10.1177/0272989X04273137MEDICAL DECISION MAKING/MNTH–MNTH XXXXSTAHL AND OTHERSORIGINAL ARTICLESOPTIMALLY REORGANIZING LIVER TRANSPLANT REGIONSJAN–FEB

A Methodological Framework for OptimallyReorganizing Liver Transplant Regions

James E. Stahl, MD, CM, MPH, Nan Kong, MS, Steven M. Shechter, MS,Andrew J. Schaefer, PhD, Mark S. Roberts, MD, MPP

Background. The United States is divided currently into 11transplant regions, which vary in area and number of organprocurement organizations (OPOs). Region size affects organtravel time and organ viability at transplant. Purpose. To de-velop a methodologic framework for determining optimalconfigurations of regions maximizing transplant allocationefficiency and geographic parity. Methods. An integer pro-gram was designed to maximize a weighted combination of 2objectives: 1) intraregional transplants, 2) geographicparity—maximizing the lowest intraregional transplant rateacross all OPOs. Two classes of functions relating liver travel

time to liver viability were also examined as part of the sensi-tivity analyses. Results. Preliminary results indicate that re-organizing regions, while constraining their number to 11, re-sulted in up to 17 additional transplants/year depending onthe travel-viability function; when not constrained, it resultedin up to 18/year of increase. Conclusion. Our analysis indi-cates that liver transplantation may benefit through regionreorganization. The analytic method developed here shouldbe applicable to other organs and sets of organs. Key words:transplant regions; organ procurement organizations; livertransplants. (Med Decis Making 2005;25:35–46)

End-stage liver disease, that is, chronic liver diseaseand cirrhosis, is the 10th leading cause of death

among adults in the United States, accounting for morethan 25,000 deaths in 1998 alone.1 At present, the onlytherapy available is liver transplantation. Fortunately,patients at almost any stage of their disease receiving aliver transplant can expect an 80% to 90% 5-year sur-vival rates.2,3 Unfortunately, liver transplantation isboth costly and limited by the supply of viable donororgans. The acute hospitalization alone has been esti-mated between $145,000 and $287,000.4–9

The number of patients registered with the UnitedNetwork for Organ Sharing (UNOS) for liver transplan-tation has increased over 12-fold in the past decade tomore than 16,000 in 2002. During this same period, thepool of donated organs only doubled, and supply hasnot been able to keep pace with demand10 (www.

UNOS.org). In 2002, even though more than 5300 peo-ple received a liver transplant, nearly 7000 died whilewaiting for one. It should be noted, however, that eventhough the registration rate for liver transplant is risingfaster than the transplant rate, fewer that 2000 of thoseregistered died from causes directly attributed to theirend-stage liver disease. Fewer than 400 of these deathswere patients who were critically ill. Currently, the do-nation rate and transplant rate are closely matched, andthe deaths per year has leveled off (see Figure 1).

Surprisingly, many donated livers are never actuallytransplanted. Between 1988 and 1998, more than 3600livers were lost to transplant because of poor matching,refusal by transplant centers, or allocation delays, re-sulting in time-related loss of organ viability. On aver-age, procured livers remain viable for only 12 to 18 honce harvested. Therefore, reducing the time needed tomatch and transport livers should increase the numberof viable livers available for transplantation. Thisshould then improve the net survival in patientssuffering from end-stage liver disease.

Because of the severe shortage of livers and theirfleeting nature, both allocation procedure and policy

MEDICAL DECISION MAKING/JAN–FEB 2005 35

Received 26 February 2003 from the Department of Radiology & De-partment of Medicine, Massachusetts General Hospital, Harvard Med-ical School, Boston (JES); Department of Industrial Engineering, Uni-versity of Pittsburgh, Pennsylvania (NK, SMS, AJS, MSR); Departmentof Medicine, University of Pittsburgh Medical Center, Pennsylvania(AJS, MSR). Supported by grant DMI-0223084 from the National Sci-ence Foundation and partially by grants R01-HS09694 and1K08HS011637-01A1 from the National Institutes of Health. Revisionaccepted for publication 27 September 2004.

