ContentslistsavailableatScienceDirect
European
Journal
of
Operational
Research
journalhomepage:www.elsevier.com/locate/ejor
Modelling
and
optimisation
in
European
Kidney
Exchange
Programmes
Péter
Biró
a,
Joris
van
de
Klundert
b,c,∗,
David
Manlove
d,
William
Pettersson
d,
Tommy
Andersson
e,
Lisa
Burnapp
f,
Pavel
Chromy
g,
Pablo
Delgado
h,
Piotr
Dworczak
i,
Bernadette
Haase
j,
Aline
Hemke
j,
Rachel
Johnson
k,
Xenia
Klimentova
l,
Dirk
Kuypers
m,
Alessandro
Nanni
Costa
n,
Bart
Smeulders
o,
Frits
Spieksma
p,
María
O.
Valentín
q,
Ana
Viana
ra Hungarian Academy of Sciences, Corvinus University of Budapest, MTA KRTK, 1097 Budapest, Tóth Kálmán utca 17, Hungary b Prince Mohammad bin Salman College, 7682 Hejaz Boulevard, King Abdullah Economic City 23965-2609, Kingdom of Saudi Arabia c Erasmus Universiteit Rotterdam, Burgemeester Oudlaan 50, 3062 PA Rotterdam, The Netherlands
d School of Computing Science, University of Glasgow, Sir Alwyn Williams Building, Glasgow G12 8QQ, United Kingdom e Department of Economics, Lund University, SE-220 07 Lund, Sweden
f National Health Service, Blood and Transplant, 500, North Bristol Park, Filton, Bristol, BS34 7QH, United Kingdom g IKEM, Videnská 1958/9, 140 21 Praha 4, Czechia
h Organización Nacional de Trasplantes (ONT), Calle Sinesio Delgado, 8, Madrid 28029, Spain i Northwestern University, Global Hub 3389, Evanston, IL 60208, USA
j Dutch Transplantation Foundation, Plesmanlaan 100, CB Leiden 2332, the Netherlands
k National Health Service Blood and Transplant, 500, North Bristol Park, Filton, Bristol, BS34 7QH, United Kingdom l INESC TEC, Campus da Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, Porto 4200-465, Portugal m KU Leuven, UZ Herestraat 49 - box 7003 11, Leuven 30 0 0, Belgium
n Italian National Transplant Centre, Via Giano della Bella, 34, Roma RM 00162, Italy o HEC Management School, Université de Liege, N1 - rue Louvrex 14 - 40 0 0 Liége, Belgium
p Department of Mathematics and Computer Science, Eindhoven University of Technology, Groene Loper 3, 5612 AE Eindhoven, The Netherlands q Organización Nacional de Trasplantes (ONT), Calle Sinesio Delgado, 8, Madrid 28029, Spain
r INESC TEC and Polytechnic of Porto, Campus da Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
a
r
t
i
c
l
e
i
n
f
o
Article history: Received 7 February 2019 Accepted 1 September 2019 Available online xxx Keywords: OR in health services Ethics in OR Kidney exchange OR in practicea
b
s
t
r
a
c
t
Thecomplexmulti-criteriaoptimisationproblemsarisinginKidneyExchangeProgrammeshavereceived considerableattentionbothinpracticeandinthescientificliterature.Whereastheoreticaladvancements arewellreviewedandsynthesised,thisisnotthecaseforpractice.Wepresentasynthesisofmodelsand methodsappliedinpresentEuropeanKidneyExchangeProgrammes,whichisbasedondetailed descrip-tionswecreatedforthispurpose.Mostdescriptionsaddressnationalprogrammes,yetwealsopresent findingsonemergingcross-nationalprogrammes. Thesynthesisprovidesasystematicanddetailed de-scriptionofthemodelsandmethodstheprogrammesuse,revealingimportantcommonalitiesaswellas considerablevariationamongthem.Ratherthandistillingasinglebestpracticefromtheseresults,we findthatthevariationinmodelsandmethodsarisesbecauseofvariationincountrycharacteristics, poli-cies,andethics.Thesynthesisedstateoftheartmaybenefitfuturenationalandcross-nationalinitiatives anddirectfuturetheoreticalcontributionswithinandacrosstheboundariesoftheOperationsResearch discipline.
© 2019PublishedbyElsevierB.V.
∗ Corresponding author.
E-mail addresses: peter.biro@krtk.mta.hu (P. Biró), jklundert@mbsc.edu.sa , vandeklundert@eshpm.eur.nl (J. van de Klundert), david.manlove@glasgow.ac.uk
(D. Manlove), william.pettersson@glasgow.ac.uk (W. Pettersson), tommy.andersson@nek.lu.se (T. Andersson), lisa.burnapp@nhsbt.nhs.uk
(L. Burnapp), pavel.chromy@gmail.com (P. Chromy),pdelgadom@mscbs.es (P. Delgado),piotr.dworczak@northwestern.edu (P. Dworczak),b.haase@
transplantatiestichting.nl (B. Haase), a.hemke@transplantatiestichting.nl (A. Hemke), rachel.johnson@nhsbt.nhs.uk (R. Johnson), xenia.klimentova@inescporto.pt (X.
1. Introduction
Since the seminal work of Rapaport (1986), theproblems oc-curringinlivingdonor kidneyexchange programmes(KEPs) have received considerable attention in the fields of medicine, health
Klimentova), dirk.kuypers@uzleuven.be (D. Kuypers), cnt@iss.it (A. Nanni Costa), bart.smeulders@ulg.ac.be (B. Smeulders), f.c.r.spieksma@tue.nl (F. Spieksma), mvalentin@mscbs.es (M.O. Valentín), aviana@inesctec.pt (A. Viana).
https://doi.org/10.1016/j.ejor.2019.09.006 0377-2217/© 2019 Published by Elsevier B.V.
policy, economics, computer science, mathematics, and in the OperationsResearchliterature. Thisscientificactivityadvanced as existingKEPsdevelopedandnewKEPsemerged.Itfocusedonthe complex and multi-faceted dilemmas which present themselves when deciding which patients will receive a kidney - and may improvetheir healthandlongevity-andwhichpatientswillnot, asyet.Shouldafirstaimbetohelpasmanypatientsaspossible? Should longest waiting patients take preference? Or the sickest, theyoungest? These are justa few ofthe ethical questions that KEPsraise and are being considered across scientific disciplines. Moreover,thesecomplexdelicatequestionsariseinpracticewhere theanswersprovidedhavefarreachingconsequences.
Inview ofthesensitivityofthesequestionsandtheimpactof the decisions forthe individuals involved, operationsresearchers and other scientists have treated the resulting allocation prob-lems with the greatest care. This placed new demands on their craftsmanshipwhendevelopingmodelsandsolutionmethods. Of-ten, such scientific advancement occurred in the relative safety ofpurely advancingtheory.Glorie, Haase-Kromwijk,vande Klun-dert,Wagelmans,andWeimar(2014a)andMak-Hau(2017)provide overviewsofsuch primarily theory orientedmodels andsolution methods.
