European Journal of Operational Research ARTICLE IN PRESS

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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

^r

aHungarian Academy of Sciences, Corvinus University of Budapest, MTA KRTK, 1097 Budapest, Tóth Kálmán utca 17, Hungary

bPrince Mohammad bin Salman College, 7682 Hejaz Boulevard, King Abdullah Economic City 23965-2609, Kingdom of Saudi Arabia

cErasmus Universiteit Rotterdam, Burgemeester Oudlaan 50, 3062 PA Rotterdam, The Netherlands

dSchool of Computing Science, University of Glasgow, Sir Alwyn Williams Building, Glasgow G12 8QQ, United Kingdom

eDepartment of Economics, Lund University, SE-220 07 Lund, Sweden

fNational Health Service, Blood and Transplant, 500, North Bristol Park, Filton, Bristol, BS34 7QH, United Kingdom

gIKEM, Videnská 1958/9, 140 21 Praha 4, Czechia

hOrganización Nacional de Trasplantes (ONT), Calle Sinesio Delgado, 8, Madrid 28029, Spain

iNorthwestern University, Global Hub 3389, Evanston, IL 60208, USA

jDutch Transplantation Foundation, Plesmanlaan 100, CB Leiden 2332, the Netherlands

kNational Health Service Blood and Transplant, 500, North Bristol Park, Filton, Bristol, BS34 7QH, United Kingdom

lINESC TEC, Campus da Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, Porto 4200-465, Portugal

mKU Leuven, UZ Herestraat 49 - box 7003 11, Leuven 30 0 0, Belgium

nItalian National Transplant Centre, Via Giano della Bella, 34, Roma RM 00162, Italy

oHEC Management School, Université de Liege, N1 - rue Louvrex 14 - 40 0 0 Liége, Belgium

pDepartment of Mathematics and Computer Science, Eindhoven University of Technology, Groene Loper 3, 5612 AE Eindhoven, The Netherlands

qOrganización Nacional de Trasplantes (ONT), Calle Sinesio Delgado, 8, Madrid 28029, Spain

rINESC TEC and Polytechnic of Porto, Campus da Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

a rt 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 practice

a b s t r a c t

Thecomplexmulti-criteriaoptimisationproblemsarisinginKidneyExchangeProgrammeshavereceived considerableattentionbothinpracticeandinthescientificliterature.Whereastheoreticaladvancements arewellreviewedandsynthesised,thisisnotthecaseforpractice.Wepresentasynthesisofmodelsand methodsappliedinpresentEuropeanKidneyExchangeProgrammes,whichisbasedondetaileddescrip- tionswecreatedforthispurpose.Mostdescriptionsaddressnationalprogrammes,yetwealsopresent findingsonemergingcross-nationalprogrammes. Thesynthesisprovidesasystematicanddetailedde- scriptionofthemodelsandmethodstheprogrammesuse,revealingimportantcommonalitiesaswellas considerablevariationamongthem.Ratherthandistillingasinglebestpracticefromtheseresults,we findthatthevariationinmodelsandmethodsarisesbecauseofvariationincountrycharacteristics,policies,andethics.Thesynthesisedstateoftheartmaybenefitfuturenationalandcross-nationalinitiatives anddirectfuturetheoreticalcontributionswithinandacrosstheboundariesoftheOperationsResearch discipline.

∗ 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).

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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 problems 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 methods 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 andsyn- thesis of all models (including objectives and constraints) and solutionmethods(includingalgorithmictechniques) activelyprac- 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) eachoneisalsoclearlydevelopedwithinadifferentnationalcon- 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

The2016GlobalBurdenofDiseaseStudyidentiﬁeschronickid- neydisease(CKD)asthe11thmostcommoncauseofdeathglob- 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 operationalef- fectivenessoftheprogrammesandthehealthcaresystemsingen- 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- portedwaitinglistlengthsareforinstance2208patientsinScan- 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 donation (LKD) hasbetter long-termpatient andtransplantoutcomes (Hartetal.,2017;MacNeill,Casula,Shaw,&Castledine,2016;Wolfe etal.,2010)(werefer toReese,Boudville,& Garg,2015 foradis- cussion of donor outcomes). This, in combination with the relatively poor outcomes of dialysis and the shortage of deceased donororgans,ledtotheestablishmentofLKDprogrammesinEu- rope,tocomplementexistingDKDprogrammes.In2017,thenum- 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-

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tientanddonorasapairandalsorefertothepatientasarecipient, orasthespeciﬁedrecipientofthedonor.