DOI: 10.1177/0272989X04273137

Address correspondence and reprint requests to James E. Stahl, MD,CM, MPH, Institute for Technology Assessment, MassachusettsGeneral Hospital, 101 Merimac St, 10th floor, Boston, MA 02114; e-mail: James@mgh-ita.org.

are critically important for the welfare of end-stageliver disease patients. Livers, as well as other types oforgans, are currently allocated through a hierarchicalgeographic allocation system with 3 levels: Organ pro-curement organization (OPO) level, regional level, andnational level. There are approximately 300 hospitalsthat perform transplantation. These are supplied by 59OPOs organized into 11 regions (Figure 2). Specifically,a liver is first offered to the most critical patients (status1)—first at the OPO level and then at the regional level.If no match is found, then the liver is offered to the re-maining patients in the local OPO level followed by thesame region, in descending order of urgency. Finally, ifstill no match is found, the liver is offered to the mosturgent patient in the rest of the country (Figure 3).

The organ shortage also makes the control and eq-uity of access to organs critical.11 Since the late 1990s, a

vigorous debate has existed between the federal gov-ernment, which has advocated for a national allocationsystem, and states such as Louisiana, Wisconsin,Texas, Arizona, Oklahoma, Tennessee, and South Caro-lina, which have either sued the government12 or intro-duced legislation requiring organs procured within astate to be used within the state.13 Since 1989, the per-centage of procured organs used locally has increasedfrom 31.5% to more than 65%, whereas the number oforgans shared across procurement areas has decreasedfrom 59% to 25%10 (www.UNOS.org).

The size of the region affects the distance organsmust travel and thus the cold-ischemia time (CIT) ofthe procured organ, which in turn affects the viabilityof the organ at transplant. Consequently, region size po-tentially affects the number of successful transplants.The current size and configuration of the transplant re-gions do not explicitly consider organ type, the dura-tion of organ viability, and intermediate technologiessuch as dialysis in their design, but rather were devel-oped for administrative purposes (www.UNOS.org).The regions currently encompass varying numbers ofOPOs (median = 5, range = 2 to 11), from which theorgans are supplied.

A regional structure that optimizes the tradeoff be-tween CIT and administrative demands may maximizeintraregional matches, help alleviate demand, and re-duce the risk of losing organs. Currently, an organ maybe offered to many patients, with each step consumingadministrative and logistical time. Because their via-bility decreases over time, unless regions are carefullydesigned, patients may lose potentially life-saving or-gans. Larger regions can share more organs, within re-gion, which increases the chance of having an intra-regional match, but increased size may result in moretravel time. On the other hand, smaller regions requireless travel time but have smaller donor pools, makingan intraregional match less likely. This increases thelikelihood that an extraregional match must be sought,which costs administrative and logistical time.

The purpose of this research is to provide a basicmodeling framework for improving organ allocationand optimizing transplant allocation efficiency andgeographic equity, focusing strictly on region design.

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0

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1995 1996 1997 1998 1999 2000 2001 2002 2003

Year

Cadaveric donations Transplants Died on waiting list

Figure 1 Number of transplants, donations, and deaths by year.

67

5 8

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Figure 2 Current system of regions.

Local Status

Regional

Local Other

Regional Other

National

National Other

Figure 3 Current allocation scheme.

This initial modeling framework focuses on 2 objec-tives. The 1st objective is to maximize the total numberof intraregional transplants. This measure serves as aproxy for transplant allocation efficiency. The 2nd ob-jective is to maximize the minimum OPO intraregionaltransplant rate. This rate is defined as the number ofintraregional transplants in an OPO divided by thenumber of patients on the OPO waiting list. This rateserves as a proxy measure for geographic equity.

To explain our concept of geographic equity further,we assume that improving geographic equity, that is,making patient access to organs as equal as possibleacross all OPOs, can be accomplished by maximizingthe intraregional transplantation rate of the OPO withthe lowest intraregional rate. Specifically, across allOPOs, improving the lowest ratio of intraregionaltransplants in an OPO to the number of patients on thewaiting lists of that OPO will result in a more equita-ble distribution of organs. In other words, if 2 regiondesigns produce equal numbers of successful intra-regional transplants, but 1 produces a higher minimumintraregional transplant rate across OPOs, that designwould be preferred.