While the theoretical advances arethus well synthesised, this is less true of the practical advances in modelling and solving KEPs.Practicaladvancesoftendifferfromthetheorydevelopment asnotalltheoreticaladvances inmodellingandsolutionmethods are suitable in practice. Conversely, practical models and meth-ods have been influenced by medical and policy developments, not all of which have reached the theoretical discourse. The recently emerging cross-national KEPs and the policy dilemmas they present illustrate such disconnects. Our research aim is to extendthepolicyorientedoverviewofnationalandcross-national European KEP practices by Biró et al.(2019) into the operations research domain. Tothis purpose,we present a review and syn-thesis of all models (including objectives and constraints) and solutionmethods(includingalgorithmictechniques) actively prac-ticedinEuropean KEPs, assurveyedvia aquestionnairethat was jointly developed for this purpose (original data are reported in
Andersson,Biró,&etal.,2019).
For anumber ofreasons,European KEPsare ofparticular rel-evance. Many European countries have advanced transplantation programmesand have existing or newly developing KEPs. While theseKEPsfollowcommon regulations(see European Committee onOrganTransplantation(CD-P-TO)oftheCouncilofEurope,2018) eachoneisalsoclearlydevelopedwithinadifferentnational con-text and with different norms and values. An overview of mod-elsandmethodspracticed inEuropean KEPsthereforeprovidesa richimpressionofrelatedyetvaried stateoftheartprogrammes. Moreover,wedescribeanddiscussthemostrecentadvancements incross-nationalKEPsasthey arise inEurope.Theoverviewmay informtheoreticalresearch,aswellaspracticesinothercountries, Europeanornot,who aredeveloping(joint)programmes,models, andmethods.
This research results from EU COST Action 15210, the Euro-peanNetwork for Collaboration in Kidney Exchange Programmes (COST, 2017; European Network for Collaboration in Kidney Eex-changeProgrammes,2017).
2. Background
2.1.ContextandprinciplesforKEPs
The2016GlobalBurdenofDiseaseStudyidentifieschronic kid-neydisease(CKD)asthe11thmostcommoncauseofdeath glob-ally, accounting for almost 1.2 million deaths worldwide (2.17%)
(InstituteforHealthMetrics&Evaluation,2018).InEurope,Chronic Kidney Disease accounts for 1.52% of all deaths. The number of deathsresultingfromCKDgrowsboth inabsoluteandinrelative terms, and has almost doubled globally since 1990 (Institute for HealthMetrics&Evaluation,2018).
No cure exists at presentfor Chronic Kidney Disease. It may progress over severalstages, the last one ofwhich is called End StageRenalDisease(ESRD).ThemostcommontreatmentforESRD is dialysis, which is costly. Recent UK evidence estimates yearly costsperpatientat15,000to35,000GBP(Baboolaletal.,2008).
Compared to dialysis, the alternative of transplantation of-fers longer life expectancy, better quality of life, andlower aver-agetreatment costs (Axelrodet al.,2018;Haller,Gutjahr, Kramar, Harnoncourt, & Oberbauer, 2011; Sánchez-Escuredo et al., 2015; Wolfe, Roys, & Merion, 2010). Hence, transplantation ispreferred asatreatmentacrossEurope.
In Europe, transplantation treatments are often provided throughdedicatedandwellorganisedtransplantationprogrammes. Initiallytheseprogrammesweresetuptotransplantkidneysfrom deceased kidney donors (DKD). At present, mostEuropean coun-triesoperatenationallyorganisedDKDprogrammesthroughwhich the majorityoftransplants areconducted. There is agreat varia-tioninthevolumesofDKDprogrammesacrossEurope,depending onethical andlegal regulations,aswell asonthe operational ef-fectivenessoftheprogrammesandthehealthcaresystemsin gen-eral(European Directorate fortheQualityofMedicines, 2017).By wayof illustration, the Spanish DKD programmehas thehighest deceased donorkidney transplantationrateat57.6per million in 2016. Germany,on theother hand,hasa deceased donation rate of 27.4 per million. From an operational perspective, variation is more limited, as manynational programmes followinternational protocols and standards, and collaborate across national borders to improve effectiveness. The organisations Eurotransplant and Scandiatransplantmanage such internationalDKDprogrammesin Europe.
InmanyEuropeancountries,thedemandforkidneytransplants increasinglyexceedsthesupplyofkidneysretrievedfromdeceased donors. Hence, DKD programmes have waiting lists. Recently re-portedwaitinglistlengthsareforinstance2208patientsin Scan-dinavia(January2018)and5033 patientsintheUK(March2018). Bothofthesenumbersconsiderably exceedtheyearly numberof transplants performed in these countries, and the same applies across Europe and beyond (European Directorate forthe Quality ofMedicines, 2017). Patients onthe waitinglistsare typically on dialysis,anditisnot uncommonforpatientstobecometooillto transplantortodiewhileonthewaitinglist,aswitnessedbythe deathratespresentedabove.
In addition to deceased donation, kidneys can be donated by living donors, as the human body has two kidneys, while com-monly onesuffices. ComparedtoDKD,living donorkidney dona-tion (LKD) hasbetter long-termpatient andtransplantoutcomes (Hartetal.,2017;MacNeill,Casula,Shaw,&Castledine,2016;Wolfe etal.,2010)(werefer toReese,Boudville,& Garg,2015 fora dis-cussion of donor outcomes). This, in combination with the rel-atively poor outcomes of dialysis and the shortage of deceased donororgans,ledtotheestablishmentofLKDprogrammesin Eu-rope,tocomplementexistingDKDprogrammes.In2017,the num-beroftransplantsresultingfromLKDasapercentageofthetotal numberoftransplantinEuropewereforinstance,5%inGermany, 10%inSpain,26.4%inScandinavia,30%intheUK,andslightlyover 50% in the Netherlands (European Directorate for the Quality of Medicines,2017).
The defaultprocedureto enableLKDis forapatient to finda livingpersonwillingtodonateakidney,andreceiveakidneyfrom thisspecified donor.In theremainder we will refer tosuch a
pa-tientanddonorasapairandalsorefertothepatientasarecipient, orasthespecifiedrecipientofthedonor.
Even when a patient finds a specified donor, however, trans-plantation of the kidney from this donor to the patient may be unfeasiblebecausethepatient(recipient)anddonorarenot medi-callycompatible(areincompatible).Belowweexplaintheformsand definitions of compatibility,while notingalready that thesehave changed over time. (The reader may further note that they also applyincaseofDKD.)Compatibilitymaytakeintoaccount:
• ABO-Compatibilityreferstothebloodtypes,A,B,AB,andO. TypeO donorscan donate toall recipients. Type Adonors candonatetoTypeAandTypeABrecipients.TypeBdonors can donate to Type B and Type AB recipients. Type AB donorscan onlydonatetoTypeAB recipients.Adonorand recipientaresaidtobeABO-compatibleifthebloodtypeof thedonorandrecipientaresuchthatthedonorcandonate totherecipient.
• HLA-Match,whichrefers totheextentto whichtheHuman Leukocyte Antigens oftherecipientandthe donorarealike. Themoretheyarealike,themorecompatiblefroma view-point of HLA matching.When fully alike (e.g. foridentical twins)wespeakofaperfectHLAmatch(Fulleretal.,2004; Terasaki,Cecka,Gjertson,&Takemoto,1995).