Even when a patient finds a specified donor, however, transplantation of the kidney from this donor to the patient may be unfeasiblebecausethepatient(recipient)anddonorarenotmedi- 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,themorecompatiblefromaview- point of HLA matching.When fully alike (e.g. foridentical twins)wespeakofaperfectHLAmatch(Fulleretal.,2004;

Terasaki,Cecka,Gjertson,&Takemoto,1995).

• HLA-Crossmatchreferstothetesttodecidewhetherarecip- ienthasantibodiestotheHLAofthedonor(insigniﬁcantly highconcentration).Ifthisisindeedthecase,onespeaksof apositivecrossmatchwhichisseenasanindicationthatthe transplantationwillnotbesuccessful.Apositivecrossmatch isonlypossiblewhentheHLAmatchisnotperfect.(Below wealsoelaborateonvirtualcrossmatches.)

Originally,recipientsanddonorswereonlyconsideredcompat- 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-compatiblewhenthecross- matchis negativeandtransplantationbetween donorandrecipi- ent requiresdesensitisationto overcomeABO-incompatibility. Re- centmetaanalysesconcludethat outcomesofstateoftheartde- 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.SomeKEPsgivehighlysensitisedpatientspri- oritytoimprovetheirchancesofreceivingatransplant.

Otherfactorsthatinﬂuencetransplantationoutcomes,inpartic- ulargraftsurvival,relatetothequalityofthekidney-whichisfor instancecorrelated withdonorage-andthehealth oftherecip- ient-which inturniscorrelatedwithrecipient ageandtimeon dialysis.

Withthesereflectionsonoutcomesanddefinitionsofcompat- ibility athand, we cannow consider thepurpose ofKEPs. Origi- nally,KEPswereinitiatedtoproviderecipientswho donot match with their specified donor access to a compatible donor by exchange 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 morecom- 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 theseexten- 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,...,p_n

}

^ofrecipient-donorpairsp_i=(^ri,d_i),i=1,...,n wherer_idenotestherecipientandd_ithedonorofpairp_i.

• Exchange cycles: Initially KEPs sought to match recipient- donorpairstogether.Moreprecisely,theysoughttoidentify twopairs p_i=(^ri,d_i)^and^pj=(^rj,d_j)^such^that^recipient^ri

iscompatible withdonor d_j and -vice versa - recipient r_j iscompatiblewithdonord_i.Suchexchangesarecalledpair- 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 pairwise 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, p_i=(^ri,d_i), p_j=(^rj,d_j) and p_k=(^rk,d_k), and r_i is matched with d_j, r_j with d_k, and r_k with d_i. 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 (second)patientandsoon,withthelastkidneydonatedtothe waitinglist.Thus,altruisticdonorsmayinitiateanexchange involvingmultiplepairs,whichformachain.Aswasthecase forcycles,KEPsmayimposelimitson themaximumnum- 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

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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,whenonerecip- ientintheKEPreceivesakidneyfromadeceaseddonorand thenhis/herdonorstartsachain(asinthecaseofanaltru- isticchain)withthelastdonordonatingtoarecipientinthe DKDprogramme. Threesuchchainswere startedinItaly in thesummerof2018.The mostcommonmodalityforinter- actionwiththeDKDprogrammeisthroughaltruisticchains, asdescribedabove.