Therefore, this preliminary analysis has 2 importantobjectives: 1) maximize total intraregional transplantsnationwide and 2) provide equitable access to trans-plants across the country, avoiding significant dispari-ties. Hence, we do not necessarily want to optimize to-tal intraregional transplants alone. At the same time,we do not want to penalize regions for high transplantallocation efficiency for the sake of equity, either.Therefore, another objective is to address the tradeoffbetween total intraregional transplants and the lowestOPO intraregional transplant rate across all OPOs.

METHODS

We considered 2 strategies for reorganizing the ex-isting OPOs into transplant regions. First, we con-strained the number of transplant regions to 11, thesame number as in the current system. This strategy ex-amined the hypothesis that the current size and config-uration of regions were already optimal with regard toeither of our objectives. Second, we relaxed this con-straint and allowed the number of regions to vary. Thisexamined the hypothesis that the optimal number ofregions should be 11.

For the purposes of these basic analyses, we mod-eled the country as a simple network (Figure 4). In thismodel, OPOs are represented as the nodes of a network.Arcs connecting OPOs indicate that they are contigu-ous. We made several further simplifying assumptions.First, we defined a hypothetical region as a set of no

more than 9 contiguous OPOs. This limit was chosenbecause of computational constraints—the time tocomplete an analytic run and the needed computermemory rises very rapidly as the maximum number ofOPOs allowed within a region increases. Second, anOPO was only allowed to supply livers to the regioncontaining it. Third, each OPO was assumed to haveonly 1 transplant center. This simplified the estimationof the distance an organ would need to travel, thoughmultiple transplant centers could be easily incorpo-rated in the model. We also assumed that there were 59OPOs as being representative of the period of time fromwhich our data were derived (1993–2000). Finally, forthis analysis, we assumed that both transplant alloca-tion efficiency and geographic equity could be rep-resented as factors in a function linking CIT and livertransport distance (LTD).

Livers may be rejected and thus lost based on a com-bination of objective and subjective criteria used bytransplant surgeons at the recipient institution, for ex-ample, poor quality, which results in a high risk of pri-mary nonfunction. Because UNOS has limited dataavailable on the reasons a surgeon or transplant centerrejects a liver, we made the simplifying assumptionthat the probability of rejecting a liver, measured by theliver’s viability, was solely dependent on its CIT. Be-cause the probability of primary nonfunction is posi-tively correlated with CIT and CIT is positively corre-lated with LTD, the distance a liver must travel affects

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OPTIMALLY REORGANIZING LIVER TRANSPLANT REGIONS

The set partitioning matrix for this

Regions OPO 1 2 3 4 5 6 7 8 9 10 11 12

1 1 0 0 0 1 1 0 0 1 1 0 1 2 0 1 0 0 1 0 1 0 1 0 1 1 3 0 0 1 0 0 1 1 1 1 1 1 1 4 0 0 0 1 0 0 0 1 0 1 1 1

1

2

3 4

OPO

Contiguous relationship

Region

Figure 4 Example of simple network and region matrix. 1 = organprocurement organization (OPO) included in the region; 0 = OPO notincluded in the region. For example, hypothetical region 1 only in-cludes OPO 1, hypothetical region 5 encompasses OPOs 1 and 2, andhypothetical region 9 encompasses OPOs 1, 2, and 3. OPOs may onlybelong to 1 region in any given partitioning scheme. However, theymay be adjacent to OPOs in other regions.

its viability and the probability that it is rejected, andthis in turn is affected ultimately by the size and config-uration of regions.14 We also assumed that any liver of-fered to a transplant center that was not rejected wastransplanted. Therefore, the size and configuration oftransplant regions affect both factors considered in thisstudy that are influenced by liver viability. Because thefunctional relationship of liver viability to CIT is notcompletely known, we tested 2 different functional re-lationships between primary nonfunction and CIT: lin-ear and polynomial (Figure 5). These 2 functions werefitted from data obtained through a meta-analytic re-view of the literature15–45 (search terms: livertransplant, cold-ischemia time, graft dysfunction,patient and graft survival, etc.; Medline, 1966 topresent).

