A new data driven model for post transplant antibody dynamics in high risk kidney transplantation

warwick.ac.uk/lib-publications

Original citation:

Zhang, Yan, Briggs, David, Lowe, David Philip, Mitchell, Daniel Anthony, Daga, Sunil,

Krishnan, Nithya, Higgins, Robert and Khovanova, N. A.. (2017) A new data-driven model for

post-transplant antibody dynamics in high risk kidney transplantation. Mathematical

Biosciences, 284 . pp. 3-11.

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ContentslistsavailableatScienceDirect

Mathematical

Biosciences

journalhomepage:www.elsevier.com/locate/mbs

A

new

data-driven

model

for

post-transplant

antibody

dynamics

in

high

risk

kidney

transplantation

Yan

Zhang

a

_,

_David

_Briggs

b

_,

_David

_Lowe

b,c,d

_,

_Daniel

_Mitchell

d

_,

_Sunil

_Daga

d,e

_,

Nithya

Krishnan

e

_,

_Robert

_Higgins

e

_,

_Natasha

_Khovanova

a,∗

a_{SchoolofEngineering,UniversityofWarwick,UK} b_{NHSBloodandTransplant,Birmingham,UK}

c_{DepartmentofHistocompatibilityandImmunogenetics,RoyalLiverpoolUniversityHospital,Liverpool,UK} d_{ClinicalSciencesResearchLaboratories,UniversityofWarwick,Coventry,UK}

e_{RenalUnit,UniversityHospitalsCoventryandWarwickshire,Coventry,UK}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Availableonline14May2016

Keywords:

Kidneytransplantation Antibodydynamics

Ordinarydifferentialequations Datadrivenmodel

Eigenvalue

VariationalBayesianinference

a

b

s

t

r

a

c

t

Thedynamicsofdonorspecifichumanleukocyteantigenantibodiesduringearlystageafterkidney trans-plantationareofgreatclinicalinterestastheseantibodiesareconsideredtobeassociatedwithshortand longtermclinicaloutcomes.Thelimitednumberofantibodytimeseriesandtheirdiversepatternshave madethetaskofmodellingdifficult.Focusingononetypicalpost-transplantdynamicpatternwithrapid fallsandstablesettlinglevels,anoveldata-drivenmodelhasbeendevelopedforthefirsttime.A varia-tionalBayesianinferencemethodhasbeenappliedtoselectthebestmodelandlearnitsparametersfor 39timeseriesfromtwogroupsofgraftrecipients,i.e.patientswithandwithoutacuteantibody-mediated rejection(AMR)episodes.Linearandnonlineardynamicmodelsofdifferentorderwereattemptedtofit thetimeseries,andthethirdorderlinearmodelprovidedthebestdescriptionofthecommonfeatures inbothgroups.Both deterministicandstochastic parametersarefoundtobesignificantlydifferentin theAMRand no-AMRgroupsshowingthatthetimeseriesintheAMRgrouphavesignificantlyhigher frequencyofoscillationsandfasterdissipationrates.Thisresearchmaypotentiallyleadtobetter under-standingoftheimmunologicalmechanismsinvolvedinkidneytransplantation.

© 2016TheAuthors.PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Kidney transplantation is proven to be the best treatment for renal failureandsuccessis dependenton thereactionofthe im-mune system primarily against human leukocyte antigen (HLA) proteins ofthetransplant.The HLAsystemisextremely complex; itisunusualtofindtwounrelatedindividualswiththesameHLA type and only a minority of the transplants in the UK are fully matched forHLA tissue proteins [1]. Conventionaltransplantation isfacilitatedbyimmunosuppressionwhichtargetscellular compo-nentsoftheimmunesystem.

AsignificantnumberofpatientsdevelopantibodiestoHLA fol-lowing exposure to non-self HLA from pregnancy, blood trans-fusion or previous kidney graft [2,3]. These antibodies exist as multiple isoforms but it is Immunoglobulin G (IgG) which is deemed to be most detrimental to transplant outcome [4]. Such

∗ _{Correspondingauthor.Tel.:+442476528242.}

E-mailaddress:n.khovanova@warwick.ac.uk (N. Khovanova).

IgG,termeddonor-specificantibody(DSA),whendirectedata cur-rentorprospectivedonorHLA,canpersistforyearsandarea bar-rier to transplantation because they can cause immediate, early, andlaterejection.Safetransplantationofpotentialrecipientswith highlevels ofcirculatingDSAsisan ongoingproblemresultingin prolonged waiting times fortransplantation [5]. Ideally, such re-cipients shouldreceive atransplant froman antibodycompatible donorbutbecauseofadonorshortagethisisseldompossible.

Innovativeclinical protocols and techniques have been devel-oped [5–7] toallow transplantation ofsuch highlysensitised pa-tientsbyremovalofDSAsimmediatelybeforethetransplant[2,8]. CompleteeliminationofpreformedHLADSAsisnotpossibleand, becauseofimmunologicalmemory,post-transplantDSA resynthe-sis can still result in severe acute antibody-mediated rejection (AMR) and an increased risk of graft loss. The mechanisms un-derlyingthecontrolofantibodyproductionarepoorly understood andtreatments giventopatients withAMRcan be ineffective. In recentyears,anumberofpublications[9–11]haveconfirmedthat HLAantibodiesarethemajorcauseofacuteAMRandchronicgraft

http://dx.doi.org/10.1016/j.mbs.2016.04.008

4 Y.Zhangetal./MathematicalBiosciences284(2017)3–11

failure. Eventhough the risk of acuterejection andchronic graft failure is positively correlated with high DSAlevels, the associa-tioncan varybetweenpatients. In theacute setting, transplanta-tionacrossveryhighDSAlevels mayresultin50% graftloss,but databasedonthecurrentlyusedantibodydetectionassayscannot reliablypredict theoutcome[12].Likewise, inthe chronicsetting thereisnotalwaysaclearrelationshipbetweentheoccurrenceof AMRandthedetectionofcirculatingDSAs[13,14].