• Inclusion of compatible pairs: Some European KEPs explicitly restrictregistrationto incompatiblepairs, e.g.,Belgium, France andPortugal.Other countriesallow orevenencour- age compatible pairs to participate, to improve outcomes forthemselvesand/orforotherrecipients.KEPsthatenable such participationoftenprovideadditionalarrangementsto ensure thatcorresponding patientsarematched toa donor

whoisatleastasgood(tobedefinedbelow)astheirspeci- 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 systems,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 desensitisation. As a result, these countrieshavelarger pools which increases match probabilities and reduces waiting timesforpatients (Biró etal., 2019). Intheseprogrammes,

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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,onlyoneofthecorrespond- 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 simultaneityincyclesandhavesuccessfullyconductednon- simultaneous exchanges(suchas a7-way exchange conductedin theCzechRepublic).Asparalleltransplantationmaybringcapacity andlogisticschallenges,a simultaneityrequirementposesrestric- 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.1foranoverviewofchainandcyclelengthlimitationsimple- mentedbyEuropeanKEPs.

KEPsadmittinglongercyclesforwhichtheyperformalltrans- plants simultaneously may needor prefer to spread thesetrans- plantsacrossmultiplecentresbecauseofcapacitylimitations,asis thecasefortheDutchprogramme.

Anonymity mayform another reasonto involvemultiple centres. SeveralEuropean countries(e.g.the Netherlands,Spain, UK) requireanonymity,eitherlegallyorbyprotocol. Anonymityisdif- ﬁcult to ensure when performing surgeries formultiple pairsin- 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 labcross- 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.Threeexamplesforcrossmatchtestingandre- 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 exchanges,such that anembeddedpairwise exchange can go ahead ifthe three-way exchange cannot proceed (Manlove

&O’Malley,article2.6,21pp,2014).Afterthecrossmatchre- sultshavebeen obtained,theKEP performsasmanytrans- plantsaspossiblefromthematchrunsolution.

• Spain:Thereare multipleHLAlabsandtworounds oftest- 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.WhenassessingthewaysKEPsareor- ganised and exchanges are identiﬁed at match runs, it is there- foreappropriatetotakealongertermperspective.Thisholdspar- ticularly true whenincluding altruistic chains that span multiple matchruns.

Longer-termperformancecriteriaconsideredbyEuropeanKEPs are: total numberoftransplants performed, percentageof recipients inthe KEP who have received a transplant, average waiting timeuntilbeingmatched,qualityoflifeforrecipientsanddonors aftertransplant, graftsurvival times, recipient survival rates,and

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donorsurvivalrates(Anderssonetal.,2019).Moreover,theseout- comesare consideredovervarious recipientsub-populations,e.g., perbloodtype,orwithregardtohighlysensitisedpatients,toalso assessequityandfairnessconsiderations.

2.6.Internationalcollaborations

Asdescribedabove,internationalcollaborationiscommonprac- 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.

ThesameapproachisusedinthecooperationbetweenAus- 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(wherethepairsfromthelat- tercountriesjointheKEPsoftheformer).

3. Matchingmodels

This section presents the models andmethods used to ﬁnda solutionforasinglematchrunofa KEP.The objectivesandcon- straintsinthemodels,aswellasthedesignofthesolutionmeth- 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 recipient 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.Asecondclassisformedfromintegerprogrammingfor- mulations.Theseformulationshavebeenparticularlyhelpfultoad- vancesolutionmethods.Bothclasses,andthecorrespondingmod- elsandmethodsareconsideredinmoredetailbelow.

The classical model that is used to formulate the problem of ﬁndingsolutions foramatchrun that takecompatibilityinto ac- countistheso-calledcompatibilitygraph.ItisadirectedgraphD(N,

A),inwhichthereisanoden_i∈Nforeachrecipient-donorpairfor i=1,...,n.Thereisanarc(i,j)fromnoden_i∈N tonoden_j∈Nif the donor ofpair n_i is compatiblewith the patient of pairn_j.If ABOitransplantsarealsoconsidered,wemaydistinguishaspecial setofarcs A⊆Arepresentinghalf-compatibledonor-patient pairs.

Aselfloop,i.e.anarc(i,i)wheren_i∈N,emanatesfromeach(half-) compatiblerecipientdonorpair. Thecompatibilitygraphiscalled virtualcompatibilitygraphifthecompatibilitiesandweightsarees- timatedbasedonvirtualcrossmatchtestresults.

Multiple donors may also be registered for one patient, in whichcasewe have anarc (i, j) ifsome ofthe donors ofpairn_i iscompatiblewiththepatientofn_j.