• HLA-Crossmatchreferstothetesttodecidewhethera recip-ienthasantibodiestotheHLAofthedonor(insignificantly highconcentration).Ifthisisindeedthecase,onespeaksof apositivecrossmatchwhichisseenasanindicationthatthe transplantationwillnotbesuccessful.Apositivecrossmatch isonlypossiblewhentheHLAmatchisnotperfect.(Below wealsoelaborateonvirtualcrossmatches.)
Originally,recipientsanddonorswereonlyconsidered compat-iblewhentheywereABO-compatible,andtherewasaperfectHLA match (and hence the crossmatch is negative).The development of immunosuppressant drugs made donation possible in case of lessthanperfectHLA-match.Recentadvancements withso called desensitisationalsomakeABO-incompatibleandHLA-incompatible transplantspossible(Halloran,2004). Thesedynamicshaveledto the definition of half-compatibility (Andersson & Kratz, 2016). A recipient-donorpairissaidtobehalf-compatiblewhenthe cross-matchis negativeandtransplantationbetween donorand recipi-ent requiresdesensitisationto overcomeABO-incompatibility. Re-centmetaanalysesconcludethat outcomesofstateoftheart de-sensitisationtreatment resultsaregoodbutsome aresignificantly worsethanoutcomesofABO-compatibletransplants(DeWeerd& Betjes,2018;Scurtetal.,2019).
Highly sensitised patients are less likely to find compatible donors due to the presence of high titres of HLA antibodies in theirblood.DifferentKEPsusedifferentparameterstocharacterise (high)sensitisation.SomeKEPsgivehighlysensitisedpatients pri-oritytoimprovetheirchancesofreceivingatransplant.
Otherfactorsthatinfluencetransplantationoutcomes,in partic-ulargraftsurvival,relatetothequalityofthekidney-whichisfor instancecorrelated withdonorage-andthehealth ofthe recip-ient-which inturniscorrelatedwithrecipient ageandtimeon dialysis.
Withthesereflectionsonoutcomesanddefinitionsof compat-ibility athand, we cannow consider thepurpose ofKEPs. Origi-nally,KEPswereinitiatedtoproviderecipientswho donot match with their specified donor access to a compatible donor by ex-change of donors between recipients. In later years, the scope of KEPshas been extended.Because of advances in immunosup-pression and desensitisation,recipient-donor pairs that are (half-)compatible mightstillchoose toparticipatetofinda more com-patible donor. Furthermore, altruistic donors, i.e. donors without an intended recipient,mayvolunteer toparticipate. Fourth, some
KEPs integrate their LKD programme with the DKD programme, e.g. by starting chains with deceased donors. All of these exten-sionsarefurthercoveredbelow.
2.2.KEPdesignvariations
Below we synthesise the variations in the design of KEPs as identified from the detailed data collected for the purpose of our research by existing European KEPs (Andersson et al., 2019). Detailed country level comparisons can be found in Table 1 of Biró et al. (2019) and in Fig. 1 below. KEPs register a set P=
{
p1,p2,...,pn}
ofrecipient-donorpairspi=(
ri,di)
,i=1,...,n whereridenotestherecipientanddithedonorofpairpi.• Exchange cycles: Initially KEPs sought to match recipient-donorpairstogether.Moreprecisely,theysoughttoidentify twopairs pi=
(
ri,di)
andpj=(
rj,dj)
suchthatrecipientri iscompatible withdonor dj and -vice versa - recipient rj iscompatiblewithdonordi.Suchexchangesarecalled pair-wiseexchanges.The KEPsofsome European countries,such asFrance,arebasedonpairwiseexchanges.Othercountries with larger programmes (UK, Spain, the Netherlands) also prioritisepairwiseexchanges(seebelow).Most of the European KEPs have advanced beyond pair-wise exchanges, orare preparingtodo so.As pairwise ex-changescanbeviewedasformingacycleoflengthtwo, be-tweenpairsiandj,anaturalextensionistocyclesoflength three, also known as three-way exchanges. In a three-way exchange, there are three pairs, pi=
(
ri,di)
, pj=(
rj,dj)
and pk=(
rk,dk)
, and ri is matched with dj, rj with dk, and rk with di. At present, the Spanish and the UK KEPs onlyconsiderpairwiseandthree-wayexchanges(forreasons explained below). Four-way exchanges are allowed in the Netherlands.WithinEurope,longerexchangesareallowedin severalcountries(suchasBelgiumandPortugal),yetso far theyhaveonlybeenconductedintheCzechRepublic. • Altruisticchains:AltruisticdonationisallowedinsomeEuro-peancountries. ItislegallyforbiddeninFrance,Polandand Portugal.Whereallowed,itcantaketheformofdonatingto a recipientin the DKD programme(bydefault inBelgium, but occasionally also in other countries). the Netherlands, Spain,andtheUK explicitlyincorporatedaltruisticdonation intheir KEPs. An altruistic donor then donates to a (first) patient, whose specified donor can then donate to a (sec-ond)patientandsoon,withthelastkidneydonatedtothe waitinglist.Thus,altruisticdonorsmayinitiateanexchange involvingmultiplepairs,whichformachain.Aswasthecase forcycles,KEPsmayimposelimitson themaximum num-berofpairsinvolvedinchains.Such limitationsvaryacross European KEPs, roughly following the limitations on cycle lengthdescribedabove.
Alternatively,donationbythelast donormaybepostponed to continue the chain at a later moment. The last donor isthen calledbridgedonor. Bridge donorscan makechains longerorevenneverending.EuropeanKEPshavenoexplicit arrangementsyetforneverendingchains.
• Timing of match runs: A match run is the process of con-structing cycles and chains among donors and recipients participatingin a KEP. Many European KEPs organisetheir matchrunsperiodically.Polandorganisesamatchrunevery month; the Netherlands,Portugal, and the UK have match runseverythreemonths;whilstSpainhasmatchrunsevery fourmonths. Spain also organises match runs whenevera newaltruistic donorarrives.Other countriesalsochoose to organisetheirmatchrunsonoccasion,ratherthanfollowing
Fig. 1. Objectives and constraints in European KEPs. Numbers reflect hierarchy level, w denotes the criterion is weighed against other criteria at the lowest level. periodic patterns. For instance,in Belgium the match runs
areorganisedwhenrequestedbyatransplantcentre. • RelationshipswithDKD:Asit avoidshealthrisksfordonors,
deceaseddonationisoftenconsideredthedefaulttreatment. Patients registered ina KEP thereforemayalsoregister for theDKDwaitinglist.
Some countriesallow andutilise interaction withthe DKD programmeintheformofdeceasedchains,whenone recip-ientintheKEPreceivesakidneyfromadeceaseddonorand thenhis/herdonorstartsachain(asinthecaseofan altru-isticchain)withthelastdonordonatingtoarecipientinthe DKDprogramme. Threesuchchainswere startedinItaly in thesummerof2018.The mostcommonmodalityfor inter-actionwiththeDKDprogrammeisthroughaltruisticchains, asdescribedabove.
