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Modeling the Allocation System: Principles for Robust Design Before Restructuring

Mehrotra, Sanjay1,2,3; Kilambi, Vikram1,2; Gilroy, Richard4; Ladner, Daniela P3; Klintmalm, Goran B5; Kaplan, Bruce4

doi: 10.1097/TP.0000000000000656
Commentary
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The United Network for Organ Sharing is poised to resolve geographic disparity in liver transplantation and promote allocation based on medical urgency. At the time of writing, United Network for Organ Sharing is considering redistricting the organ procurement and transplantation network so that patient model for end-stage liver disease scores at transplant is more uniform across regions.

We review the proposal with a systems-engineering focus and find that although the proposal is promising, it currently lacks evidence that it would perform effectively under realistic departures from its underlying data and assumptions. Moreover, we caution against prematurely focusing on redistricting as the only method to mitigate disparity. We describe system modeling principles which, if followed, will ensure that the redesigned allocation system is effective and efficient in achieving the intended goals.

Geographic disparities and liver allocation, simulation testing reality: This review describes the basic mathematics and computational hurdles in deriving a robust liver allocation system. Supplemental digital content is available in the text.

1 Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL.

2 Center for Engineering and Health, Institute for Public Health and Medicine, Northwestern University Feinberg School of Medicine, Chicago, IL.

3 Northwestern University Transplant Outcomes Research Collaborative (NUTORC), Comprehensive Transplant Center, Northwestern University Feinberg School of Medicine, Chicago, IL.

4 University of Kansas Medical Center, Kansas City, KS.

5 Baylor University, Annette C. and Harold C. Simmons Transplant Institute, Dallas, TX.

Received 26 November 2014. Revision requested 1 December 2014.

Accepted 12 December 2014.

The authors disclose no conflicts of interest.

Correspondence: Sanjay Mehrotra, PhD, Professor of Industrial Engineering and Management Sciences, 2145 Sheridan Road, Tech C246, Evanston, IL 60208.(mehrotra@.northwestern.edu).

Supplemental digital content (SDC) is available for this article. Direct URL citations appear in the printed text, and links to the digital files are provided in the HTML text of this article on the journal’s Web site (www.transplantjournal.com).

The United Network for Organ Sharing (UNOS) is contending with geographic disparities in deceased donor liver transplantation in the US. In accordance with the U.S. Department of Health’s Final Rule, the UNOS liver committee is considering proposals to ameliorate the problem on the basis of medical urgency by redistricting the current organ procurement and transplantation network (OPTN).1,2 The OPTN system currently divides the countries into 11 regions and further subdivides each region into 58 donation service areas (DSAs), where each has a corresponding organ procurement organization (OPO). Prioritization of liver distribution is based on medical urgency, which is measured in adult liver transplantation using the model for end-stage liver disease (MELD). With some notable exceptions, liver allocation generally proceeds by prioritizing patients with MELD greater than 35 in the region of the procuring DSA, next patients in the same DSA by MELD, thereafter, other patients in the region by MELD, and ultimately patients nationally. The plans under consideration by the liver committee involve reorganizing the UNOS regions into 4 to 8 contiguous districts that each contain at least 6 transplant centers and require on average no more than 4 hours of travel time from organs to various DSAs.

Understanding the OPTN requires a systems-engineering approach—and assessments of policy changes ought to anticipate the subsequent effects on system performance. Redistricting is a natural, but substantive change to the liver transplantation system and thus requires that the remapping of districts not only be based on sound methodology but also be robust in implementation as well as being in accordance with the strategic aims of the UNOS liver committee. Below, we briefly review the redistricting proposal and offer several comments.

