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Analyzing the effect of APOE on Alzheimer's disease progression using an event-based

model for stratified populations

Alzheimer's Disease Neuroimaging Initiative

DOI

10.1016/j.neuroimage.2020.117646

Publication date

2020

Document Version

Final published version

Published in

NeuroImage

Citation (APA)

Alzheimer's Disease Neuroimaging Initiative (2020). Analyzing the effect of APOE on Alzheimer's disease

progression using an event-based model for stratified populations. NeuroImage, 227, [117646].

https://doi.org/10.1016/j.neuroimage.2020.117646

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ContentslistsavailableatScienceDirect

NeuroImage

journalhomepage:www.elsevier.com/locate/neuroimage

Analyzing

the

effect

of

APOE

on

Alzheimer’s

disease

progression

using

an

event-based

model

for

stratified

populations

Vikram

Venkatraghavan

a,∗

,

Stefan

Klein

a

,

Lana

Fani

c

,

Leontine

S.

Ham

a

,

Henri

Vrooman

a

,

M.

Kamran

Ikram

c,d

,

Wiro

J.

Niessen

a,b

,

Esther

E.

Bron

a

,

for

the

Alzheimer’s

Disease

Neuroimaging

Initiative

1

a Biomedical Imaging Group Rotterdam, Department of Radiology & Nuclear Medicine, Erasmus MC, University Medical Center Rotterdam, the Netherlands b Quantitative Imaging Group, Dept. of Imaging Physics, Faculty of Applied Sciences, Delft University of Technology, Delft, the Netherlands

c Department of Epidemiology, Erasmus MC, University Medical Center Rotterdam, the Netherlands d Department of Neurology, Erasmus MC, University Medical Center Rotterdam, the Netherlands

a

r

t

i

c

l

e

i

n

f

o

Keywords:

Disease Progression Modeling Event-Based Model Alzheimer’s Disease

APOE

a

b

s

t

r

a

c

t

Alzheimer’sdisease(AD)isthemostcommonformofdementiaandisphenotypicallyheterogeneous.APOEisa

triallelicgenewhichcorrelateswithphenotypicheterogeneityinAD.Inthiswork,wedeterminedtheeffectof

APOEallelesonthediseaseprogressiontimelineofADusingadiscriminativeevent-basedmodel(DEBM).Since

DEBMisadata-drivenmodel,stratificationintosmallerdiseasesubgroupswouldleadtomoreinaccuratemodels

ascomparedtofittingthemodelontheentiredataset.Henceoursecondaryaimistoproposeandevaluatenovel

approachesinwhichwesplitthedifferentstepsofDEBMintogroup-aspecificandgroup-specificparts,where

theentiredatasetisusedtotrainthegroup-aspecificpartsandonlythedatafromaspecificgroupisusedto

trainthegroup-specificpartsoftheDEBM.Weperformedsimulationexperimentstobenchmarktheaccuracyof

theproposedapproachesandtoselecttheoptimalapproach.Subsequently,thechosenapproachwasappliedto

thebaselinedataof417cognitivelynormal,235mildcognitivelyimpairedwhoconverttoADwithin3years,

and342ADpatientsfromtheAlzheimersDiseaseNeuroimagingInitiative(ADNI)datasettogainnewinsights

intotheeffectofAPOEcarriershiponthediseaseprogressiontimelineofAD.Inthe𝜀4carriergroup,themodel

predictedwithhighconfidencethatCSFAmyloid𝛽42andthecognitivescoreofAlzheimer’sDiseaseAssessment

Scale(ADAS)areearlybiomarkers.Hippocampuswastheearliestvolumetricbiomarkertobecomeabnormal,

closelyfollowedbytheCSFPhosphorylatedTau181(PTAU)biomarker.Inthehomozygous𝜀3carriergroup,the

modelpredictedasimilarorderingamongCSFbiomarkers.However,thevolumeofthefusiformgyruswas

identifiedasoneoftheearliestvolumetricbiomarker.Whilethefindingsinthe𝜀4carrierandthehomozygous

𝜀3carriergroupsfitthecurrentunderstandingofprogressionofAD,thefindinginthe𝜀2carriergroupdidnot.

Themodelpredicted,withrelativelylowconfidence,CSFNeurograninasoneoftheearliestbiomarkersalong

withcognitivescoreofMini-MentalStateExamination(MMSE).Amyloid𝛽42wasfoundtobecomeabnormal

afterPTAU.Thepresentedmodelscouldaidunderstandingofthedisease,andinselectinghomogeneousgroup

ofpresymptomaticsubjectsat-riskofdevelopingsymptomsforclinicaltrials.