• Inclusion of compatible pairs: Some European KEPs explic-itly restrictregistrationto incompatiblepairs, e.g.,Belgium, France andPortugal.Other countriesallow oreven encour-age compatible pairs to participate, to improve outcomes forthemselvesand/orforotherrecipients.KEPsthatenable such participationoftenprovideadditionalarrangementsto ensure thatcorresponding patientsarematched toa donor
whoisatleastasgood(tobedefinedbelow)astheir speci-fieddonor.TheKEPsoftheCzechRepublic,Scandinaviaand Spain have explicit arrangements for this purpose. Notice that these KEPsmay endup simply matching such recipi-entstotheirspecifieddonors.
• DesensitisationasanalternativeofKEP:ForABO-incompatible pairs transplantation from the specified donor to his/her recipient is possible with desensitisation. Hence, these pairsdo not needto registerina KEPandbe transplanted through an exchange. The recommended pathwayforsuch pairs differs across Europe depending on healthcare sys-tems,traditions andalsothesizeandeffectivenessoftheir KEPs. If the default treatment is desensitisation and, as a consequence,theKEPpoolissmall,thenthisgivesanother reason for the patients not to register in the KEP due to the relatively limited chance of finding exchange partners. This is the case in France and Italy, for example. In the countrieswiththe longeststanding KEPs(the Netherlands, UK, and Spain) the policy is to prefer exchange over de-sensitisation. As a result, these countrieshavelarger pools which increases match probabilities and reduces waiting timesforpatients (Biró etal., 2019). Intheseprogrammes,
theABO-incompatiblepairswhicharenotmatchedwithina reasonableperiodoftime(e.g.twomatchruns)areadvised toconsiderdesensitisation.
• Allowing ABOi transplants in exchanges: When a recipient is not even half-compatiblewiththe specified donor, KEPs mayconsidermatchingwithhalf-compatibledonorsthrough donor exchange. The Czech,Scandinavian, Spanish,and UK KEPspresentlyfacilitatesuchmatches.
• Multipledonorsregisteringforonerecipient:Thisisallowedin mostKEPsbutnotyetinBelgium,FranceandNetherlands.It likelyincreasesthechancesofthe recipienttobematched. Whentherecipientismatched,onlyoneofthe correspond-ingspecifieddonorsdonatestoanotherrecipient.
2.3. Logisticsandorganisation
Most KEPs require transplantations for all donors and pairs in a same cycle to occur simultaneously to avoid withdrawal of donors after their specified recipients have received kidneys but beforedonatingthemselves(seee.g. Cowan,Gritsch, Nassiri, Sina-core, & Veale, 2019). The Czech Republicand Polanddo not en-force simultaneityincyclesandhavesuccessfullyconducted non-simultaneous exchanges(suchas a7-way exchange conductedin theCzechRepublic).Asparalleltransplantationmaybringcapacity andlogisticschallenges,a simultaneityrequirementposes restric-tionsonexchangecyclelength.
Inthecaseofaltruisticdonation,simultaneitymaybelessofa strict requirement, as recipients can receive kidneys before their specified donor donates, thus avoiding the risk of leaving a re-cipientunmatchedandwithoutdonor.Thus,themaximumlength KEPsallowforchainsmayexceedthemaximumlengthforcycles. Infact,whenthelastdonorrepeatedlyinitiatesanewchaininthe next matchrun,the chaincanbecome neverending.We referto
Fig.1foranoverviewofchainandcyclelengthlimitations imple-mentedbyEuropeanKEPs.
KEPsadmittinglongercyclesforwhichtheyperformall trans-plants simultaneously may needor prefer to spread these trans-plantsacrossmultiplecentresbecauseofcapacitylimitations,asis thecasefortheDutchprogramme.
Anonymity mayform another reasonto involvemultiple cen-tres. SeveralEuropean countries(e.g.the Netherlands,Spain, UK) requireanonymity,eitherlegallyorbyprotocol. Anonymityis dif-ficult to ensure when performing surgeries formultiple pairs in-volved in an exchange in the same hospital or when the donor travelstothehospitalofthepatientforthetransplant.
Large travel distances for donors or kidneys can be consid-eredundesirable.Spainforexample,preferstomatchrecipientsto donorsfromthesameregion.
2.4. HLA-testingandre-optimisation
Before a transplant is conducted, laboratory tests for HLA matching and cross matching must be done. Depending on the labresults,the transplantcanbe consideredinfeasible,or requir-ingimmunosuppressionand/ordesensitisation.TheEuropeanKEPs varyconsiderablyintheirorganisationsofthelabtestingandthe integrationofthelabtestswiththematchruns.
HLA matching requires the HLA profile of each recipient and each donor to be determined. HLA matches can subsequently be determined by comparing the HLA profiles of donors and recipients. HLA cross matching requires to determine whether a recipient has antibodiesagainst the specific HLA ofa donor. The compatibility check of a pair is done first via so-called virtual crossmatchtestsbycomparingtheABOtypesandtheHLAdataof the patient anddonor. Forpairs that are matchedand estimated tobe compatible,alaboratorycrossmatch testmustsubsequently
bedone beforetransplantationisapproved.The timingofthelab crossmatch testsdiffer acrosscountries, asthesecan be costlyor time consuming, especially if multiple HLA labs are involved in thetesting.
For KEPs with smaller numbers of participating pairs and in smallercountriesit maybe feasible toconduct all the lab cross-matchtestsprior to executing the matchrun. Polandand Portu-galhave adoptedthis practice.Forlarger KEPs, complete a-priori crossmatchingisoftenconsideredundesirable.Thelabcrossmatch testingisthendoneafterthematchrunonthevirtualcrossmatch input hasresulted in a setof cycles(and chains) toconsider for transplantation.Now,anycycleorchaincanonlybeexecutedifall transplantsimpliedbythecycle(orchain)arebetweenarecipient anddonorforwhichthelabcrossmatchisnegative.
European KEPs have different procedures to advance in case therearepositivecrossmatchesforoneormoretransplantsinthe proposedexchanges.Threeexamplesforcrossmatchtestingand re-optimisationstrategiesinlargeKEPsare:
• UK:TheUK KEPusesmultipleHLA-labsandconsiders only onesolutionper periodicmatchrun. Toimprovethe likeli-hoodthatalltransplantsinacyclewillproceedtheUKKEP onlyallows pairwiseandthree wayexchangesto minimise theriskofimmunological,clinicalorlogisticalreasons pre-ventingtransplantsproceedingtoplan.Moreover, itprefers three-way exchanges that contain embedded pairwise ex-changes,such that anembeddedpairwise exchange can go ahead ifthe three-way exchange cannot proceed (Manlove &O’Malley,article2.6,21pp,2014).Afterthecrossmatch re-sultshavebeen obtained,theKEP performsasmany trans-plantsaspossiblefromthematchrunsolution.
• Spain:Thereare multipleHLAlabsandtworounds of test-ing.In the firstround an optimalsolution istested andin the second round an alternative solution, where the can-celledcyclesareintendedtoberepaired.
• Netherlands:OnecentralHLAlabisresponsiblefortheHLA testingandthecrossmatchtests. Ifapositivecrossmatch is found the matchrun will be repeatedby the coordinators tofind a next-best solution, until the crossmatch testsare negative.
2.5.KEPsasdynamicsystemsandtheirlongtermperformance After each match run, some recipients may have received a transplant,andhencethesepairsandsomeorallofthealtruistic donorsleavetheKEP.Over time,newrecipient-donorpairs arrive andregister.Additionally,pairsmayleave,forexampleasaresult ofreceivingatransplantfromtheDKDprogramme,findinga (half-)compatibledonor,preferringadonation involvingdesensitisation (i.e.thepresenceofhightitresofHLAantibodiesintheirblood)or becomingtoosicktobetransplanted.
