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

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 importance depends on the type of organ involved. These include compatibility, region, 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 Procurement ... 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

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 rejection 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 cannot 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, satisfying 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 transplantation. 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 bottomup 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 valuable when given to young recipients. x + 0.5 : This is a stylized adoption of the idea that transplants given to terminally 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 decreasing the success rate of operations at centers that do not perform a threshold number of operations per year. * With small centers, kidneys are allocated on a more local basis,, which mininmizes deterioration of organs in transportation. We simulate this by applying a penalty when kidneys are moved to larger regional centers, and also when kidneys are moved between centers. 121