Our group has investigated early DSAdynamics inthese high risk transplants because the nature of their response is likely to profoundlyaffectclinicaloutcomes[2,8,15].Wehaveobservedthat thedynamicbehaviourofpost-transplantDSAsvariesfromcaseto case,andevendifferentDSAsinthesamepatient(targeting differ-entHLA)show diversepatterns.Developmentofa strong mathe-maticalapproachtodescribethedynamicsofthepreformed DSAs hasnotyetbeenattempted.Thisisbecausewhiteboxmodels,i.e. physiological models, are not yet feasible dueto the complexity ofunderlyingimmunologicalresponsestotransplants.Data-driven models, on the other hand, require both an accurate method of measuringtheDSAsinhumanseraandanappropriate mathemat-icalframeworkforthedevelopmentofthemodelfromlimitedand complexsetsofdata.

The possible mechanisms underlying changes in the levels of DSAsarecomplexandtheDSAslevels cannoteasily bemeasured inthelaboratory.DSAlevelsmaychangebecauseofrisesandfalls inthe rate ofproduction. Thisitself could be relatedto changes inthe populations of antibody-producing cells(plasma cells and memory B lymphocytes), and these cells could be formed pre-transplantand/or recruitedfromlessmature lymphocyte popula-tionspost-transplant[16].FallsinthelevelsofDSApost-transplant areveryinteresting,asthesemayoccurmuchfasterthanthe ‘nat-ural’rateofantibodyclearancefromthebody (thoughttohavea halflifeofabout20–30days[17]).Mechanismsassociatedwith re-ductionsinantibodylevelscouldincludeabsorptionofantibodies ontoHLAmoleculesonthegraft[18]– itisknownthatthelevels ofHLAonagraftmayincreasepost-transplant,butthiscannotyet bequantified.Some HLAisshedbythegraft,so antibodiescould beabsorbedinthecirculation.It isknownthat onephysiological methodusedbythebody tocontrolantibodylevelsistoproduce antibodiesthat block other antibodies(idiotypic antibodies), and productionof idiotypic antibodiescould explain the falls in DSA post-transplant [19]. However, aswith other potential regulatory mechanisms,it is currently hard to measure idiotypic antibodies accurately.Thus,mathematicalmodellingofchangesinDSAlevels mayindicatewheretheeffortsinvolvedindevelopingnew labora-toryassaysmightbebestdirected,andonceappropriateassaysare available,themodellingmayhelpintheinterpretationofresultsof theassaysatdifferenttime points.Thiscould be particularly im-portantinrelationtofallsinDSAlevels,sincethisisakeyclinical objectivethatiscurrentlynotachievableinclinicalpractice.

It has recently been recognised by the transplant commu-nity[2,20] that post-transplant screening foranti-HLA antibodies couldbeanimportanttoolformonitoringoftransplantrecipients. Highly sensitive and specific assays using purified HLA protein havebeendevelopedin recentyears.This developmentinassays meets the increasing need for monitoring post-transplant DSAs

[21]andopensupopportunitiestodevelopdata-driven mathemat-icalmodelsfortheevolutionofantibodiesaftertransplantation.

A unique dataset with detailedantibody measurements span-ningthree to sixmonths, starting around ten days before trans-plantation has been obtained by our group. A previous analysis

[2] of these data revealed various patterns of antibody dynam-ics,bothwithorwithoutacuteAMR.SomeDSAtimeseriesshow a rapid rise duringthe first two weeks followed by a rapid fall toalmost undetectable levels, which then remain low.This find-ingisstriking: inmanyofthesepatients, theDSAshadpersisted

formany yearsbefore transplantation, andtherapies used exper-imentally have been unable to stop antibody production before transplantation.Abetterunderstandingofthisphenomenoncould thereforehavepracticalbenefits.

The aim of this work is therefore to describe the pathologi-cal early antibody response in mathematical terms and we hy-pothesize that thisapproach mightenablea more intelligent ap-plicationoflaboratorytestingandsuggest therapeuticapproaches to selectively control this antibody response and improve clini-cal transplantoutcomes. Totake full advantageof thedata avail-able, we havedeveloped a data-driven model based on differen-tialequations that reflects the continuousnature ofthe underly-ing immunological process [22].The usefulness of the modelfor classification between patients with and without AMR was also investigated.

Datafromthepatientsinthisserieswere analysedinrelation toasingleoutcomemeasure,namelytheoccurrenceofearlyacute AMR.Thisisakeyearly outcomeinantibodyincompatible trans-plantation(AIT),asitisassociatedwiththelevelsof immunosup-pression required in theearly post-transplant period, andis also associatedwithshortandlongtermgraftsurvival.

The structure of the paper is as follows. Section 2 gives de-tails on the data and presents visual analysis of the variety of dynamicantibodyresponses totransplantation.Section3explains themethodologyformodelformulationandparameterestimation.

Section4presentsthefinalmodelanddetailedanalysisofsystems parameters.Section5summarisestheresults,justifiestheneedfor further work andoutlines the relevance of the modelfor kidney transplantmanagement.

2. Datadescriptionandvisualanalysisofdynamicpatterns

Data from twenty-three patients who underwent renal AIT atUniversity Hospitals CoventryandWarwickshire (UK)between 2003 and2012were analysed inthis study.The data were com-prisedoftimeseriesofDSAevolution overaperiodofaboutten daysbefore andsixmonths aftertransplantation. Serumsamples forDSAanalysisweretakenalmostdailyinthefirstthreetofour weeks,asmostdynamicbehaviouroccursduringthatperiod,and sampling became more sparse later when the antibodiestended to be morestable. Antibodylevels were measured using the mi-crobeadassaymanufacturedbyOneLambdaInc(CanogaPark,CA, USA), analysed on the Luminex platform (XMap 200, Austin, TX, USA). The assay measures the Mean Fluorescence Intensity(MFI) whichcorrespondstoantibodylevelalthoughtheirrelationshipis linearonlyoveralimitedrange.Asdescribedin[2],whentheMFI value ishigherthan10,000AU(ArbitraryUnits)andbelowabout 1000AU,thelinearcorrelationbreaks.