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METHODOLOGY CURRENTLY BEING USED FOR REDISTRICTING

Analysis by Gentry et al3,4 and the references cited therein form the argument of the liver committee’s proposals. The methodology is centered on solving a special type of deterministic optimization problem known as an integer program (IP). A deterministic IP assumes that all model data are known precisely, that is, it has no uncertainty. The model decision variables are required to take integer values. A system of equations and an objective function is used to model the decision problem. Qualitatively, the IP used in Gentry et al minimizes the spread of MELD scores at transplant across regions, or across districts as described in the references. Districts are created by requiring that the IP assign each DSA to a unique district. The constraints of the IP require that the MELDs of the patient being offered the last organ procured in each district be as close as possible (the supply of organs/number of candidates in each DSA are treated as known parameters); average demand-weighted transport time be within a threshold (4 hours); and that districts be fixed in number (4‐8) and contiguous. The model was solved using 2010 OPTN data.

After solving the IP, the studies used the Liver Simulated Allocation Model (LSAM) to simulate 2010 liver allocation and transplantation using the regions/districts newly defined by the IP. The simulation results indicated that, within the LSAM paradigm and assumptions, the standard deviation of patient MELD at transplant decreased by approximately 30% to 40% in 2010 depending on the number of created districts. A decrease in the standard deviation indicates that the average MELDs at transplant are more similar across districts, not necessarily that the average MELD at transplant increases or decreases (average MELD at transplant is expected to increase).

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METHODOLOGY CONSIDERATIONS BEFORE IMPLEMENTING ALLOCATION SYSTEM CHANGES

The underlying findings of the UNOS liver committee’s proposal are remarkable—the aforementioned studies succeeded in obtaining a redistricting scheme that promoted a more uniform allocation that prioritized medical urgency as stipulated by the guidelines. However, there are several limitations that need be discussed before potentially imposing substantive change to the national liver distribution system.

In this study, we review the methodology used underlying the particular redistricting proposal presently discussed nationally, at 3 levels: (1) the suitability of redistricting as part of UNOS strategy to mitigate disparity; (2) the methodology justifying redistricting; (3) and practicability of implementing redistricting effectively. Our approach is system-driven, and in particular, the levels emphasize 3 logical principles of modifying large systems: identifying the nature and type of intervention, estimating the performance of the intervention with realistic changes to the initial data and assumptions, and monitoring or adjusting the intervention for its intended effect.

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(1) The Strategy of Redistricting

The UNOS must first determine whether it has already committed itself to redistricting or a similar solution. Other conceptual approaches for discovering policies that mitigate geographic disparity have been proposed. A policy for liver distribution, a term which we use in the technical sense from the optimization literature, is a set of precise rules that determine how organs are allocated and to whom and where organs are allocated. Policies related to geographic disparity may encompass the following types of characteristics:

  • (A) Modification of membership relationships of transplant centers to DSAs or UNOS regions or the geographic extent of DSAs or UNOS regions with no other changes.
  • (B) Prioritization of organ allocation based on donor characteristics and patient characteristics (e.g., MELD-based allocation).
  • (C) Modification of allocation hierarchy and prioritization based on characteristics not limited to donor or patient (e.g., statewide sharing).

Each of these policies can be applied at various levels in the OPTN system: patient, transplant center, DSA, region, or nation. The redistricting strategy mainly combines elements of both A and B and is applied at the regional level because it reassigns DSAs and transplant centers to new regions/districts while also enforcing MELD prioritization. Earlier enactments, such as Share 35, were mostly of type B and applied at the regional or DSA levels.

