transplants, and the length of.wait time. To illustrate our model, We use data specific to kidney transplants, and also examine the policy of Eurotransplant for ideas on improving the current U.S. system. We then construct a model of list paired donation to determine how to maximize the number of exchanges while maintaining compatibility. Finally, we analyze the implications of our model for patient and donor decisions, taking note of important ethical and political issues. Generic U.S. Transplant Network Overview We model the generic U.S. transplant network as a rooted tree (growing downward). The root represents the entire network, and its immediate children represent the regions. Each.node represents some kind of organization, whether an Organ Procurement Center, a state organization, or a interstate region. At each node, there is a patient wait list, the concatenation of the wait list of the node's children. We approximate the network's functioning as a discrete-time process, in which each time step is one day with four phases: o In phase I, patients are added to the leaf nodes. We approximate the rate of wait list addition by a Poisson process; doing so is valid because we can reasonably assume that the arrivals are independent, identically distributed, and approximately constant from year to year. Suppose that this number of candidates added to the wait list at time t is additiont, then we model n a ik Pr(additiont+1 - additiont = k) = ek!

120 The UMAP Journal 28.2 (2007) For the rate constant A, we use the number of organ applicants in a given year,AA(number of new applicants)/365.25. o In phase I, we add cadaver organs to the leaf nodes. As with patients,we model cadaver arrivals as a Poisson process, with rate the average number of cadaver organs added in a given year. o In phase Ill, we allocate organs based on bottom-up priority rules. A bottom- up priority rule is a recursive allocation process propagated up from the bottom of the tree, which requires any organ-patient match to meet some minimum priority standard. For example, for kidney allocation, the first priority rule is to allocate kidneys to patients who match the blood type and HLa profile exactly. within this restriction, oPtN dictates that kidneys be allocated locally first, then regionally, then nationally. In our model, this corresponds to moving from the- leaves up the tree Matched organ patient pairs undergo transplantation, which has a success rate dependent on the quality of the match. (In later sections, we explore the success, rate as also a function of the experience of the doctors at the center and the quality of the kidney. o In phase iv, we simulate the death of patients on the waiting list. We treat the death rate k of a patient as a linear function aT+b of the persons wait timeT.Hence,calculating from time 0, a persons chance of survival to time Tis e-AT=e-(aT'+b)T Under this mathematical model, our problem becomes finding a good tree structure and an appropriate set of bottom-up priority rules Simulation To study this model, we average results sover na ny simulations of kidney transplant network. Our simulation works as follows: At every time round, in phase l, we generate a number according to the Poisson distribution of the numberofnew candidates. Foreach new patient added, we randomly generate the person's race and age according to data on race and age distributions Usingthe srace, we generate the person'sblood type and HLA makeup, according to known distributions, and the patient's PRA, based on probabilities published by the OPtN Similarly, in Phase I, we generate a list of donor organs according to known distributions of blood type and HLA makeup. Moreover, we record where the rgan was generated, so we can study the effect of having to move the organ before transplantation, the time for which lowers its quality In Phase Ill, we implement recursive routines that traverse the tree from he bottom up, following the OPTN system for kidneys. To model the success rate of an operation, we use the statistics published by the OPTN; our main method of determining whether an operation is successful is the number of

120 The UMAP Journal 28.2 (2007) For the rate constant A, we use the number of organ applicants in a given year, A : (number of new applicants)/365.25. "In phase II, we add cadaver organs to the leaf nodes. As with patients, we model cadaver arrivals as a Poisson process, with rate the average number of cadaver organs added in a given year. " In phase RiL, we allocate organs based on bottom-up prioritj rules. A bottom￾up priority rule is a recursive allocation process propagated up from the bottom of the tree, which requires any organ-patient match to meet some minimum priority standard. For example, for kidney allocation, the first priority rule is to allocate kidneys to patients who match the blood type and HLA profile exactly. Within this restriction, OPTN dictates that kidneys be allocated locally first, then regionally, then nationally. In our model, this corresponds to moving from the-leaves up the tree. Matched organ-patient pairs undergo transplantation, which has a success rate dependent on the quality of the match. (In later sections, we explore the success.rate as also a function of the experience of the doctors at the center and the quality of the kidney.) " In phase IV, we simulate the death of patients on the waiting list. We treat the death rate k of a patient as a linear function aT + b of the person's wait time T. Hence, calculating from time 0, a person's chance of survival to time T is e-kT = e-(aT+b)T. Under this mathematical model, our problem becomes finding a good tree structure and an appropriate set of bottom-up priority rules. Simulation To study this model, we average results over many simulations of the kidney transplant network. Odr simulation works as follows: At every time round, in phase I, we generate a number according to the Poisson distribution of the number of new candidates. For each new patient added, we randomly generate the person's race and age according to data on race and age distributions. Using the person's race, we generate the person's blood type and I-ILA makeup, according to known distributions, and the patient's PRA, based on probabilities published by the OPTN. Similarly, in Phase HI, we generate a list of donor organs according to known distributions of blood type and HLA makeup. Moreover, we record where the organ was generated, so we can study the effect of having to move the organ before transplantation, the time for which lowers its quality. In Phase 1II, we implement recursive routines that traverse the tree from the bottom up, following the OPTN system for kidneys. To model the success rate of an operation, we use the statistics published by the OPTN; our main method of determining whether an operation is successful is the number of