These functional forms were believed, by expertopinion, to be clinically reasonable and would boundthe true function, modeling the relationship betweenliver viability and CIT. For the relationship betweenCIT and LTD, we used CIT (in hours) = 9.895 + 0.003 ×LTD (in miles).14 Geographic data, such as the numberand current configuration of regions, OPOs and trans-plant centers, were used from publicly availablesources.10,11

The next step in the analysis was to identify all pos-sible regional configurations. A depth-first search(DFS) algorithm46 was performed to identify all hypo-thetical regions where each individual region com-prised 9 or fewer contiguous OPOs (see appendix).Given 59 existing OPOs, there are more than 311,000hypothetical regions that could be constructed, with

between 1 and 9 OPOs per region. For any OPO withina given region, the number of transplants contributingto the intraregional transplant traffic was the number oflivers offered to that OPO from other OPOs within theregion minus those lost in transport. For any given re-gion, the number of intraregional transplants was thesum of the inter-OPO transplants within that region(see appendix). This estimation was done for all hypo-thetical regions. The optimization problem, however,is even more complex. There are billions of possibleways to combine these regions into potential solutionsthat span the entire United States. Two sets of integerprograms47 were solved over these regions to find theoptimal transplant regional configuration with respectto the proposed objectives in our experiments. Integerprograms are standard models in operations researchfor making decisions over discrete variables.48 Integerprogramming is particularly useful when trying tomodel packing, partitioning, logistical, or travel prob-lems when there are a finite number of options tochoose from. Our analysis is, in essence, a partitioningproblem, but the method may be applied in other areas.For example, integer programming has been usefullyapplied in policy areas as diverse as political redistrict-ing,49 crew scheduling,50,51 EMS location,52 optimizingsurgical scheduling,53 and network design.54

In our analysis, we first examined the effect of maxi-mizing transplant allocation efficiency on regionalconfiguration. We solved an integer program to find theoptimal set of regions such that the total number ofintraregional transplants was maximized (see appen-dix). Next we considered the effect of geographic eq-uity on transplant allocation efficiency. As discussedabove, this meant for this analysis, the integer programobjective function was maximizing the intraregionaltransplant rate of the worst-off region, that is, the onewith the lowest rate. The intraregional transplant ratefor an OPO, denoted by λ, was defined as the number ofintraregional transplants preserved in an OPO dividedby the number of patients in the OPO waiting for trans-plants. We denoted λmin to be the intraregional trans-plant rate of the worst-off OPO, that is, the minimum λacross all OPOs. Therefore, the objective of the geo-graphic equity analysis is to maximize λmin acrossOPOs.

There is a tradeoff between transplant allocation ef-ficiency and geographic equity. Improving one oftencomes at the expense of the other. To quantify thistradeoff, we defined a new constant ρ and thensearched for a set of regions that maximizes the numberof intraregional transplants plus ρ, multiplied by theminimum intraregional transplant rate across OPOs,that is, the number of intraregional transplants + ρ •

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Figure 5 Primary nonfunction (PNF) v. cold-ischemia time(CIT)—Functions modeling the relationship between the probability of liverprimary nonfunction and the cold ischemia time. PNF1 = .905 +.433*CIT; PNF2 = –1.5545 + 1.17799*CIT – 0.03451*CIT^2 +0004*CIT^3.

λmin (see appendix). The value assigned to ρ is based onhow much importance we place on the minimumtransplant rate across OPOs versus intraregional trans-plants (higher values of ρ mean more weight is given toλmin). Hence, ρ provides a mathematical means of bal-ancing the 2 conflicting factors, transplant allocationefficiency and geographic equity. ρ may be considereda function of the total interregional waiting list, andany changes in ρ represent changes in the efficiency ofthe waiting list. For any value of ρ, a decision makerwould be indifferent between increasing the total intra-regional transplants by 1 and increasing the intra-regional transplant rate in the worst OPO by 1/ρ.