The purpose of these categories is to reveal where UNOS has implicitly focused its policy discussions. Policies of type C are less well-studied or amenable to standard models such as LSAM, but may possess attractive properties nonetheless. For example, enactment of statewide-sharing variances was linked to reductions in geographic disparity for kidney transplantation.5 Such variances modify the allocation hierarchy by having state allocation preempt regional allocation. Moreover, these variances are likely less costly to implement than redistricting. Although statewide sharing itself may not be the ideal strategy or even be superior to redistricting, UNOS would nevertheless be prudent to develop and assess a counter proposal, if only to better reevaluate redistricting subsequently. Because redistricting is a substantive and potentially disruptive change to the OPTN system, similar reductions in disparity may be achieved with finer and less costly changes to the system. The redistricting proposal under consideration is estimated to increase the annual total cost of recovering and transplanting organs by approximately $50 to $70 million, but still expected to yield a net savings of about $150 million. The savings are achieved presumably by the concurrent reductions in the pretransplant costs borne by the payer (e.g., insurance).1,6,7 However, it is unclear how redistricting produces such reductions in pretransplant costs. The conclusions of the underlying studies were that pretransplant costs were positively correlated with individual patient MELD. Assuming that this finding holds also for the case of redistricting, the resulting increase in average MELD at transplant prompted by the elimination of local allocation and MELD prioritization would suggest that sicker patients are being transplanted and that pretransplant costs would henceforth increase if median waiting times in the districts increased due to an overall increase in average MELD before transplant.

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(2)The Methodology of Redistricting

Redistricting potentially carries unintended consequences if the methodology is not scrutinized for robustness. The IP achieves an as-close-to-proportional allocation of organs as possible based on MELD at the regional level. There is no inherent flaw in the formulation of the optimization model itself, but it is not necessarily the best practice in the field of optimization. Foremost, the data are based only on the year 2010. Deterministic mathematical optimization problems can be very sensitive to small changes in the model data. The modeling inputs to the IP are estimates of the number of organs, the number of candidates, and travel times between DSAs. Changes in these data may arise from both statistical errors in estimating these parameters or from practice when these observed quantities inevitably depart from their historical values. Even small changes on the order of 0% to 5% can dramatically deter the quality of the solution. The appendix (SDC, http://links.lww.com/TP/B127) provides an illustrative example of this phenomenon. In practice, this would mean that small departures from the 2010 observations of organ/patient arrivals could possibly result in suboptimal mappings. Fortunately, more robust techniques are available and are highly recommended before using redistricting in practice.

The LSAM model is also a very helpful tool for assessing allocation rules for redistricting or other schemes based on patient and organ characteristics, but it is limited in predicting overall system performance, and for the purposes of geographic disparity, is not designed to analyze policies that modify the allocation hierarchy (type C). Several of these limitations have already been raised, but include LSAM’s reliance on historical data and assumptions of no change in transplant center or OPO behavior. The former criticism will likely apply to any solution methodology using simulation, but its impact can be perhaps be blunted by developing a robust forecasting solution or sensitivity analyses. The latter threatens the validity of predictions regarding redistricting. Because transplant centers and OPOs are largely autonomous, by imposing greater uniformity in the MELD range of organ recipients, transplant centers with different distributions of patients by MELD will necessarily respond differently to the pooling of organs. It is unclear whether this can be predicted precisely beforehand, but sensitivity analysis on the coefficient for patient MELD in the organ acceptance model employed in LSAM would help, as would including transplant center heterogeneity in the acceptance models (LSAM currently features one acceptance model for all transplant centers). As mentioned above, this particular redistricting proposal focuses on modifying membership relationships among transplant centers, DSAs, and regions while promoting MELD-based allocation. However, because it applies at the regional level and does not alter transplant center behavior uniformly, it is unclear whether redistricting would have its full intended effect over time.

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The Practicality of Redistricting

Redistricting is a static strategy. A good strategy should be dynamic and somewhat self-correcting—that is, the plan should be resilient to changes in the numbers of organs procured and patients registered, and also be relatively inexpensive to adjust over time. Unfortunately, the robustness of the current redistricting proposal is difficult to claim given the limited sensitivity analyses and the nature of the underlying optimization problem. Even if the underlying optimization models are perfect, changes in the distribution of organs, patient listing behavior, and MELD scores by region will over time diminish the quality of the redistricting. An improved redistricting strategy should have plans for regular updates to the districts—which may be potentially costly or disruptive to stakeholders of the transplantation system. Other strategies to resolve geographic disparity might be easier to adjust routinely and be more robust to changes in organ supply and demand. For example, in response to the success of statewide sharing agreements in kidney transplantation, sharing partnerships among DSAs have been proposed.8 These sharing partnerships preempt regional and national allocation by redirecting a specified number of organ offers to previously matched DSAs with the objective of reducing geographic disparity. The matches are potentially less expensive to implement and can be adjusted periodically in addition to modifying the number of organs redirected. Although we do not necessarily advocate partnerships over redistricting currently, the example demonstrates that other promising and effective strategies are worth the committee’s consideration before undergoing the substantive changes prompted by redistricting.