Fromthe aboveit isevident thatthe effectivenessofaKEP is not determined by the quality of the solution found fora single matchrun,butbyitscontributiontoaddressthelong-termhealth problemsoftherecipientsregisteringoveraperiodoftime,andin relationtoalternativesolutions,such astheDKDprogrammeand desensitisationprogrammes.WhenassessingthewaysKEPsare or-ganised and exchanges are identified at match runs, it is there-foreappropriatetotakealongertermperspective.Thisholds par-ticularly true whenincluding altruistic chains that span multiple matchruns.
Longer-termperformancecriteriaconsideredbyEuropeanKEPs are: total numberoftransplants performed, percentageof recipi-ents inthe KEP who have received a transplant, average waiting timeuntilbeingmatched,qualityoflifeforrecipientsanddonors aftertransplant, graftsurvival times, recipient survival rates,and
donorsurvivalrates(Anderssonetal.,2019).Moreover,these out-comesare consideredovervarious recipientsub-populations,e.g., perbloodtype,orwithregardtohighlysensitisedpatients,toalso assessequityandfairnessconsiderations.
2.6.Internationalcollaborations
Asdescribedabove,internationalcollaborationiscommon prac-ticeforDKDprogrammesandincreasinglypractisedbyKEPs.Such collaborationcan result infinding better matches andin match-ingmorerecipients.TheresultinginternationalKEPstypicallyalign withnational KEPs already in place,rather than replacing them. Wecategorisethepossiblecooperationsasfollows:
• Merged pools In the most advanced mode of cooperation, the pools are merged anda solution is found for a match run involvingthemergedset ofpairs.Inthisvariant,there are no national match runs. Still, the countries involved mayhavedifferentconstraintsandobjectives.Anexampleof thisapproachisSTEPorganisedbyScandiatransplant,which startedinSweden andnowincludes DenmarkandNorway. Thesameapproachisusedinthecooperationbetween Aus-triaandCzechRepublicsince2016(Böhmigetal.,2017). • Consecutive runs In the cooperation of Portugal, Spain and
Italy each country firstconducts its national matchingrun, afterwhichtheremainingpatient-donorpairsparticipatein aninternationalmatchrun.
• Outside registrations A(large) country mayextend registra-tionofpairstoitsKEPstopairsfromanothercountry.Such arrangements exist between the UK and Ireland, and be-tweenFranceandSwitzerland(wherethepairsfromthe lat-tercountriesjointheKEPsoftheformer).
3. Matchingmodels
This section presents the models andmethods used to finda solutionforasinglematchrunofa KEP.The objectivesand con-straintsinthemodels,aswellasthedesignofthesolution meth-odsarecloselybasedonthecontextualconsiderationsdescribedin theprevioussection.
Before going into the details of the models and optimisation methodshowever, let usmentionthat mostEuropean KEPs have adoptedproceduresthroughwhichintheendcliniciansdecideon theactual matching.This is done with thepurpose of takingall relevant medical considerations into account, as well asto have feasibility explicitly checked by all centres involved. As a result, thetechnologiesappliedmaynot beoptimisationmethods inthe classicalsense. Forinstance,themethods practisedby theCzech, PolishandPortugueseKEPsdeliverarankedlistofsolutions,from whichclinicianschoose.InSpain, onceasolutionisobtainedand thecentresare informed,thosecentres shareclinical information andcoordinatethecrossmatchtests.TheyinformtheSpanishKEP whethertheyadvancewiththetransplantationsresultingfromthe match run. In the Netherlands the donor is assessed in the re-cipient centre which ultimately decides on the suitability of the donorforthepatient(sometimescrossmatchestestsaretherefore repeatedintherecipientcentre).
There are two broad classes of models used to describe the problemoffinding an optimal setof exchangesfora matchrun. Thefirstclassencompassesgraphmodels,whichareintuitiveand insightful.Asecondclassisformedfromintegerprogramming for-mulations.Theseformulationshavebeenparticularlyhelpfulto ad-vancesolutionmethods.Bothclasses,andthecorresponding mod-elsandmethodsareconsideredinmoredetailbelow.
The classical model that is used to formulate the problem of findingsolutions foramatchrun that takecompatibilityinto ac-countistheso-calledcompatibilitygraph.ItisadirectedgraphD(N,
A),inwhichthereisanodeni∈Nforeachrecipient-donorpairfor i=1,...,n.Thereisanarc(i,j)fromnodeni∈N tonodenj∈Nif the donor ofpair ni is compatiblewith the patient of pairnj.If ABOitransplantsarealsoconsidered,wemaydistinguishaspecial setofarcs A⊆Arepresentinghalf-compatibledonor-patient pairs. Aselfloop,i.e.anarc(i,i)whereni∈N,emanatesfromeach(half-) compatiblerecipientdonorpair. Thecompatibilitygraphiscalled virtualcompatibilitygraphifthecompatibilitiesandweightsare es-timatedbasedonvirtualcrossmatchtestresults.
Multiple donors may also be registered for one patient, in whichcasewe have anarc (i, j) ifsome ofthe donors ofpairni iscompatiblewiththepatientofnj.
WemaydistinguishaspecialclassofnodesNa⊆Ntorepresent altruistic donors. A possible wayof simplifying the modelling is toassume that altruisticdonors havespecified dummyrecipients who are compatible withall donors, except foraltruistic donors. Hence thealtruisticdonor nodesni∈Na haveincomingarcsfrom everynodeni∈NࢨNa.Anychainemanatingfromanaltruisticdonor node cannow betriviallyextendedto formacycleby addingan arcfromthelastnodeonthechaintothealtruisticdonornode.
After the modification ofD(N,A) foraltruistic donor nodes,a matchrun solutionconsistsofa setofcycles inD(N,A).As each donorandrecipientcanparticipateinatmostone transplant,the problemoffindingamatchrunsolutioncannowbeinterpretedas anode-disjointcyclepackingproblem. Moreover,amaximum car-dinalitycyclepackinginD(N,A)nowreferstoamatchrunsolution withthehighestpossiblenumberoftransplants.
Each arc (i, j), can have a weight wi, j to represent the util-ity of matching the donor frompair ni∈N with the recipient of pair nj∈N. This value can include clinical considerations as well as other priority-based contributions, such as matching type O donorstotypeOpatients.Moreover,nodesni∈Ncanhaveweights to distinguishprioritiesamong recipients, forinstancedepending on waiting time or sensitisation. Further, cycles can have cycle weights,forinstanceaccordingtocyclelength ortothestructure of the subgraph induced by the nodes included inthe cycle. An example of such a subgraph property is the number of pairwise exchangesintheinduced subgraphofathree-wayexchange (also referred to asthenumber ofback-arcsin a three-way exchange). Allsuchweightsenablesolutionstobedistinguishedbasedon per-formancecriteriafortheKEPsandsubsequentlypermatchrun.
Asiscommon,thegraphmodelpresentedabovecanbe formu-latedasan integerprogram.Here we presenttwobasic formula-tions,referredtoasthearcformulationandthecycleformulation (Abraham,Blum,&Sandholm,2007).