WemaydistinguishaspecialclassofnodesN^a⊆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 nodesn_i∈N^a haveincomingarcsfrom everynoden_i∈NࢨN^a.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,amaximumcar- dinalitycyclepackinginD(N,A)nowreferstoamatchrunsolution withthehighestpossiblenumberoftransplants.

Each arc (i, j), can have a weight w_i_,_j to represent the util- ity of matching the donor frompair n_i∈N with the recipient of pair n_j∈N. This value can include clinical considerations as well as other priority-based contributions, such as matching type O donorstotypeOpatients.Moreover,nodesn_i∈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).

Allsuchweightsenablesolutionstobedistinguishedbasedonper- formancecriteriafortheKEPsandsubsequentlypermatchrun.

Asiscommon,thegraphmodelpresentedabovecanbeformu- latedasan integerprogram.Here we presenttwobasicformula- tions,referredtoasthearcformulationandthecycleformulation (Abraham,Blum,&Sandholm,2007).

Thearcformulationhasabinaryvariabley_i_,_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 y_i_,_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

w_i,_jy_i,_j (1)

s.t.

j

y_i_,_j−

j

y_j_,_i=0,

∀

ⁱ^∈^V ⁽²⁾

j

y_i,_j≤1,

∀

ⁱ^∈^V ⁽³⁾

yi1,i2+yi2,i3+· · · +yi_K−1,iK ≤K−1, foreachdirected

chainoflengthK (4) The cycleformulationuses a binaryvariable xc foreach cycle (and chain,seeabove) cof length atmostK. We denotethisset

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ofcycles byC^K.The weight ofacyclec isdenoted by wc, which can betakenasthesumoftheedge-weightsin thecycle,orcan bedeﬁneddifferentlyasmentionedabove.Findingallcyclesinthe graphcanbedoneusinge.g.Johnson’salgorithm.

max

c∈C^K

wcxc (5)

s.t.

c∈C^K,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 cycles 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 appliesdifferentupperboundsforchainsandcyclesarepre- 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 exchanges or to pairwise and three-way exchanges (Scandinavia, Spain, andthe UK).Other KEPsmay fixthese parameters explicitly 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,afterwhichconstraintsfol- 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 constraint.Wealsoincludeashortexplanationofasampleimplemen- 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 objectivefunc- 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 max

wcxc wherewcisthelengthofcyclec.

2. Minimising the length of the longest selected cycle, as longer cycles are more likely to result in positive crossmatches,andthusintransplantsthatwillnotbeperformed.

Selectingshorter cyclesis alsoimportantfor logisticalrea- sons(NL,UK).

Mathematically, this is formulated as minmaxwcxc where wc is thelength ofcyclec.This canbe implementedinan ILPbyaddingavariablem_iforeachi∈

{

¹,...,L

}

^where^L^is

thelengthofthelongestcycle. Constraintswillensurethat m_iwill takethevalue 1ifandonlyifacycleoflength ≥i isselected. Theseconstraints are implemented by ﬁrstlet- tingC_i bethe setofcycleswithlengthsatmosti(soC₃ is thesetofall cyclesof length2 or3),andthen addingthe constraints

m_i≥

c∈Cixc

|

^Ci

|

foreach i∈

{

¹,...,L

}

^. ^We ^can ^then ^minimise ^the ^function ^mitominimisethelengthofthelongestcycle.

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- mainingafterdeletingpositivecrossmatchesfromthesolu- tion,andalsoforlogisticalreasons.(ES,UK).

This can be implemented by the objective function max

wcxc 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 embeddedtwo- wayexchange.

Improvingtheoverallqualityofthetransplants:

6. Minimisingthenumberofimplied desensitisations inKEPs thatallowABOiand/orHLAitransplants(CZ,SE).

This can be implemented with the objective function min

wcxc wherewc isthenumberofimplieddesensitisa- tionsincyclec.

7. Maximisingthe(weighted)sumoftheHLA-matchingscores (CZ,PL,UK)withfocusingonDR-antigeninparticular(CZ).

wcxcwherewcistheweightedsumofHLA-matching scoresacrossalltransplantsincyclec.