The above discussion maintained the assumption that MELD is indicative of medical urgency. In preparation for delivering an opinion on the redistricting issue, the UNOS committee is not considering changes to MELD scoring or its validity for use in liver allocation at this time.1 The MELD is a very good predictor of 3-month mortality for cirrhotic patients (c-statistic approximately 0.80).9 However, the quality of the posttransplant patient survival prediction using multivariable cox models involving patient age, donor age, sex, and pretransplantation MELD diminishes (c-statistic approximately 0.65).10 Although MELD serves its intended purpose very well, it was never intended to be used to predict global utility before and after transplantation and might not be robust enough for such a use.

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CONCLUSIONS

We believe redistricting is a sound concept for mitigating geographic disparities in liver transplantation. However, we are concerned that redistricting has not been presented with alternative strategies, and whether redistricting, as proposed by the liver committee, is flexible and robust to inevitable changes in transplant system behavior. Additionally, we recommend additional discussion about whether changes in transplant center or OPO performance would deter or enhance any potential solution. We recommend that UNOS formally consider alternative solutions and a more rigorous methodology before deciding on redistricting. Table 1 gives a list of system design principles that, if followed, would improve the redesigned allocation system.

TABLE 1

TABLE 1

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REFERENCES

1. OPTN/UNOS Liver and Intestinal Organ Transplantation Committee ( 2014), Redesigning Liver Distribution to Reduce Variation in Access to Liver Transplantation: http://optn.transplant.hrsa.gov/ContentDocuments/Liver_Concepts_2014.pdf
2. Editorial, Organ distribution by redesign. The Lancet. 2014; 384: 929.
3. Gentry SE, Massie AB, Cheek SW, et al. Addressing geographic disparities in liver transplantation through redistricting. Am J Transplant 2013; 13: 2052. doi: 10.1111/ajt.12301.
4. Gentry SE. Addressing Geographic Disparities in Organ Availability. Scientific Registry of Transplant Recipients: http://www.srtr.org/publications/pdf/pres/2013/Addressing_Geographic_Disparities_in_Organ_Availability.pdf. 2014.
5. Davis A, Mehrotra S, Kilambi V, et al. The effects of the statewide sharing variance on geographic disparity in kidney transplantation in the United States. Clin J Am Soc Nephrol 2014; 9: 1449. doi: 10.2215/CJN.05350513.
6. Buchanan P, Dzebisashvili N, Lentine KL, Axelrod DA, Schnitzler MA, Salvalaggio PR. Liver transplantation cost in the model for end-stage liver disease era: looking beyond the transplant admission. Liver Transpl 2009; 15: 1270. doi: 10.1002/lt.21802.
7. Salvalaggio PR, Dzebisashvili N, MacLeod KE, et al. The interaction among donor characteristics, severity of liver disease, and the cost of liver transplantation. Liver Transpl 2011; 17: 233. doi: 10.1002/lt.22230.
8. Davis AE, Mehrotra S, Friedewald JJ, et al. Improving geographic equity in kidney transplantation using alternative kidney sharing and optimization modeling. J Med Decis Making 2014; in print.
9. Kamath PS, Wiesner RH, Malinchoc M, et al. A model to predict survival in patients with end-stage liver disease. Hepatology 2001; 33: 464–470. doi: 10.1053/jhep.2001.22172.
10. Habib S, Berk B, Chang CCH, et al. MELD and prediction of post–liver transplantation survival. Liver Transpl 2006; 12: 440. doi: 10.1002/lt.20721.

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