We denoted ρc to be the number of intraregionaltransplants divided by the minimum λ with respect tothe current regional configuration. Then with the valueof ρc, a decision maker would be indifferent betweenincreasing the total intraregional transplants by 1 andincreasing the minimum λ by 1/ρc. We chose ρ = 0 andρc in the analysis. In fact, the case ρ = 0 is equivalent tothe 1st experiment in which we did not incorporategeographic equity.

RESULTS

Results without theConsideration of Geographic Equity

Table 1 displays the preliminary results for bothcases in which the number of regions in the optimalconfiguration was fixed at 11 and when the numberwas unrestricted. In the 2nd column of the table, theintraregional transplants increase per year is presentedfor the former case. The intraregional transplants in-crease per year, and the number of regions in the opti-mal regional configuration is presented in the 3rd and4th columns of the table, respectively, for the lattercase.

It appears that regardless of the liver viability func-tion chosen (linear or polynomial), the optimal re-

gional configuration obtained from our model resultedin more intraregional transplants. It should be notedthat the regions chosen by our search strategy were notthe same regions that currently exist (see Figure 6). Re-laxing the constraint on the number of regions slightlyincreased the number of intraregional transplants overthe scenario where the number of regions was fixedto 11.

Results with theConsideration of Geographic Equity

Table 2 shows the preliminary results for 2 cases in-corporating the factor modeling geographic equity.Again, the 1st case is when the number of regions in theoptimal configuration was restricted; the 2nd is whenthe number was unrestricted. For the 1st case, the 2ndcolumn presents the primary outcome—theintraregional transplants increase per year. For the 2ndcase, the 4th and 5th columns present the primary out-come and the number of regions in the optimal regionalconfiguration. In addition, we report the minimumintraregional transplant rate across OPOs for bothcases, shown in the 3rd and 6th columns, respectively.With regard to the primary outcome, similar conclu-sions were drawn as in the cases where geographic eq-uity was not considered. It should be noted that theminimum intraregional transplant rate across OPOswas significantly higher than that in the current re-gional configuration given the ρ value.

Comparing the 2 sets of experiments reflected in Ta-bles 1 and 2 and Figures 7–9, we observed that the in-crease in intraregional transplants over the current

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OPTIMALLY REORGANIZING LIVER TRANSPLANT REGIONS

Table 1 Base Case Primary Outcome Results(without the consideration of geographic equity)

Number of Number ofRegions Fixed Regions Not Fixed

Number of Number of Number ofPNF Function Increase Increase Regions

Linear 17.1/year 17.9/year 22Polynomial 17.2/year 18.4/year 23

Note: PNF = probability of liver primary nonfunction.

Figure 6 Example of alternative region configuration. Solid linesoutline current regions. Shaded areas denote optimized regions withlinear organ decay function, 11-region constraint without consider-ation of geographic parity.

configuration diminished as the value of ρ increased.However, the minimum intraregional transplant rateacross OPOs increased as the value of ρ increased. Thisobservation reflects the tradeoff between transplantallocation efficiency and geographic equity. As theweight assigned to the equity increases, equity be-comes a more important issue for region reorganizationas opposed to efficiency.

Sensitivity Analyses

Because the liver viability is a key parameter of ourmodel, we conducted sensitivity analyses in which we

varied the underlying function modeling the relation-ship between CIT and primary nonfunction and inwhich we systematically varied the parameters in themodels within ±2 standard errors (see Table 3). Thesensitivity analyses indicated that the primary out-come was not very sensitive to the varying underlyingrelationship between CIT and primary nonfunction, re-gardless of whether the number of regions in the opti-mal configuration was restricted to 11 (see Figure 7).We also observed that the optimal regional configura-tion was nearly identical in most of the sensitivity anal-yses to the base case optimal configuration, though notthe same as the current configuration.

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Table 2 Base Case Primary Outcome Results (with consideration for geographic equity)

Number of Regions Fixed Number of Regions Not Fixed

PNF Function Number of Increase Minimum λ Number of Increase Number of Regions Minimum λ

Linear 13.0/year 0.345 13.0/year 10 0.345Polynomial 12.2/year 0.337 12.2/year 10 0.337

Note: OPOs = organ procurement organizations; PNF = probability of liver primary nonfunction. For the consideration of geographic equity, the minimumintraregional transplant rate across OPOs (minimum λ) in the current regional configuration is 0.060 for the linear PNF function and is 0.059 for the polynomialPNF function; the weighting parameter ρ was assigned to be ρc, and thus its value is 154,160 for the linear PNF function and is 153,978 for the polynomial PNFfunction.