Fig.1. MeasuredtimeseriesillustratingindividualDSAchangesintheno-AMRgroup.Markerscorrespondtoeachmeasurementpoint.MFI=meanfluorescentintensity.

Fig.2. MeasuredtimeseriesillustratingindividualDSAchangesintheAMRgroup.Markerscorrespondtoeachmeasurementpoint.MFI=meanfluorescenceintensity.

Visual examinationofthe time seriesrevealsdiverse dynamic behaviour ofDSAs.Figs.1and2showsome examplesofthe pat-ternsfromtheno-AMRgroupandtheAMRgroup,respectively.As some patientshad multiple DSAs, the casenumber in these fig-ures andin the corresponding text isfollowed by theDSA type. Forexample,inFig.2,patient36hadtwoDSAs,HLA-A24and HLA-DR17,comprisingtwodifferenttimeseries:case36HLA-A24(case 36A24forshort)andcase36HLA-DR17 (case36DR17forshort). Pretransplant antibody removalcan be seen to reduce total DSA levels dueto cyclesofdouble filtration plasmapheresis.Typically, betweentwoandfivealternatedaysessionswereperformed.

The initial drop is typically followed by a rapid rise in DSA which usually occurs witha lag of a few days after transplanta-tionandiscausedbytwofactors:plasmapheresisstoppingandan increasedrateofDSAsynthesisduetoan immunologicalmemory response. After thepeak levels adiversity ofdynamicpatternsis noticeable: antibody levels do not follow a commonroute, vary-ingfromcasetocase,andevendifferingfordifferentDSAsinthe samepatient.InsomecasesthereisarapidfallinDSAtoasteady state, correspondingto alow (almostzero)levelofDSA,andthis is typically reached within the first month after operation. Such patterns are observedin both no-AMRand AMR groups:case34

(bothB62 andB60) and case28in Fig.1, andcase36 DR 17in

Fig.2.Inothercasesthedynamicsofthefallafterthepeakare fol-lowedby anotherrise,andantibodiesdonotsettleatalow level within the first three months after operation: case 59 in Fig. 1, case36 A24 andcases 61 and69 in Fig. 2. They either demon-stratea slowdynamicaround acertain constantlevel(case61 in

Fig.2)orchangedramaticallyoverthefirstthreemonths(case59 inFig.1andcase69inFig.2).Thereisnoobviousrelationship be-tweenthesedynamicalpatterns,steadystatelevelsandthe occur-renceofAMRepisodes.Insomecases,asshownabove,lowsteady state levels are observed inthe no-AMR group andhigherlevels ordramatic changes are noticeablein the AMR group. There are alsocaseswiththeabsenceofAMRdespitehighlevelsofDSA,or presence of AMR despite low DSA levels. Finally, some patients (e.g.case36inFig.2)rejectedthekidney,buthadmultipleDSAs withonetype thatrose aftertheinitialfall post-transplant(A24) andanother type that kept falling to a low steady level (DR17). Thisvisualanalysisdemonstratesthat thereisnocertain associa-tionbetweenhigherlevels ofpost-transplant DSAandthe occur-renceoftherejectionepisodes.

6 Y.Zhangetal./MathematicalBiosciences284(2017)3–11

betweenthepatientswithandwithout theincidence ofAMR. In thisstudy,we areparticularlyinterestedintheDSAdynamics af-terthefirst peakvalue downtoan almostzerolevel,i.e.focusis onthetypicalpatternofarapidfallthatoccursinmostofthe pa-tientswithandwithoutAMRepisodes.Fallsintheserumlevelsof HLADSAsafterkidneytransplantationareofgreatclinicalinterest, asthey are associatedwithresolution ofrejectionandgoodlong termoutcomesinpatientsathighriskofgraftloss[15].

3. Modelsandmethods

3.1.Datafittingandmodelselection

As seen from the preliminary observations of the dynamic patterns, the HLA antibody response to the transplanted kidney is a complex immunological process, nonlinear and stochastic in general. Time series available for analysis are complex and one-dimensional:only one variableasa function oftime (MFI levels) isavailablerepresentingaresponseoftheentirebiologicalsystem toexternalstimuli.Thesecharacteristicsposeasetofchallenging questionswithrespecttotheorderofthesystemandthenumber ofparameters tobeusedinthemodel.Itisalsounclearwhether the system equation should be linear or nonlinear, stochastic or deterministic,andwhatwouldbethemostappropriate modelling approachtoidentifysystemparametersinthesituationwhereno preliminaryknowledgeofthemodelisavailable.Althoughweonly considerthefallingpartofMFIleveldynamics,alltheabove ques-tionsremain.

3.1.1. Exponentialfitting

ItcanbenoticedthatthefallingMFIdynamicsofHLADSAafter thepeakvalueisarelaxationprocess,thesimplesttheoretical de-scriptionofwhichisanexponentiallaw.Initiallythecurve fitting tool(Cftool)inMatlab[24]wasusedtofiteachofthethirty-nine DSAs.Someofthetimeserieswerecorrectlydescribedbythis ap-proach;however,theuseofsuperpositionofexponentialfunctions couldnotcorrectlydescribeallthecaseswithandwithoutAMRin ourcohort.Asthenext step,insteadofexponentialfunctions, i.e.

solutionsof dynamicequations, dynamicmathematical models in

theformofdifferentialequationswereconsidered.