Thearcformulationhasabinaryvariableyi,jforeacharc(i,j). Finding a maximum (value) solution with cycles and chains of length atmostK canbe solved through
(
1)
−(
4)
,where(4) en-sures The arc formulation has a binary variable yi,j for each arc (i,j).Finding amaximum (value)solution withcyclesandchains of length at most K can be solved through(
1)
−(
4)
, where (4) ensuresthatnocyclelongerthanKexistsinthesolution.max i, j wi, jyi, j (1) s.t. j yi, j− j yj,i=0,
∀
i∈V (2) j yi, j≤ 1,∀
i∈V (3)yi1,i2+yi2,i3+· · · +yiK−1 ,iK ≤ K− 1, foreachdirected
chainoflengthK (4) The cycleformulationuses a binaryvariable xc foreach cycle (and chain,seeabove) cof length atmostK. We denotethisset
ofcycles byCK.The weight ofacyclec isdenoted by w c, which can betakenasthesumoftheedge-weightsin thecycle,orcan bedefineddifferentlyasmentionedabove.Findingallcyclesinthe graphcanbedoneusinge.g.Johnson’salgorithm.
max c∈CK wcxc (5) s.t. c∈CK,i∈c xc≤ 1
∀
i∈V (6)As not all of the listed objectives can be expressed using the edge-formulation,thecycle-formulation appears morerobust. For instance, objectives 2.-5. that prioritise shorter cycles or cy-cles with more back-arcs require a cycle formulation. However, the edge-formulation can have additional value, for instance to find long (never-ending) chains, sometimes in combination with cycle-variables. We refer to Anderson, Ashlagi, Gamarnik, and Roth(2015)andDickerson,Manlove,Plaut,Sandholm,andTrimble (2016)for recent relatedresults.Refinementsforthecasewherea KEP appliesdifferentupperboundsforchainsandcyclesare pre-sented(Andersonetal.,2015;Dickersonetal.,2016;Glorie,vande Klundert,&Wagelmans,2014b).
3.1. Dataandparameters
Mostdatarequiredtosolveactualmatchrunscanbecollected frommedicalrecordsofrecipientsanddonors.Theserecordsneed to include the ABOtypes andHLA profiles asobtained fromthe labtest.Theindividual labtestswillalsoprovidethedataneeded for virtual crossmatch tests. Theselab testscan be done accord-ing to different methods and with different degrees of accuracy andhencethecorrectnessofthevirtualcrossmatchtestsmayvary amongKEPs.TheymayevenvarywithinKEPsincasemultiplelabs areinvolved,asisparticularlyrelevantforinternationalKEPs.
The policy related parameters may be set implicitly, as is for instance the casewhen restrictingthe matching to pairwise ex-changes or to pairwise and three-way exchanges (Scandinavia, Spain, andthe UK).Other KEPsmay fixthese parameters explic-itly in policies or leave them to be set manually (as in Poland andPortugal).Obviously,otherconstraintparametersandobjective functionscoefficientsmustalsobeset.Withfewexceptions,these parameters are fixed asthey areset inlegislatoryframeworks or formalisedpolicies.
3.2. Objectives
Below, we synthesisemodel variationsamong European KEPs. Objectivefunctionsarepresentedfirst,afterwhichconstraints fol-low.Theperformance measures, orcriteria, consideredinthe ob-jectivefunctioncanoftenalsobemodelledinconstraintsandvice versa. Forexample,agedifferencesmaybe weighed asan objec-tivefunctioncomponentorboundedviaaconstraint.Thesynthesis groups the criteriathematically. Where helpful,brief motivations andinterpretations providefurtherclarification.Betweenbrackets we listthe countries that include the objective criterion or con-straint.Wealsoincludeashortexplanationofasample implemen-tationofeachcriterionforacycleformulation.
All European KEPshave formulated multiple-criteria objective functions. In fact, many adopt a hierarchical objective function. For instance, the Czech KEP prioritises to maximise the number of transplants, and within all such matchings, chooses one that maximises the number ofcycles, constituting a hierarchyof two criteria.Ashierarchicalobjectivefunctionscanbereformulatedas weightedobjectivefunctions, wemayconsiderall objective func-tions as weighted. However, as the hierarchies are often distin-guished inthesolutionmethods,the hierarchicalmodels arealso explicitlypresentedassuch.Fig.1summarisesthefindings.
Optimisingnumberofactualtransplantstoperform: 1. Maximisingthenumberoftransplants(All).
This can be implemented with the objective function maxwcxc wherewcisthelengthofcyclec.
2. Minimising the length of the longest selected cycle, as longer cycles are more likely to result in positive cross-matches,andthusintransplantsthatwillnotbeperformed. Selectingshorter cyclesis alsoimportantfor logistical rea-sons(NL,UK).
Mathematically, this is formulated as minmaxwcxc where wc is thelength ofcyclec.This canbe implementedinan ILPbyaddingavariablemiforeachi∈
{
1,...,L}
whereLis thelengthofthelongestcycle. Constraintswillensurethat miwill takethevalue 1ifandonlyifacycleoflength ≥ i isselected. Theseconstraints are implemented by first let-tingCi bethe setofcycleswithlengthsatmosti(soC3 is thesetofall cyclesof length2 or3),andthen addingthe constraintsmi≥
c∈Cixc
|
Ci|
foreach i∈
{
1,...,L}
. We can then minimise the functionmitominimisethelengthofthelongestcycle.
3. Maximising the number of cycles selected, (which in turn reducestheaveragelengthsofthecycles)(CZ,ES,UK). This can be implemented with the objective function max
xc.
4. Maximisingthe number ofpairwise exchanges inthe sub-graphsinduced bythree-way exchanges, i.e.the numberof back-arcs, in order to improve the number ofmatches re-mainingafterdeletingpositivecrossmatchesfromthe solu-tion,andalsoforlogisticalreasons.(ES,UK).
This can be implemented by the objective function maxwcxc where wc is the number of back-arcsin cycle c.
5. Maximising the number of pairwise exchanges and three-wayexchangeswithembeddedtwo-wayexchanges(UK). This can be implemented with the objective function maxwcxc wherewc=1ifandonlyifcyclecisapairwise exchangeorathree-wayexchange withan embedded two-wayexchange.
Improvingtheoverallqualityofthetransplants:
6. Minimisingthenumberofimplied desensitisations inKEPs thatallowABOiand/orHLAitransplants(CZ,SE).
This can be implemented with the objective function minwcxc wherewc isthenumberofimplied desensitisa-tionsincyclec.
7. Maximisingthe(weighted)sumoftheHLA-matchingscores (CZ,PL,UK)withfocusingonDR-antigeninparticular(CZ). This can be implemented with the objective function maxwcxcwherewcistheweightedsumofHLA-matching scoresacrossalltransplantsincyclec.
8. Minimisingagedifferencesbetweenthedonorsandpatients (BE,PL,ES).
This can be implemented with the objective function minwcxc wherewc isthesumoftheagedifferences be-tweendonorsandpatientsintransplantsincyclec. 9. Prioritisingpaediatricpatients(ES).
This can be implemented with the objective function maxwcxc wherewc is thenumber ofpaediatric patients involvedinthecycle.