Figure 7 Additional transplants per year with reconfigured regions. * may be considered functionally related to the total national waiting list.

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OPTIMALLY REORGANIZING LIVER TRANSPLANT REGIONS

Figure 9 Optimal number of regions v. . * may be considered functionally related to the total national waiting list.

Figure 8 Minimum intraregional transplant rate v. . * may be considered functionally related to the total national waiting list. * minimumintraregional transplants/intraregional waiting list size.

DISCUSSION

Although our model is necessarily a simplificationof reality, this framework gives us a means for estimat-ing the effect of various strategies for organizing OPOsinto regions. In this model, the optimal sets of regionstend to group densely populated areas. For example,unlike the current system, in each of the optimal sets,the New York metropolitan area and New Jersey appearin the same region. This is due to the fact that the bene-fit of grouping denser populated regions, that is, the re-sulting increase of intraregional transplants in thedenser populated regions, tends to outweigh any lossesfrom having larger, less densely populated regions. Inaddition, the optimal transplant regions based on therelationship between liver viability and transport dis-tance modeled in our preliminary analysis appear to bedifferent from the current system of regions regardlessof which relationship was assumed, and they appear toimprove the number of intraregional transplants. Fi-nally, incorporating considerations of geographic eq-uity may reduce the total number of intraregionaltransplants and vice versa. This implies that regula-tions aimed at transplant allocation efficiency (i.e., in-creasing the total transplants by restricting their trans-port) and those aimed at equity should be examinedcarefully before being considered for implementation.In the future, we will make refinements to this modelby including more accurate and comprehensivedemographic data.

The main purpose of this study is to establish meth-ods and principles that can be applied to a more de-tailed and wide-ranging study of region design. There-fore, we only focused here on liver transplantation.However, we believe the optimal regional configura-tions produced from this framework may differ by or-gan type, owing to the availability and viability of theorgan. With this framework, it may be feasible to gener-ate an optimal regional configuration for each organtype. It may also be feasible to generate an overall opti-

mal regional configuration based on the weight as-signed to each type of organ, considering the scarcity ofthe organ and the availability of any replacement ther-apy. For example, organs such as livers that are rela-tively rare and have no replacement therapy availablemay be weighted more than organs such as kidneys,which have an intermediate therapy in dialysis.

In our models, we only considered the uncertaintyon 1 factor, organ viability as a function of distance theorgan needs to travel. There are many other areas of un-certainty in our models for which we did not performsensitivity analyses. However, because the main pointof this study was to establish an analytic frameworkrather than a definitive answer, we did not elaborate onthe sensitivity analysis. Our models also do not cur-rently build in health care costs, which might have aneffect on the tradeoff between region size and the will-ingness to transplant organs across OPOs. We felt this areasonable omission in that the cost of administering aregion is negligible when compared to the cost of a sin-gle transplant. We also only considered liver transplan-tation within the United States. Although it is possiblethat organs might be transported across national bor-ders, this is very unusual. We therefore felt that thiswas reasonable to exclude. This being said, any appli-cation of this framework to a multinational situationsuch as the European Union would have to considerthis possibility.

One major limiting factor in our study was comput-ing capacity. We were unable to exactly solve problemscontaining some hypothetical regions consisting ofmore than 9 OPOs. This precluded comparisons to anyhypothetical configurations containing larger regions.We are currently exploring methods for approximatelysolving larger problems. However, considering largerregions will only improve our solutions, which alreadyseem to indicate that the existing configuration is notoptimal.