3.1.2. Formofthemodel:linear/nonlinearandstochasticterms A generalformofan nthordernonlinear differentialequation withcoefficientsinthe formofa polynomial functionhave been considered.Initially two stochasticterms wereincluded to repre-sent noise inthe systemequations. Measurement noise isadded dueto uncertaintyin measured data,and thedynamic noise ac-countsforanyotherhiddenpropertiesnotcapturedbythemodel. Thus,DSAfalls aftertheinitial rise (to a peak level)in theearly post-transplantperiodcanbedescribedbythefollowingmodel:

dn

dtnxt+ n−1

i=0

fi+1

(

xt

)

di

dtixt+f0

(

xt

)

=

η

t (1)

yt=xt+

ε

t (2)

Eq.(1)isanevolutionequationofnthorder,wherextisafunction oft thatdescribestheMFIdynamics,andyt isthemeasured MFI timeseries.

η

tissystemnoise,and

ε

t ismeasurementnoise.Each noisewasmodelledasGaussian-distributedwhitenoisewithzero mean and intensity (variance) of I_η and I_ε, respectively. f_i+1

(

xt

)

(

i=0,1,...,n−1

)

are polynomial functionsof xt. The derivative of order zero of xt is defined to be xt itself. f0(xt) is defined as

−

θ0

forconvenience. Theorderofthesystemequation nistobe decidedtogetherwithunknownparametersoffunctionsfi+1(xt). n initialconditionsarerequiredtoobtainaclosedformsolution.

ModelMn constituting Eqs.(1)and(2)covers avariety of dy-namic patternsdependingon the orderof the systemn. Amore complex model maybe able to explain a wider range ofsystem behaviourinthedataattheriskofoverfitting.

3.2. Modelandparameteridentification

In the current work, for DSA time series, nonlinear and lin-ear stochastic dynamic hierarchical models were developed us-inga variationalBayesianinferenceapproach [25]forbothmodel andparameteridentification.Boththeformandparametersofthe modelswere identified usingtheSPM9toolbox [26] (freely avail-able online) for MATLAB [24]. This variational Bayesian toolbox

[26]allowsaccountingforbothtypesofstochasticterms: measure-mentnoiseandsystemnoise.

Starting from the first ordermodel M1 (n=1 in Eq.(1)), the

order n wasincreased until the model Mn fitted the data suffi-ciently well – satisfy the criteriagiven in Section 3.3. The varia-tionalBayesian learningalgorithm [25] wasmodifiedto our spe-cific data to calculate probabilities p(yt|M) (where M is M1, M2,

…,Mn) ofobservingthe time seriesyt givendifferentmodels M, so thatthe modelwiththe highestvalueof p(yt|M) could be se-lected forthat specific DSA time series.Attention hasto be paid to the features in the dataset that can be explained by a model witha higherorder butcannot be explained by the modelwith alower order,andtodecideifthefeaturesaregeneralenoughto makethefinaldecisionontheorderforallDSAtimeseriesunder investigation.

Foreachmodelcandidate,the valueoftheprobabilityp(yt|M), which is also referred to as the model evidence [27], was ap-proached by iteratively optimising the states of the system and model parameters until a local maximum value of p(yt|M) was reached. This procedure is embeddedinto the variation Bayesian optimisation algorithm[27].Briefly,to infer multipleelementsof the hierarchical model, i.e. system states, parameters (related to thedeterministictermsintheequation),andhyperparameters (re-lated to the stochastic terms) each element is optimised one by one whilethe restofthe elementsare kept fixed.There are two stepsinthisoptimisationprocedure.Firstly,assumingtheelements (parametersandthestates)are conditionallyindependentofeach other, the combined distribution can be factorised into indepen-dent partitions of each element distribution, which is known as themean-fieldapproximation,i.e.thecombineddistributionofall theelementswasapproximatedby theproduct ofindividual ele-mentdistributions[27].Secondly,thedistributionofeach individ-ual parameter/statewas approximated by the first two moments (mean and variance) known as Laplace approximation [27]. The mean-fieldapproximationandtheLaplaceapproximationallowfor an iterativeupdate of the parameters andthe statesby applying variationalcalculus.Thelogarithmofp(yt|M)isknownas‘free en-ergy’F

(

θ

,yt

)

,a termborrowedfromstatisticalphysics [25].The freeenergywasmaximised,and,amongothercriteria(normalised rootmeansquare errorandthestabilityoftheimmuneresponse, bothofwhicharediscussedinthenextsection),definedthe good-nessoffit.

This Bayesian approach not only provides the most probable values of the parameters but also accountsfor the uncertainties intheparameters.Thepriorinformationregardingtheparameters isalso takenintoaccount. Such informationon possible parame-ter valueswasnot available tous,andthereforethemeanvalues ofthe parameter priorswere set to zero.Toallow the algorithm to search in a relatively wide region forthe optimal parameters, allvariancesweresettobe104_{,i.e.priorswithwidedistributions}

informative Jeffreys priors, asdescribed in[28], were chosen for the precisionsofthe noise,withboth shape andrateparameters setto 1.The initialconditionswere allmodelledasGaussian dis-tributions. Theprior means oftheinitial conditionswere defined from themeasurement time series,and theprior variances were setto104_.

3.3. Modelselectioncriteria

Thefollowingfourcriteriawereappliedtoidentifythebest fit-tingmodel.

1. ThefreeenergyFhasbeenmaximisedbytuningsystem

pa-rameters in an iterativemanner foreach model.Note that decisionmakingbasedonthecomparisonoffree energyof anytwomodels withdifferentorders couldbe problematic duetotheheavy penalisationofthemodelcomplexity em-bedded inthe variationalBayesian method asexplainedin

[27]. Increasingthe order ofthe systemby one would not onlyincreasethedegreeoffreedomintheparameterspace, but also increase the dimension of the system states.This leadstoadramaticdecreaseinthefreeenergy,whichcould bean order(orseveralorders)greaterthanthefree energy differencebetweenmodelsofthesameorder.Therefore,the freeenergycriteriawasonlyusedtocomparethemodelsof the sameorder. Formodels with differentorders, criterion 2,asbelow,wasutilised.