Optimizing thre effectiveness of Organ Allocation 121 HLA mismatches. Success is affected by the sensitivity of the person, measured by the persons PRA, which we model by. adding a linear term to the success rate. Moreover, we reduce the success rate by 5 percentage points if the orga is not procured from the same center as the patient; 5% is the average effect on the success rate of increasing the delay by 10-20 hrs, according to optn data. In Phase IV, we regress the coefficients a and b of the previous section, and use this formula to calculate the probability of death. To adjust the parameters for this model, we use the OPtN national data for the national active wait list for cadaver kidneys from 1995 to 2004 and feedinto our model the number of donations for each year. Results of the basic model To quantify the quality of a network, we use: a set of objective functions, which represent various ideas about the desirability of policy outcomes. For these functions, let a be the number of"healthy"patients to receive a successful transplant each y be the corresponding number of"sick"patients(those with some terminal illness or serious medical condition) to receive a successful transplant, be the average age of the transplant recipients, and m be the maximum wait time in the queue. We examine the following objective functions a+y: This is simply the number of successful transplants per year. (100-a) x(+y): This considers the premise that transplants are more valu- able when given to young recipient a+0.5 y: This is a stylized adoption of the idea that transplants given to ter- minally ill recipients are less valuable (=+y)/max(9, m): This incorporates queue wait time. We also include a proposed tradeoff between big and small centers In a big center, the doctors are more experienced. We simulate this by de- creasing the success rate of operations at centers that do not perform a thresh- old number of operations per year. with small centers, kidneys are allocated on a more local basis, which mini- mizes deterioration of organs in transportation. We simulate this by apply ing a penalty when kidneys are moved to larger regional centers, and also when kidneys are moved between centers

Opthnizing the Effectiveness of Organ Allocation HLA mismatches. Success is affected by the sensitivity of theperson, measured by the person's PRA, which we model by adding a linear term to the success rate. Moreover, we reduce the success-rate'by. percentage points if the organ is not procured from the same center as'the patient; 5%,is the average effect on the success rate of increasing the delay by 10-20 hrs,,according to OPTN data. In Phase IV, we-regress,the coefficients a and b of the previous section, and use this formula to calculate'the probability of death. To adjust the parameters for this model, we use the OPTN national data for the national active wait listlfor cadaver kidneys from 1995 to 20041and feed into our model the number of donations for each year. Results of the Basic Model To quantify the quality of a network, We use aý set of objective functions, which represent various ideas about the desirability of policy outcomes. For these functions, let x be the number of "healthy" patients to receive 'a successful transplant each year, y be the corresponding number of'"sick" patients (those with some terminal illness or serious medical condition) to receive a successful transplant, a be the average age, of the transplant.recipients, and m be themaximum wait time in the queue.. We examine the-following objective functions: x + y : This is simply the number of successful transplants per year. (100 - a) x (x +-y) : This considers the premise thattransplants are more valu￾able when given to young recipients. x + 0.5 : This is a stylized adoption of the idea that transplants given to ter￾minally ill recipients are less valuable. (x + y)/ max(9, m) : This incorporates queue walt time. We also include a proposed tradeoff between big and small centers: . In a big center, the doctors are more experienced. We simulate this by de￾creasing the success rate of operations at centers that do not perform a thresh￾old number of operations per year. * With small centers, kidneys are allocated on a more local basis,, which mini￾nmizes deterioration of organs in transportation. We simulate this by apply￾ing a penalty when kidneys are moved to larger regional centers, and also when kidneys are moved between centers. 121

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 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 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 kid￾ney? 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 reason￾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 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 callfit￾ture 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 fu￾ture alternative and delays transplant, survival is governed by a default survival function Sd

Optimizing the effectiveness of organ Allocation 125 Summary of Assumptions The patient can choose either a transplant now(the immediate alternative Ao), or from a finite set of transplants(Al, A2,..., An )in the hypothetical future(the future alternatives) Each alternative has a corresponding survival function si(to, t), which de- scribes the chance of survival until time t as a function of t and transplant starting time to. Survival functions have value I at time 0, are continuous, and have limit zero at infinit Each future alternative Ai has a corresponding constant probability p: of becoming available at any given time. Hence, the probability at time t of the alternative not yet having become available is e-pit. a default survival function sa(t)defines the chance of survival when there has not yet been a transplant. To maintain continuity, sa(to)=si(to, t). The patient can have only one transplant. The patient attempts to maximize expected lifespan; in case of a tie in ex- pected values, the patient chooses an option that provides a kidney more Thesurvival functions mustbehave consistently; they cannotbecome wildly better or worse-performing relative to each other. We propose a formal defini- tion to capture this concept A separable survival function si(to, t)is one that can be expressed as the product of two functions, one a function of only to and the other a function i(to, t)=ai(to)bi (t-to) We stipulate that b(0)= 1. In a separable set of survival functions, all functions are individually separable with the same function a(to) Is it be reasonable to assume that forany patient, the set of survival functions is separable? It is not an entirely natural condition, and indeed there are cases where it does not seem quite right-for instance, when some to is high, so that higher values of t approach extreme old age, where survival decreases rapidly and the patient is less likely to survive than the product of a and b predicts But in this case, the absolute error is small anyway: a(to)accounts for the probability of survival that stems from waiting for a kidney until time to, and thus if to is large, a(to)b(t-to) is likely to be quite tiny as well Moreover, separability is intuitively reasonable for modeling the effects of a delayed kidney donation. The function a(to)measures the decrease in survival rate that results from waiting for an organ transplant. This should be consistent across all survival functions for a given patient; we express this notion in the