In the current hierarchical system employed byUNOS, there are local, or OPO-level transplants, re-

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Table 3 Second-Order Sensitivity Analysis (without the consideration of geographic equity)

PNF Function Base Case Coefficient (SD) Range of Increasea Range of Increaseb

Linear a0 = 0.905 (1.03) 16.7 ~ 17.4/year 17.6 ~ 18.3/yeara1 = 0.433 (0.05) 16.8 ~ 17.3/year 17.5 ~ 18.3/year

Polynomial a0 = –1.5545 (1.049) 16.9 ~ 17.6/year 18.0 ~ 18.7/yeara1 = 1.17799 (0.419) 16.3 ~ 22.0/year 16.6 ~ 22.0/yeara2 = –0.03451 (0.022) 16.8 ~ 25.9/year 16.9 ~ 26.0/yeara3 = 0.0004 (0.0003) 16.4 ~ 18.0/year 17.2 ~ 19.5/year

Note: Linear: PNF = .905 + .433*CIT; Polynomial: PNF = –1.5545 + 1.17799*CIT – 0.03451*CIT^2 + 0004*CIT^3.a. Analysis restricted to 11 transplant regions.b. Analysis not restricted to 11 transplant regions.

gional transplants, and national transplants. We as-sumed that the number of local transplants was unaf-fected by the regional configuration. However, giventhe current allocation scheme, the effect of region de-sign on the transplantation at the local level may not benegligible but is difficult to model. Although the frame-work provided in this article is able to maximize totalintraregional transplants, it is unable to consider thenumber of national transplants. This is important be-cause more intraregional transplants result in fewer na-tional transplants. These issues will be addressed infuture research.

In conclusion, liver transplantation provides a rela-tively simple example with which to test the validity ofthis modeling framework: 1) livers can remain outsidethe body for up to 18 h, 2) the CIT increases as the LTDincreases, and 3) the liver quality decreases as the CITincreases. In comparison, hearts and lungs must betransplanted almost immediately, and kidneys have amuch longer CIT and an intermediate technology in di-alysis to relieve the time pressure. The lessons learnedhere, however, should be applicable to other individ-ual organ types and the organ allocation system in gen-eral. This framework will also provide the means to es-timate the effect of new preservation and organ

replacement technologies that may alter the con-straints of the allocation system considerably. In addi-tion, even apparently modest gains resulting from reor-ganizing the regions may substantially change thetransplant environment. Annually, there are approxi-mately 1200 patients in the most urgent clinical catego-ries on the waiting list and less than 400 with a pro-jected life expectancy of less than 1 month. On anygiven day, there are typically less than 20 patients inthis most urgent category.10 This has been relatively sta-ble for the past 5 years. An additional 15 to 25 organs/year could save the lives of 5% or more of those in mosturgent need and shift the profile of the overall trans-plant waiting list toward the less ill and start reducingthe waiting list. However, because there will undoubt-edly be monetary and nonmonetary costs associatedwith a regional reorganization, which are as yet un-known, this preliminary analysis cannot yet answerwhether this is cost-effective relative to other means tobalance supply and demand, such as improvedprocurement, alternate technologies, or improvedpreservation techniques, but this framework gives usthe means to do so. This will be the focus of futurework.

GLOSSARY

Cold Ischemia Time (CIT): The time interval beginning whenan organ is cooled with a perfusion solution at the organprocurement surgery and ending when the organ is re-perfused at implantation.

Organ Procurement and Transplantation Network (OPTN):The unified transplant network established by the USCongress.

Organ Procurement Organization (OPO): OPOs serve asthe vital link between the donor and recipient and are re-sponsible for the identification of donors and the re-trieval, preservation, and transportation of organs fortransplantation.

Organ Transplant Region: The national UNOS membershipis divided into 11 geographic regions. This region struc-ture was developed to facilitate organ allocation andprovide individuals with the opportunity to identifyconcerns regarding organ procurement, allocation, andtransplantation.

United Network for Organ Sharing (UNOS): A private, non-profit organization to administer OPTN under contract

with Health and Human Services. It establishes an organ-sharing system that maximizes the efficient use of organsthrough fair and timely allocation.

Depth-First Search (DFS): Any search algorithm, which con-siders outgoing edges of a vertex before any neighbors ofthe vertex, that is, outgoing edges of the vertex’s predeces-sor in the search.

Integer Program: An optimization in which the decision vari-ables are required to be integer-valued.

Objective Function: The function to be optimized in an opti-mization problem.

Local Transplant Rate (λ): The rate that a patient receivesintraregional transplant in the perspective OPO.