1. Introduction

Dementiaaffectsroughly 5%of theworld’selderlypopulationof whom60−70%areaffectedbyAlzheimer’sDisease(AD),whichisthe mostcommonformofdementia(Organization,2017).Thereareseveral neurobiologicalsubtypesofAD(Ferreiraetal.,2020)andeachsubtype potentiallyneedsadifferentstrategytopreventorslowtheprogression

Correspondingauthor.

E-mailaddress:v.venkatraghavan@erasmusmc.nl(V.Venkatraghavan).

1 DatausedinpreparationofthisarticlewereobtainedfromtheAlzheimer’sDiseaseNeuroimagingInitiative(ADNI)database(adni.loni.usc.edu).Assuch,the

investigatorswithintheADNIcontributedtothedesignandimplementationofADNIand/orprovideddatabutdidnotparticipateinanalysisorwritingofthis

report.AcompletelistingofADNIinvestigatorscanbefoundat:http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf

of AD.Understandingthepathophysiologicalprocesses inADis thus crucialforselectingnovelpreventiveortherapeutictargetsforclinical trialsofdiseasemodifyingtreatments,identifyingtargetgroupsforsuch trialsandtrackingthediseaseprogressioninpatients.

While several studies have looked into the pathophysiology of AD (Bloom,2014; JackJr. etal., 2013; Weigandetal., 2019),itis stillnotcompletelyunderstood.AlthoughithasbeenobservedthatAD

https://doi.org/10.1016/j.neuroimage.2020.117646

Received15September2020;Receivedinrevisedform12November2020;Accepted10December2020

Availableonline16December2020

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Fig.1. OverviewofthestepsinvolvedinDEBM.InputfortheDEBMmodelisacross-sectionaldataset𝑋 with𝑀 subjectsandvariousbiomarkers(𝐴,𝐵,𝐶 and𝐷)

representingdifferentaspectsofneuro-degeneration.UsingGaussianmixturemodeling(GMM),mixingparameters(𝜃𝑖)andprobabilitydensityfunctionsofnormal

(𝑝(𝑥⋅,𝑖|¬𝐸𝑖))andabnormal(𝑝(𝑥⋅,𝑖|𝐸𝑖))levelsareestimatedforeachbiomarker.Thisisfollowedbytheestimationofsubject-specificorderings(𝑠𝑗),foreachsubjectin thedataset.Diseaseprogressiontimelineconsistingofcentralordering(𝑆)andevent-centers(𝜆)areestimatedbasedonthesesubject-specificorderings.Basedon theconstructeddiseaseprogressiontimeline,patientstages(Υ𝑗)ofsubjectsinanindependenttest-setcanbeestimated.

isphenotypicallyheterogeneous(Auetal.,2015;Murrayetal.,2011; Patterson,2018)withpotentiallydifferentpathwaysfordisease progres-sion,thesepathwaysremainunclear.Thereishenceaneedto under-standthephenotypicheterogeneityinADwhileleveraging neuroimag-ing,fluidandcognitivebiomarkers.

APOEisatriallelicgeneinwhichthe𝜀2allelereducestheriskof AD(vanderLeeetal.,2018),the𝜀3alleleactsasareferencealleleand the𝜀4alleleisamajorgeneticriskfactorofAD(Geninetal.,2011;Kim etal.,2009;Saundersetal.,1993).APOEhasbeenshowntocorrelate withphenotypicheterogeneityinAD(Weintraubetal.,2019).Hencewe hypothesizethatthepathophysiologyofADcanbebetterunderstood whenconsideringtheeffectofAPOEcarriershiponbiomarkerchanges. Inthecontextofdata-drivenmethodsforunderstandingAD patho-physiology,diseaseprogressionmodelshavebeenusedtostudythe tra-jectoriesofindividualbiomarkers(Jedynaketal.,2012;Lorenzietal., 2019;Schirattietal.,2015)aswellastheirprogressionwithrespectto eachother(Fonteijnetal.,2012;HuangandAlexander,2012; Venka-traghavan etal., 2017; Young etal., 2014). Unliketypical machine learningapproaches,thesemodelsareinterpretablebydesignand pro-videinsightforunderstandingthemechanismsofdiseaseprogression. Event-basedmodels(EBMs)areaclassofsuchinterpretabledisease pro-gressionmodelsthatestimatethetimelineofneuropathologicchange duringADprogressionusingcross-sectionaldata(Fonteijnetal.,2012; Venkatraghavanetal.,2019a).