2. Normalised Root Mean Squared Error (NRMSE) was used to

comparethe modelswithdifferentordersforeach individ-ualtimeseries.Inferredparameters

θ

iwereappliedbackto thesystemequationtogeneratetimeserieswithout stochas-tic terms,i.e.deterministic solution. Notethat because pa-rameters were identified in the form of normal distribu-tions, the most probable(mean) value of parameters were plugged into the system equation. Root Mean Squared Er-ror(RMSE)betweenthemeasurementMFItimeseriesytand theinferreddeterministictimeseriesyˆt canbecalculatedas follows:

RMSE=

_n

t=1

(

yˆt−yt

)

2

n (3)

NRMSE accountsfor the different heightsof the peaks for

each DSA time series and is found by dividing the RMSE by themaximalMFIvalue foragivenDSAtimeseries.The model with the lowest value of NRMSE describesthe data most accurately. For the model to be deemed satisfactory, NRMSEshouldnotexceedthevalueof0.15(or15%)asitis knownthat theinter-assaycoefficientofvariabilityforDSA measurementsisaround10–30%[29].

3. Genericform.Astheentireaimwastofindamodelcapable ofcapturingthecommonpatternsinalltimeseries,amodel that could onlydescribe some of theDSA time series was disregarded.

4. System stability. The model has to have a unique stable

steady state, which impliesthat the system’s response de-cays with time. This has been checked via calculations of the real parts of corresponding eigenvalues which have to benegativeforstability.Note,eventhoughthesteadystate oftheimmunehomoeostasiswasdisturbedby transplanta-tion,theantibodylevelssettledrapidlytoanewsteadystate exceptfortheextremecases(examplecase69HLA-DR53in

Fig.2),butconsiderationofsuchcasesisoutofthescopeof thiswork.

3.4.Statisticalanalysis

Statistical analysis of model parameters was performed using theWilcoxon ranksumtest.The nullhypothesis ofnodifference betweenthe groupsof interest wastestedat the 5% significance level,andtheresultsarepresentedasp-values.Statisticalanalysis ofthedifferences inNRMSE betweentwomodels wasperformed usingtheonesamplet-test,whichassesses thenormalityofdata withzeromeanandunknownvarianceatthe5%significancelevel. Theresultispresentedasp-values.

4. Resultsanddiscussions

4.1. Modelselection

4.1.1. Comparisonoflinearmodelsofdifferentorders

Linearmodelswithdifferentsystemordersareconsideredfirst.

Eq.(1)in Section 3.1.2 transforms intoa linear differential equa-tion when f_i+1

(

xt

)

are constants, with constant parameters

θ

i

(

i=0,1,...,n−1

)

,where fn

(

xt

)

=

θ

n,...,f2

(

xt

)

=

θ2

,f1

(

xt

)

=

θ1

. f0(xt)isdefinedas−

θ0

forconvenience.

Afirst orderlinear modelwasconsidered first,andit didnot show good performance. Then linear models withhigher system orders were investigated. In this section, we present the results ofsystemandparameteridentificationbycomparingsolutionsfor linearfirst,secondandthirdorderdynamicequationsonly:

Model1

(

M1

)

:

dxt

dt +

θ

1xt−

θ

0=0 (4)

Model2

(

M2

)

:

d2_x

t

dt2 +

θ

2

dxt

dt +

θ

1xt−

θ

0=0 (5)

Model3

(

M3

)

:

d3_x

t

dt3 +

θ

3

d2_x

t

dt2 +

θ

2

dxt

dt +

θ

1xt−

θ

0=0 (6) Noteifthethirdorderequationhadnotbeensuccessful,the pro-cedure would have continued to account for nonlinearities (pre-sented in Section 4.1.2) first and then increase the order of the systemuntilasuitablesolutionisfound.

Initiallynot onlythemeasurement noise

ε

t (asinEq.(2))but alsothesystemnoise

η

t(as inEq.(1))wasincludedinthe mod-els.ItwasfoundthatforallDSAtimeseries,modelswithout sys-temnoise havelarger freeenergycomparedwiththecounterpart mathematicalrepresentationscontainingbothtypesof stochastic-ity.The benefit– improved fitting– obtained by usingthe more complex model with system noise does not exceed the penalty introduced by adding two degrees of freedom in the parameter space.Therefore,we excluded thesystem noise fromthemodels andthisisreflectedonthezerorighthandsideoftheEqs.(4)–(6). TypicalfittingsforfourDSAtimeseries,onefromtheno-AMR groupandtheotherthreefromtheAMRgroup,bythethree sug-gestedmodels(4)–(6)are showninFig.3.Theresultsformodels M1 –M3 inFig.3(a)and(c)showawinningmodelcandidateM3.

Eventhough (a)isfromapatient intheno-AMR groupand(c)is froma patient in theAMR group,both time series show oscilla-tionsafterday30.M1failedtodescribethedynamicsofbothtime

series as indicated by large NRMSE values in Table 1: NRMSE₌ 0.272 and NRMSE=0.090. M2 successfully described the initial

fallsforboth time series,butfailedto capturetheoscillations in DSA after day 30, which is also confirmed by the large NRMSE valueof0.053 and0.096(Table1).M3 capturedsuccessfullyboth

thefallingpartandthelatertrendwithsmallerNRMSE valuesof 0.014and0.053.Fig.3(b)exhibitsdifferentdynamicswithacluster ofdataaroundday20.Thisisacommonfeature observedinthe majorityoftimeseriesinboth AMRandno-AMRgroups, and re-quiresspecialattention.Thetemporarystalloffallingcouldnotbe expressedby using M1 orM2;however, M3 successfullydepicted

8 Y.Zhangetal./MathematicalBiosciences284(2017)3–11

Fig.3. TypicalfittingresultscomparedamongthethreemodelsM1–M3for(a)HLA-B60(case52)forapatientfromtheno-AMRgroup;(b)HLA-DRB3∗01forapatient (case14)fromtheAMRgroup;(c)HLA-A32forapatient(case16)fromtheAMRgroup;(d)HLA-A2forapatient(case17)fromtheAMRgroup.Themeasuredvaluesare indicatedbycircles.