10. Prioritising patients that have not started dialysis yet (BE, PL).
This can be implemented with the objective function maxwcxc wherewc=1ifandonlyifthepatienthasnot starteddialysis.
Forthecriteriabelow,prioritisationcanbeimplementedby includinganobjectivethatmaximisesthetotalweightofthe matching(maxwcxc)andaddingasuitableweighttoa cy-cle foreach transplantof the particulartype. For instance, the Spanishsystemprioritises highlysensitisedpatientsby adding 30 pointsto the weight of a cycle for each donor withlessthana26% chanceoffindinga compatibledonor. Certain types of transplants (i.e., ones that require desen-sitisation) can be avoided by instead subtracting fromthe weightofacycle.
Forimprovingequalaccessinexpectation: 11. Prioritisehighlysensitisedrecipients(PL,ES,UK)
12. Prioritise blood-type-Orecipients,forwhichthedonorpool isthesmallest(PL)
13. Prioritise recipientsaccordingto(low)matchingprobability [seee.g.Keizeretal.2005)](BE,NL,PL,PT,ES,SE)
14.Prioritiserecipientsbasedonwaitingtime(inKEP/on dial-ysis)(ES,UK/NL,BE,PT,ES)
15.Prioritiseidenticalblood-grouptransplants(BE,NL,PT,ES) 16.Prioritisepairswithtype-ABdonors(ES)
Logistical considerations: Besides prioritising shorter cycles andchains,thefollowingobjectivescanbeexplainedby lo-gisticalreasons:
17. Prioritiserecipient-donorpairsfromthesameregion(ES) 18.Prioritisesolutionsthatinvolvemoretransplantcentres(NL)
Fairness:
19.Minimisingagedifferencesbetweendonoranddonorofthe matchedrecipient(NL,PL,PT,UK)
3.3.Constraints
Thelistofpotentialconstraintsareasfollows(seeFig.1). Toavoidcancellationorforlogisticsreasons:
20.Upperboundonthelengthofcycles(PL,PT,SE,ES,UK) 21. Upperboundonthelengthofchains(NL,UK)
Upper bounds on the lengths of cycles (chains) are imple-mented by not creating variables for cycles(chains) that are too long.
Fairnessconsiderations:
22.Providingstrictly betterdonors forcompatiblepairs, where thedefinitionsof‘better’varypercountryandrelatetoone ormoreofthecriteriamentionedabove(CZ,NL,ES,SE,UK) 23.Providingstrictlybetterdonorsforhalf-compatiblepairs(CZ,
SE)
24.Boundthedonor-donorordonor-patientagedifferences(PL, PT,UK)
25.End thealtruisticchainintheregionwherethedonor reg-istered(IT,NL,ES)
The aboverestrictions canbe implementedby notconsidering cyclesthatbreakthem.Forinstance,intheItaliansystemaltruistic chains that wouldend ina differentregion to the one in which thealtruisticdonorregistered wouldnotbe consideredatall (no variablexcwouldbecreatedforsuchachain).
3.4.Solutionmethods
Thesolutionmethodsforthemodelsformulatedforeachofthe EuropeanKEPsaregroupedandsynthesisedbelow.
• Edmonds’ algorithm. It is well known that the version in which the maximum cycle length is bounded by two, i.e., only pairwise exchanges are allowed, reduces to finding a maximum (weight) matching in an undirected graph. This problem is solvable in polynomial time, e.g. through Ed-monds’ algorithm. The Scandinavian KEP relies on the ap-plication of Edmonds’ algorithm, even though their model
is complicated by the introduction of half-compatibility (Andersson & Kratz, 2016). The UK KEP uses Edmonds’ al-gorithmasafirststep,tomaximisethenumberofpairwise exchangeswithintheselectedexchange.
• Graph heuristics. When allowing cycles (and chains) of bounded lengths greater than two, the resulting optimisa-tion problems are known to be NP-hard (Abraham et al., 2007). The Spanish KEP employs a polynomial-time, yet heuristicsolutionmethodthat searchesforcyclesoflength two andthree. The heuristic makes use ofEdmonds’ algo-rithm.ForfurtherdetailswerefertoBofilletal.(2017). • Exactmethodsusingthearcformulation.ThePolishKEPuses
thearcbasedintegerprogrammingformulationinthecase that enumeration of all cycles results in too many cycles. TheproposedapproachforthearcformulationIPwithcycle length constraint is asfollows. The cycle length constraint is relaxed and the remaining IP is solved using standard software.Ifthesolutionsatisfiesthecyclelengthconstraint it is optimal and is reported. Otherwise, constraints are addedto eliminatethe cyclesincludedin thesolution that aretoolongfromthesolutionspace,andtheresultingIPis againsolvedusingstandardtechniques.Thisprocessrepeats untilafeasiblesolutionisobtained.Moreover,itisexecuted hierarchicallytofirstobtainamaximumcardinalitysolution, and subsequently optimise a weighted objective function. The subsequent optimisation incorporates as a constraint that the matching found is of the previously determined maximumcardinality.
• Exact methods using the cycle formulation. The Portuguese KEPusesexactmethodstosolvethecycle-basedinteger pro-grammingformulation.The Polish KEP doeslikewise ifthe numberofcyclesisnottoolarge.Thisisdonehierarchically to first obtain a maximumcardinality solution, and subse-quentlyoptimiseaweightedobjectivefunction.TheUKKEP usesmultiplehierarchicallevelsthatare optimised sequen-tially(Manlove&O’Malley,article2.6,21pp,2014).Thefirst levelisoptimisedusingEdmonds’algorithm,whilelater lev-els usethe cycleformulation to potentially selectdifferent exchanges that still contain the same number of pairwise exchangesascomputed by Edmonds’ algorithm.An impor-tantpartofusingthecycleformulationisgeneratingall cy-cleswithin agraph. Thiscan be done using e.g.,Johnson’s algorithm.
• Enumerative methods.The Dutch and Czech KEPsin princi-pleenumerateallsolutions.Toreducethesearch space,the CzechKEP firstlydeterminesall stronglyconnected compo-nentsof the compatibility graph (in polynomial time) and subsequentlyenumeratesper component.BothKEPs subse-quentlypresentasortedlistofsolutions. Fromthislist,the computer program ofthe Dutch KEP only showsone opti-malsolution. The proposed exchangesinthis solutions are testedandinthecaseofapositivecrossmatchanew (next-best)solutionissought(andtested).