8. Minimisingagedifferencesbetweenthedonorsandpatients (BE,PL,ES).

This can be implemented with the objective function min

wcxc wherewc isthesumoftheagedifferencesbe- tweendonorsandpatientsintransplantsincyclec. 9. Prioritisingpaediatricpatients(ES).

wcxc wherewc is thenumber ofpaediatric patients involvedinthecycle.

10. Prioritising patients that have not started dialysis yet (BE, PL).

wcxc wherewc=1ifandonlyifthepatienthasnot starteddialysis.

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Forthecriteriabelow,prioritisationcanbeimplementedby includinganobjectivethatmaximisesthetotalweightofthe matching(max

wcxc)andaddingasuitableweighttoacy- 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 desensitisation) 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/ondial- 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,thefollowingobjectivescanbeexplainedbylo- 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 implemented 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 thealtruisticchainintheregionwherethedonorreg- 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 optimisation 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’ algorithm.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-basedintegerpro- grammingformulation.The Polish KEP doeslikewise ifthe numberofcyclesisnottoolarge.Thisisdonehierarchically to ﬁrst obtain a maximumcardinality solution, andsubse- quentlyoptimiseaweightedobjectivefunction.TheUKKEP usesmultiplehierarchicallevelsthatare optimisedsequen- tially(Manlove&O’Malley,article2.6,21pp,2014).Theﬁrst levelisoptimisedusingEdmonds’algorithm,whilelaterlev- els usethe cycleformulation to potentially selectdifferent exchanges that still contain the same number of pairwise exchangesascomputed by Edmonds’ algorithm.An impor- tantpartofusingthecycleformulationisgeneratingallcy- 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 stronglyconnectedcompo- nentsof the compatibility graph (in polynomial time) and subsequentlyenumeratesper component.BothKEPssubse- quentlypresentasortedlistofsolutions. Fromthislist,the computer program ofthe Dutch KEP only showsone optimalsolution. The proposed exchangesinthis solutions are testedandinthecaseofapositivecrossmatchanew(next- best)solutionissought(andtested).

4. Discussionandconclusion

KidneyExchangeProgrammesarecomplexanddynamichealth- 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,KEPde- signsimportantlyvarywithdifferencesinethicalandlegalframe-

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workse.g.regardingtheadmissionandintegrationofaltruisticdo- 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 empirical 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 countries makeuse ofenumerative solution methods oruse heuristic approaches.Regardless ofthesemethods,wenoticethatrelatively fewcountriestendtodirectlyimplementthesolutionsprovidedby theoptimisationmethods.Manycountriesleavethedecisionmak- ingtoacommitteebasedona(ranked)listofalternativesolutions’

toallowthecommitteetoweighinadditionalconsiderations.

A keyaimof allEuropean KEPsis tofacilitateasmanytrans- 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.duetopos- itive crossmatches),the practicalgoalis tomaximisethenumber oftransplantsthat can actuallybe conducted. Countrieshavetai- loreda varietyofapproachestoreduce thegapbetweenthecar- 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)maythere- fore allow longer cycles and need not consider minimising cycle lengthahighpriority.LargercountrieswithmultipleHLAlabscan mitigate thefailure risks by selecting solutions withback-upop- tions,e.g.,three-cycleswithembeddedtwo-cycles(UKandSpain).

Altruisticdonationcanalsogreatlybeneﬁtthenumberoftrans- 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,KEPsthenmayguaranteequalityofmatchingforrecip- 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-nationalinitiatives. Three collaborations have already started: between Aus- tria and the Czech Republic, between Italy, Portugal and Spain, andbetweenSweden,NorwayandDenmark.Fornowthesecross- national KEPshavemainly consideredpatients left unmatchedin thenationalKEPs. Wemayexpect thecross-nationalKEPstofur- ther expand their beneﬁts in the near future when initiating to mergenational patientpoolsandto optimisethe resultingcross- 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.

Suchequityconsiderationsoccurringamongcountriesformarela- tivelyunexploredarea.ThecurrentEuropeancross-nationalinitia- 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- derssonisfundedbytheJanWallanderandTomHedeliusFounda- 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.