Table1

Summary of the NRMSEvalues for 3 models correspondingto the four ex-ampledatasetsinFig. 3 .

NRMSE

M1 M2 M3

(a) 0.272 0.053 0.014 (b) 0.083 0.085 0.013 (c) 0.090 0.096 0.053 (d) 0.088 0.071 0.073

in Fig. 3(b).Further, the fittings by M1 and M2 were almost

in-distinguishableafterday70,andbothmodelsunderestimatedthe settlinglevel ofDSAs. The fittingby M3 otherwise correctly

esti-matedthe settling levelandgave a better descriptionof another clusteredregionaroundday30(seemagnifiedbox2inFig.3(b)). Thus,M1andM2wereruledoutbasedontheirincapabilityof

de-scribing the important features, and the higher order model M3

waschosen.

The sameapproach wasappliedto all theother DSAtime se-ries.In32outof39cases,theNRMSE valueofM3 wasthe

small-estamongthethreemodels.Intheother7cases,theNRMSEvalue ofM2 wascomparablewiththeNRMSE value ofM3.An example

isshowninFig.3(d)wherethefittingsbyM2andM3 were

indis-tinguishablefromeachother,withNRMSEvaluesgiveninTable1. Tocompare theNRMSE values betweenM2 and M3 across all 39

cases, the NRMSE value of M3 was subtracted from the NRMSE

of M2, and the differences for all time series are shown in the

boxplotFig. 4. The differences in NRMSE betweenthe two mod-elswere testedby the one-samplet-testat thesignificance level of0.001,andthemeanvalue wasfound tobe significantlylarger thanzero.Therefore,M3 wasselectedasthebestmodelacrossthe

cohortwiththe dynamicequation inthe form(6).Note that the NRMSEM₂−NRMSEM₃ values for the seven cases with very close

NRMSEvalues,mentionedabove,arelocatedaroundthezerovalue

inFig.4.

Fig.4. BoxplotofthedifferencebetweentheNRMSEofM2andNRMSEofM3.

4.1.2. Nonlinearversuslinear

Nonlinearity was introduced into the second order model in the form of polynomial nonlinear coefficients f1(xt) or f2(xt) in

Eq.(1)usingtheapproachof[23,30,31].Itwasacknowledgedthat a linear description is preferable over nonlinear if this does not increasethe numberofunknown parametersdramatically. Conse-quently,themaximal numberofunknownparameters inthe sec-ond order nonlinear model waskept comparable with the num-ber ofparametersofthe thirdorderlinearequation, i.e.nomore than 4.Underthiscondition, two nonlinear modelswere consid-ered:modelNM1withnonlinearityinthedampingterm(f1

(

xt

)

= k1xt+

θ1

, f2

(

xt

)

=

θ2

) and model NM2 with nonlinearity in the

xtterm(f1

(

xt

)

=

θ1

, f2

(

xt

)

=k2xt+

θ2

).Thecorrespondingsystem equationsareasfollows:

NM1:

d2_x

t

dt2 +

θ

2

dxt

dt +

(

k1xt+

θ

1

)

xt−

θ

0=0 (7)

NM2:

d2_x

t

dt2 +

(

k2xt+

θ

2

)

dxt

dt +

θ

1xt−

θ

0=0 (8)

Anexampleofthe fittingsofthetime seriesfromFig.3(c)by nonlinear models NM1 and NM2 is shown in Fig.5. From Fig.5,

neither NM1 norNM2 capturedthe dynamicfeatures ofthe time

series.TheNRMSE criterionwasappliedhereformodelsof differ-entorder:bothNRMSE values(0.080forNM1 and0.067forNM2)

Fig.5. FittingresultscomparedbetweenthetwononlinearmodelsNM1andNM2 andthelinearmodelM3forthetimeseriesshowninFig. 3 (c).Themeasuredvalues areindicatedbycircles.

by NM2 leadstoanunstable solution.Therefore,linearmodelM3

(Eq.6)outperformed bothNM1 andNM2,andwaschosenasthe

finalmodel.Thisexampleisatypical(representative)fittingforall theothertimeseriesinthecohort.

4.2. Analysisoftheinferredparameters

The inferred parameters (

θ

0,

θ

1,

θ2

and

θ3

) of the selected

model M3 have been compared between the two groups (AMR

andno-AMR)formeaningfuldifferences.Theresultsarepresented in Fig. 6(a)–(d) in the form of boxplots. For all four parameters, the ranges of the parameter values are much wider in the AMR group comparedwiththeno-AMR group,indicatingmore diverse dynamicbehaviourofDSAsintheAMR group.TheWilcoxonrank sumtestshowedstatisticallysignificantdifferencesinthemedian values betweenthe AMR and theno-AMR group forall four pa-rameters, which confirmed the results of our preliminary study

[23]withfewercases.