4. Discussionandconclusion
KidneyExchangeProgrammesarecomplexanddynamic health-care system components addressing the needs of patients with ESRD. Variation amonghealthcare systems - for instance regard-ingtheeffectivenessofthedeceaseddonationprogramme- subse-quentlytranslatesintovariationinpurposesofKEPs.Thedetailed country descriptions which we composed and synthesised show that thistranslates intovariation in KEP designs. The size ofthe countryandtheKEPpools, andthenumberofHLAlabsinvolved incrossmatchtestingalsoaffecttheKEPdesigns.Finally,KEP de-signsimportantlyvarywithdifferencesinethicalandlegal
frame-workse.g.regardingtheadmissionandintegrationofaltruistic do-nation. A first conclusion is therefore that each of the European KEPsisdesignedtofititsparticularcontextandshouldbeassessed within thiscontext.Byconsequence,an apparentlyadvanced KEP that servesits owncontextwell maybe apoormatchinanother setting. Forinstance, while some countries haveadvanced to in-crease the effectiveness of altruistic donation, altruistic donation isconsideredunethicalandforbiddenbylawinother (neighbour-ing) countries. Likewise, some countries prefer desensitisation as a treatment fora patient-donor pairover participating in a KEP, whileothercountriesconsiderparticipatingintheKEPpreferable. Still,KEPscan learnfromeachother andadvancebyadopting (and adapting) each other’s practices. Below we synthesise com-monlyapplicableadvancesasemergingfromthisstudy.Operations Researchcan contribute byprovidingsuch generaladvancements. In view of the conclusion above, however, we would caution against ‘one sizefits all’ ambitions,oragainst claimsthat certain models andmethods are better than others.Evenpresented em-pirical evidence should always be considered in context. Hence, Operations Research can also contribute by tailoringmodels and methodstothedemands,normsandvaluesofspecificcontexts.
Letusstartwithsomegeneralobservationsonthemodelsand methods used in the European KEPs. Integer Programming mod-elsappeartobethemostgenericmodelsinuse.TheUK,Portugal and Polandare robustly andeffectively solving their match runs using Integer Programming-based solution methods. Other coun-tries makeuse ofenumerative solution methods oruse heuristic approaches.Regardless ofthesemethods,wenoticethatrelatively fewcountriestendtodirectlyimplementthesolutionsprovidedby theoptimisationmethods.Manycountriesleavethedecision mak-ingtoacommitteebasedona(ranked)listofalternativesolutions’ toallowthecommitteetoweighinadditionalconsiderations.
A keyaimof allEuropean KEPsis tofacilitateasmany trans-plants as possible. Therefore, the primary goal of (most of) the models and methods proposed is to find maximum cardinality matchings.Yet,becauseoftheriskofcancellations(e.g.dueto pos-itive crossmatches),the practicalgoalis tomaximisethenumber oftransplantsthat can actuallybe conducted. Countrieshave tai-loreda varietyofapproachestoreduce thegapbetweenthe car-dinalityofthemaximummatchingandthenumberoftransplants thatcanactuallybeconducted.
Ageneric solution istorestrict the solutiontoonly consistof short cycles,ascyclesinvolving fewer pairs carry alower risk of cancellation. Hence, many programmesput upperboundson the cycle length (3 is a common bound) or prioritise shorter cycles. The usageofa singleHLA-lab can alsodecreasecancellationrisk, asitharmonisestestproceduresandenablesquickre-optimisation. CountrieswithacentralHLAlab(e.g.theNetherlands)may there-fore allow longer cycles and need not consider minimising cycle lengthahighpriority.LargercountrieswithmultipleHLAlabscan mitigate thefailure risks by selecting solutions withback-up op-tions,e.g.,three-cycleswithembeddedtwo-cycles(UKandSpain). Altruisticdonationcanalsogreatlybenefitthenumberof trans-plants.Afirstmainadvantage ofchainsovercyclesisthat simul-taneity is not considered to be required and thus chains can be longerthancycles.Asecondmainadvantageisthatthelastdonor inthealtruisticchaincanbekeptasabridgedonorforlater,thus extendingtheadvantagesintothefuture.
HLA-matching and age difference between matched donors and patientsare the two main factors influencing expectedgraft survival times. Hence, many countries maximise HLA-matchings and/or minimise age differences to improve the quality of the transplants. Some countries even include such quality considera-tionsthroughhardconstraintsonHLA-matchingoragedifferences. ForKEPswhichallowABO-incompatibletransplantation,quality can also encompass the numberof desensitisation transplantsin
theoptimisationobjectives.Matchqualityisalsoaconsiderationto includecompatiblepairs.Thepatientsfromsuchpairsmayfinda bettermatcheddonorthemselvesandtheirparticipationmayalso be beneficial toother participating pairs.To encouragetheir par-ticipation,KEPsthenmayguaranteequalityofmatchingfor recip-ientsfromsuchpairs(asisthecaseintheUK).
Inadditiontomaximisingthenumberoftransplantsandtheir quality,equityis of explicitimportance. Forinstance, manyKEPs restrict type Odonors todonate to type Opatientsto ensurean equitabletransplantprobabilityfortypeOpatients.Similarly, sev-eralKEPSprioritisehighlysensitisedpatientstoimprovetheirpoor matchprobability.Following comparablefairnessprinciples, some KEPsprioritisepatientswithlongwaitingtimes.
Ethicalconsiderations are not limitedto participants insingle matchruns.Fromapolicyperspective,theeffectivenessofKEPsis typicallyevaluatedovertimeandhenceforlargerpopulations.The relationshipsbetweenthe complexoptimisationmodels,methods andobjectivesemployedforsinglematchrunsandthelonger-term outcomesconsidered by policy makers is as yet not always well understood. This is certainly an area for further research where analysis on multiple, longitudinal, empirical data sets is called for.
Lastly,itis worthconsidering theemerging cross-national ini-tiatives. Three collaborations have already started: between Aus-tria and the Czech Republic, between Italy, Portugal and Spain, andbetweenSweden,NorwayandDenmark.Fornowthese cross-national KEPshavemainly consideredpatients left unmatchedin thenationalKEPs. Wemayexpect thecross-nationalKEPsto fur-ther expand their benefits in the near future when initiating to mergenational patientpoolsandto optimisethe resulting cross-nationalpatientpool.Thiswillbringaboutnewchallenges.Firstly, how to ensure that national regulations (constraints) and prior-ities (optimisation criteria) of each participating country are re-spected.Secondly,itshould benotedthat considerationofequity and fairness now not only apply to individual patients, but also topatient populationsfrommultiplecountries. Patientsfromone country might benefit more than patients fromanother country. Suchequityconsiderationsoccurringamongcountriesforma rela-tivelyunexploredarea.ThecurrentEuropeancross-national initia-tivescanplayaguidingroleindevelopingequitablecross-national KEPs and in the lively global scientific and policy discourses on this matter (Bozek et al., 2018; Delmonico & Ascher, 2017; European Union National CompetentAuthorities on Organ Dona-tion&Transplantation,2–18).
Acknowledgments
COST financed meetings (i.e., the travel and accommodation costs oftheauthors) through EU COSTAction 15210. Tommy An-derssonisfundedbytheJanWallanderandTomHedelius Founda-tion(P2016-0126:1)andtheRagnarSöderbergFoundation(E8/13). Péter Biró is supported by the Hungarian Academy of Sciences
(Research grants no. LP2016-3/2018 andKEP-6/2018), and by the
Hungarian Scientific Research Fund (OTKA) Grant No. K129086. Xenia Klimentova is supported by grant SFRH/BPD/101134/2014. David Manlove is supported by grant EP/P028306/1 from the
Engineeringand Physical Sciences ResearchCouncil.Ana Viana is supported by mKEP (PTDC/IIMGES/2830/2014). Frits Spieksma is supportedbyNETWORKS(GrantNo.024.002.003).
Supplementarymaterial
Supplementary material associated with this article can be found,intheonlineversion,atdoi:10.1016/j.ejor.2019.09.006.
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