Optimizing the Effectiveness of Organ Allocation Summary of Assumptions "* The patient can choose either a transplant now (the immediate alternative A0 ), or from a finite set of transplants (A1, A2,... ,A,) in the hypothetical future (the future alternatives). "* Each alternative has a corresponding survival function si(to, t), which de￾scribes the chance of survival until time t as a function of t and transplant starting time to. Survival functions,have value 1 at time 0, are continuous, and have limit zero at infinity. * Each future alternative Ai has a corresponding constant probability pi of becoming available at any given time. Hence, the probability at time t of the alternative not yet having become available is e-Pit. * A default survival function sd(t) defines the chance of survival when there has not yet been a transplant. To maintain continuity, sd(to) = si (to, t). * The patient can have only one transplant. * The patient attempts to maximize expected lifespan; in case of a tie in ex￾pected values, the patient chooses an option that provides a kidney more quickly. The survival functions mustbehave consistently; they cannotbecome wildly better or worse-performing relative to each other. We propose a formal defini￾tion to capture this concept. A separable survival function si(to, t) is one thlat can be expressed as the product of two functions, one a function of only to and the other a function of only t - to: si(to,t) = ai(to)bi(t - to). We stipulate that b(O) ' 1. In a separable set of survival functions, all functions are individually separable with the same function a(to). Is itbe reasonable to assume that for any patient, the set of survival functions is separable? It is not an entirely natural condition, and indeed there are cases where it does not seem quite right-for instance, when some to is high, so that higher values of t approach extreme old age, where survival decreases rapidly and the patient is less likely to survive than the product of a and b predicts. But in this case, the absolute error is small anyway: a(to) accounts for the probability of survival that stems from waiting for a kidney until time to, and thus if to is'large, a(to)b(t 7 to) is likely to be quite tiny as well. Moreover, separability is intuitively reasonable for modeling the effects of a delayed kidney donation. The function a(to) measures the decrease in survival rate that results from waiting for an organ transplant. This should be consistent across all survival functions for a given patient; we express this notion in the 125

126 The UMAP Journal 28. 2(2007) concept of a separable set. Meanwhile, the factor b(t- to)accounts for the decrease in survival during the time(t-to )spent with the new kidney. Consequently, we assume that our survival functions are separable. This will lead us to an explicit heuristic for lifespan-maximizing decisions, which is the goal of this section. For A; and A, two future alternatives in a separable set, we assign an order accor Ing b()≤/b()d←4≤4 We turn to the derivation of an lifespan-maximizing strategy Such a strat- egy, when presented with alternative Ai at time to, will either accept or wait for other alternatives. In fact: Theorem. If a patient's alternatives form a separable set, then the optimal strategy is either to accept an alternative Ai at all times to or to decline it at all times to. If the patient declines Ai, then the patient must decline all alternatives less than or equal to Ai in the order relation defined above. Similarly, if the patient accepts Aj, then the patient must accept all alternatives greater than or equal to A i Proof: The patient will accept the alternative or probabilistic bundle of alterna tives that the patient' s survival functions indicate gives the greatest lifespan For alternative Ai, the expected lifespan beyond time to is 8i(to, t)dt Suppose that a patient at time 0 declines this alternativein favor of some optimal set of future alternatives. Furthermore, suppose that this set includes some alternative Ak such that Ak S Ai. Then the expected lifespan from this set is (+)(+叫)间 p b(t)+pb (t) dt dto ∑+pk j ranges over all alternatives Ai in the optimalsetexcept A. This double of hartal does not mix integration variables and is therefore equal to a product (+叫)厂(+叫)间-厂 ∑P()+pkb() ∑+Pk Since Ak is less than or equal to Ai, and Ai was declined in favor of the set of alternatives that we are examining, the presence of the h term in the weighted verage under therightintegrand lowers the value of the average. The previous expression is thus less than (2+)(+时间.2购a

126 The UMAP Journal 28.2 (2007) concept of a separable set. Meanwhile, the factor b(t - to) accounts for the decrease in survival during the time (t - to) spent with the new kidney. Consequently, we assume that our survival functions are separable. This will lead us to an explicit heuristic for lifespan-maximizing decisions, which is the goal of this section. For .A and A•4 two future alternatives in a separable set, we assign an order according to: 1 0000 0 O bi(t) dt ~pj bj(t) + pk.blk(t)dt fo Ej P1