Even though the values ofthe parameters donot havedirect clinical interpretations, which is one of the main drawbacks of data-driven modellingin biomedicalresearch, a certain combina-tion ofthe parametersindicates importantfeaturesofthe system underinvestigation. The ratio

θ

0/

θ

1 fromEq.(6) definesthe

set-tling level ofDSA,which isof clinical interest.Kidney transplan-tationconstitutesamajordisturbanceintheimmunesystem, and thesystemshouldsettledowntoanewhomoeostaticequilibrium afterthe transientresponse tothetransplanted organ.A success-fultransplantationisusually characterisedbyanewstablesteady statewithlowDSAlevels(ideallyzero,orbelowthelimitof

detec-tionoftheassay).FromthecomparisonbetweentheAMRand no-AMRgroups,themajorityofthesettlingMFIvaluesinbothgroups arelessthan1000AU,indicatinglowDSAsettlinglevels.The high-estsettlinglevelintheno-AMRgroupis3862AU,comparedwith thelevel of5783AU intheAMR group. The lowestsettling level intheno-AMR groupis22AU,comparedwiththelevelof27 AU intheAMRgroup.Thereisnosignificantdifferenceinthemedian valueof

θ

0/

θ

1 betweenthegroupswitha p-value of0.5(300AU

in the no-AMR group, 425 AU in the AMR group), which means that aDSAtime series fromtheAMR group doesnot necessarily havea highersettling level. However, significant difference in

θ

0

and

θ1

separatelybetweenthegroupsshowninFig.6impliesthat thedynamicbehaviour ofDSAsinthe AMRgroup mightbe con-trolledbymorecomplexanddiverseunderlyingmechanisms.

Such detailedanalysisofthe parameters ofthe models devel-opedallows forenhancedunderstanding oftheclinical character-isticswhicharemostimportantforsuccessfuloutcomeinthishigh riskformoftransplantation.Ourfindingsmayfacilitatethe forma-tionofanaccurate pre-transplantriskprofilewhichpredictsAMR andallowsthecliniciantointerveneatamuchearlierstage.Given that AMR in theearly post-transplant period has beenshown to lead to worse long-term graft outcome any strategy to prevent earlyAMRwillbeofgreatbenefittothepatients[15].

Noiseaccountsforboth measurement errordueto inaccuracy in the MFI readings, and the perpetual actions of many unac-countedforfactorsthatinfluencetheevolutionofthesystem.The noiseintensitiesI_εwerecomparedbetweentheno-AMRandAMR groups. Our preliminary study [23] showed a smaller and more compact range of the noise intensities from the no-AMR group with9 time seriescompared with the AMR group with12 time series(shownin Fig.5of[23]). Limitedby thenumbers ofcases availablethen, theWilcoxonrank sumtest showednosignificant differenceinthe medianvaluebetweengroupswitha p-value of 0.08.Thisstudy,onalargercohortwithalmosttwiceasmanytime series,confirmedthe previousobservationwithasmaller p-value of0.01,indicatingasignificant differenceinthemedianvaluesof thenoiseintensitybetweengroups.

Thesquarerootofthenoiseintensity

I_ε,whichisanabsolute errorvalue,shares thesameunit astheMFIlevel.Intheno-AMR group with an average MFI peak height of 5716 AU, the median andrange(inbrackets)for

I_ε were159(5–353)AU.IntheAMR group withan average MFIpeak height of8502 AU, the median andrange(inbrackets)for

I_εwere 253(34–1425)AU.Asmaller noise intensity and more compact range of values across the

a

b

c

d

10 Y.Zhangetal./MathematicalBiosciences284(2017)3–11

no-AMRgroupis noticeable.Eventhough theassay usedto mea-surethe DSAlevel inbothgroupswasthesame,therelationship betweenMFI measurement andthe antibody leveldeviates from linearity asthe antibodylevel approaches10,000 AU. The higher antibody peak values in the AMR group can therefore introduce anadditionalsourceofmeasurementerrorcomparedwiththe no-AMR group,explainingthewider rangeandgreater magnitudeof noiseintensityseenintheAMR group.Anotherexplanationcould beadifferentlevelofmodelimperfectionbetweenthetwogroups. The higher level ofnoise in the AMR group could be caused by more and/or stronger factors unaccounted for by M3. Also it is

worth noticing that the priors for the noise intensity applied in theinferencemethodarechosentobeweaklyinformative.Amore informativepriormaylimittheflexibilityofthemodel,buta care-fullychoseninformativepriorcouldimprovetheestimationof de-terministicparametersandparametersrelatedtonoisedescription. Thechoiceofthepriorsisnotstraightforward,andanappropriate methodologyisunderdevelopment.

Insingleantigenbeadmeasurements,anothermeasure,termed inter-assaycoefficientofvariability(CV)isoftenusedtoindicatethe measurementuncertainty.Itisdefinedastheratioofthestandard deviationandthemeanvalueofseveralmeasurementsusing sep-arateassays.In[32],theinter-assayCVwaslargerthan20%when themeasurementsfromsevendifferentlabswerecompared.Inour model,consideringthemedianvalueof

I_εandthemedianvalue ofMFImeasurements,the medianCV is13%and14%forthe no-AMR andAMRgrouprespectively,whichislessthan20%givenin

[32].

4.3.Eigenvalues

TheevolutionequationEq.(6)canbetransformedintothethird orderlinearstatespacemodeloftheform

_x˙ t

¨

xt

...

xt

=

_{0 1 0}

0 0 1

−

θ

1 −

θ

2 −

θ

3

_x

t

˙

xt

¨

xt

+

₀

0

θ

0

(9)

ThesolutionofEq.(9)isdefinedby theeigenvalues

λ1

,

λ2

,

λ3

of the3_×3matrix,thecorrespondingeigenvectorsandthreeinitial conditions.The sumofthe eigenvalues defines thedivergence of thevectorfield(phasevolumeV(t))inthestatespace[33]:

V

(

t

)

=V0e(λ1+λ2+λ3)t=V0eRt, (10)

whereRcanbeinterpretedasthedissipationrateofDSAs.Forall thetimeseriesinthecohort,thedissipationrateislessthanzero, whichmeansthatthephasevolumeshrinks.