Weighting Parameter (ρ): The coefficient that reflects thedecision-making tradeoff between transplant allocationefficiency and geographic equity.

Integer programming suggested reading: Integer Pro-gramming by LA Wolsey, Introduction to OperationsResearch by F Hillier.

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APPENDIXDetailed Model Description

Depth-first search to enumerate all region sets:

All regions composed of up to 9 contiguous organ procure-ment organizations (OPOs) were enumerated. To enumerateall possible solutions, that is, sets of regions, a depth-firstsearch (DFS) algorithm55 was run with the OPO network toidentify all hypothetical regions, in which each individualregion was composed of less than or equal to 9 contiguousOPOs. A DFS is a type of network search algorithm that exam-ines the state-space search tree recursively, in this case, thesearch tree constructed by sets of OPOs, by proceeding down1 branch of the tree until a solution state or a dead-end state(i.e., no more moves are possible from that state) is encoun-tered. When a dead-end state is encountered, the DFS algo-rithm then backs up to the last decision it made and makesanother choice of which direction to pursue downward in thesearch tree.

Variables:

For every possible region j, once enumerated, a variable xjis created. This variable may only take on the values 0 or 1.The value xj = 1 is interpreted as region j being chosen, andthe value xj = 0 is interpreted as region j not being chosen.

Objective function:

Our 1st objective was to find the set of regions that maxi-mizes the total number of intraregional transplants. Thechance that an individual patient is able to receive a liver of-fer from other OPOs within the same region was estimatedas the ratio of patient population in the patient’s OPO to thepatient population of the entire region, excluding the OPOwhere the liver is offered.

The coefficients in the objective function are describedbelow:

Patients(i) = the number of patients who register in anywaiting list in OPO i.

Organs(i) = the number of organs procured in OPO i andshared outside the OPO.

PNF(cit) = the probability of a liver’s primary nonfunctiongiven its cold ischemia time.

CIT(i,j) = the cold ischemia time of a liver that transportsfrom OPO i to OPO j.

Assume region r contains OPO j1, j2, . . . jm. Then the to-tal number of intraregional transplants for region r maybe estimated by

cPatients j

Patients r Patients jrk

ll kl

m

k

=−≠

==

∑ ( )( ) ( )

11

m

l k lOrgans j PNF CIT j j

∑

× × −( ) ( ( ( , ))),1

and the intraregional transplant rate in OPO jk, contained inregion r, may be estimated by

fPatients r Patients jkr

ll kl

m

=−≠

=

∑ 1

1

( ) ( )

× × −Organs j PNF CIT j jl k l( ) ( ( ( , ))).1

Formulation:

Let I and J be the sets of all OPOs and hypothetical regions,respectively. The matrix A is defined by aij = 1 if region j con-tains OPO i, and 0 otherwise. The number of rows of this ma-trix is the number of OPOs, and the number of columns of thismatrix is the number of hypothetical regions. For the cases inwhich the geographic equity is not considered, we solved thefollowing integer program:

Max c xj jj J∈∑

subject to a xij ijj J

=∈∑ 1, i in I.

xj in (1), j in J.

This model maximizes the number of intraregionaltransplants while ensuring that each OPO belongs toexactly 1 region. If the number of regions was requiredto remain at 11, the constraint xj

j J

=∈∑ 11 was added.

When the geographic equity is incorporated in the ob-jective, we solve the following problem. Here the ma-trix F is composed by fkr , and λmin is denoted as the min-imal local transplant rate.

Max c xj jj J∈∑ + ρλ min

subject to a xij ijj J

=∈∑ 1, i in I.

f xij ijj J

− ≥∈∑ λ min ,0 i in I.

xj in (1), j in J.

This model maximizes a weighted sum of the numberof intraregional transplants and the minimum intra-regional transplant rate across OPOs. If the number ofregions was required to remain at 11, the constraint

xjj J

=∈∑ 11was added.

44 MEDICAL DECISION MAKING/JAN–FEB 2005

STAHL AND OTHERS

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ORIGINAL ARTICLES 45

OPTIMALLY REORGANIZING LIVER TRANSPLANT REGIONS

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