The eigenvalues foreveryDSAtime serieswere calculated us-ingthe inferred parameters

θ

1,

θ

2 and

θ

3. EachDSA time series

inthecohortischaracterisedbythreeeigenvalues,oneofwhichis real,

λ1

,andtwoof whicharecomplexconjugate,

λ2

_,3=

λ

r±i

λ

i. All eigenvalues

λ1

and the real parts of

λ2

and

λ3

were nega-tive,confirmingthatthesystemgeneratesstablesolutionsforeach DSAtype,whichsatisfiescriterion4inSection3.3.Thesystem dy-namicsforeachDSAdemonstrateadecaywithsomeoscillations, thefrequency ofwhich isdetermined by

λ

i. The dissipation rate isdetermined by thereal parts of the eigenvalues:R₌

λ1

₊2

λ

r. The characteristicdissipation rateR takes into account the over-alldecayalongthe pathfromthepeakvalue downto thesteady state.Thesteadystateofthesystemisafixedpoint,whichserves asanattractor.TovisualisethedynamicsofDSAs,phaseportraits havebeenplottedforanAMRcase(Fig.7(a)),andano-AMRcase (Fig.7(b)).Thetrajectoriesstartfromtheinferredinitialstatesand evolutetothefixedpointsinaspiralmannerinthephasespace.It canbeseenthatthedissipationrateintheAMRgroup(Fig.7(a),R

(a)=−0.81days−1)isfasterthan intheno-AMRgroup (Fig.7(b), R₍b)=−0.27days−1).

a

b

Fig.7. PhaseportraitsofthethreedimensionalsystemfortwoDSAtimeseries,(a) fromapatientintheAMRgroupand(b)fromapatientintheno-AMRgroup.The timedifferencebetweentwoconsecutivemarkersisoneday.

Thedissipationratesandfrequencies ofoscillationswere com-paredbetweentheAMRandno-AMRgroupsfor39timeseries.In theno-AMRgroup,themedianandrange(inbrackets)forRwere

−0.42(−0.66− :0.25) days−1.Inthe AMRgroup,the medianand range(in brackets) forRwere ₋0.79(₋3.88₋: 0.15) days−1_.The

comparisonof the dissipation ratesR betweenthe groupsfor all thetime seriesconfirmeda significantlyfaster dissipationrateof DSAsintheAMRgroupthanintheno-AMRgroupwithap-value of0.04.

The imaginarypartsofthe eigenvaluesbetweenAMR and no-AMR groupsalso showedsignificant differenceswithap-valueof 0.03.Intheno-AMRgroup,themedianandrange(inbrackets)for

λ

iwere 0.20(0.01: 0.34)days−1.Inthe AMR group,themedian andrange(inbrackets)for

λ

iwere 0.28(0.05:0.80)days−1.The larger valuesof the imaginaryparts inthe AMR group represent a higherfrequency of oscillation,which indicates a stronger reg-ulationduringthetransientantibodyresponseforthepatientsin the AMR group.One hypothesis ofthis regulationisthe possible productionofa secondaryantibody(such asanti-idiotype) which targetsthedramaticallyincreasedDSA,resultinginabattlingforce betweenthe DSA production andsecondary antibody production

[19].

Notethat theprevious studyby Higginsetal.[2]investigated thechangeinabsoluteMFIvaluesandinthemeanpercentagefalls intheAMRandno-AMRgroups,andsuggestedthatthefallswere greaterintheAMR groupcomparedwiththeno-AMRgroup.Our results show that not only is the difference inthe MFI level be-tweenpeakandsteadystatedifferentbetweenthetwogroups,but therateofchangeofthefallisfasterintheAMRgroup, also im-plyingastrongerregulationmechanisminthisgroup.

5. Conclusions

ob-served from the clinical data,showed no difference betweenthe AMR and the no-AMR groups, all the parameters of the model (bothdeterministic andstochastic)were found tobe significantly different between the two groups. This approach is found to be useful in capturing properties of antibody evolution from their peak concentration to final settling level and showed that the dynamic responses are different in AMR and no-AMR groups. A higher frequencyof oscillations anda faster antibody dissipation ratefortheAMRgrouphadbeenobservedfromthephaseportrait depicting thetrajectories ofthe system states,anda further test confirmedsignificantdifferencesbetweenthegroups.

The findings haveimportantimplications forthe development of laboratory assays that might define the nature of the mecha-nismsresponsibleforthefallsinDSAlevelspost-transplant,since a fullerunderstanding ofthesemechanisms mightallow for pre-transplantmanipulation of DSAlevels andimproved clinical out-comes.Thisisparticularlyimportantwithrespecttotheoscillating nature ofDSAlevels,whichmayreflect asystemslowlyreaching homeostasis,andmaybereflectedinlaboratorymeasurements.

Furtherworkmightalsoincludemodellinginrelationtomore detailedcharacteristicsoftheantibodies.Forexample,wehave al-readyshownthatthesubclassesofIgGareassociatedwithclinical outcomes,sothatmeasuringthelevelsofthesesubclassesatmore timepointsmightbevaluable[34].Theclinicaloutcomemeasures might also be extended. Since acute AMR is often treatable and is not always associated witha poor clinical outcome (especially whenthesettlinglevelofDSAisverylow),longertermgraft sur-vivalcouldalsobe consideredasanimportantoutcome level.The daytodayrenalfunctiondoesnotalwaysfollowDSAlevels[8]and ourunderstandingofhowagraftrespondstoDSAlevelsandhow AMRevolvesislimited.

Thisstudycomprisesapilotresearchondata-drivenmodel de-velopment for early post-transplant antibody dynamics, focusing on one of the typical patterns of a rapid fall following a rapid rise inDSAafterkidney transplantation.Future workwillinvolve classifyingandmodellingtheotherpatternsofthepost-transplant DSAdynamics that havebeendescribedin section2; a universal modelthat iscapableofdescribing differentdynamicsinDSAsis underdevelopment.

Details on the data used for analysis are available from the University of Warwick institutional repository at http://wrap.warwick.ac.uk/78888/.

Acknowledgment

ThisworkhasbeensupportedbyEPSRCUK(EP/K02504X/1).

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