Our primary aim is touse the discriminativeevent-based model (DEBM),whichwas showntobe moreaccuratethanpreviously pro-posedEBMs(Venkatraghavanetal.,2019a),tounderstandtheeffectof differentAPOEallelesonthediseasetimelineofAD.Toshedlighton differentaspectsofneurodegenerationandidentifytheearliestbrain regions affected, we included commonlystudied cerebrospinal fluid (CSF) biomarkers,cognitivescores, and volumetricbiomarkers from neuroimaging.

Thedefaultapproachforestimatingthediseaseprogressiontimeline wouldbetostratifythepopulationbasedontheirAPOE𝜀2−4carrier statusandindependentlytraintheDEBMmodelonthestratified pop-ulations(Youngetal., 2014).However,sinceDEBMisadata-driven model,stratification into smallergroupswouldlead tolessaccurate modelsthanthoseobtainedbytheoriginalmethodontheentiredataset. Henceoursecondaryaimistoproposeandevaluateanovelapproach inwhichwesplitthedifferentstepsofDEBMintogroup-aspecificand group-specificparts,wheretheentiredatasetisusedtotrainthe group-aspecificpartsandonlythedatafromaspecificgroupisusedtotrainthe group-specificpartsoftheDEBM.Wepresenttwodifferentvariations ofthisapproachandwehypothesizethattheoptimalsplitoftheDEBM stepsintothegroup-aspecificandgroup-specificpartswouldresultin betteraccuracyoftheestimateddiseaseprogressiontimeline.Sincethe ground-truthtimelinesareunknowninaclinicalsetting,weevaluate theaccuracyoftheproposedvariationsusingsimulationexperiments andweselecttheoptimalmethodfortheanalysisontheeffectofAPOE

ontheADprogressiontimelineonpatientdata.

Tosummarize,ourcontributionsinthispaperincludeproposingand evaluatinganovelapproachforusingDEBMinstratifiedpopulations andestimatingacomprehensivetimelineofADprogression,intermsof biomarkerchanges,inthepresenceofdifferentAPOEalleles.

2. Methods

AnintroductiontotheDEBMmodel(Venkatraghavanetal.,2019a) isprovidedinSection2.1.InSection2.2weproposeournovelapproach forusingDEBMinstratifiedpopulationswithitstwovariations.

2.1. Discriminativeevent-basedmodeling

In a cross-sectional dataset (𝑋) of 𝑀 subjects, including cogni-tivelynormalindividuals(CN),subjectswithmildcognitiveimpairment (MCI)andpatientswithAD,let𝑋𝑗 denoteameasurementof biomark-ersforsubject𝑗∈ [1,𝑀],consistingofscalarbiomarkervalues𝑥𝑗,𝑖for 𝑖∈ [1,𝑁].𝑥⋅,𝑖denotesthe𝑖thbiomarkerforanyunspecified𝑗.DEBM

estimatestheposteriorprobabilitiesofindividualbiomarkersbeing ab-normal.Theseposteriorprobabilitiesareusedtoestimatetheordering ofbiomarkerchangesforeachsubjectindependently.Thecentral order-inganddiseaseprogressiontimelinefortheentiredatasetareestimated basedonthesesubject-specificorderings.Theresultingdisease progres-siontimelineisusedforassessingtheseverityofdiseaseinanindividual basedonhis/herbiomarkervalues.Figure1showsthedifferentsteps involvedinDEBM.

Step 1-MixtureModeling:AsADischaracterizedbyacascade ofneuropathologicalchangesthatoccursoverseveralyears, presymp-tomaticCNsubjectscanhavesomeabnormalbiomarkervalues.Onthe otherhand,insomeclinicallydiagnosedADsubjects,aproportionof biomarkersmaystillhavenormalvalues,astheymightnothavean un-derlyingADpathologyorcouldhaveatypicalAD.Henceclinicallabels cannotdirectly be propagatedtoindividualbiomarkers tolabel nor-malandabnormalbiomarkervalues.Weshallrefertothisasbiomarker labelnoiseintherestofthepaper.Inordertoestimatetheposterior probabilitiesofindividualbiomarkersbeingabnormal,DEBM,similar topreviouslyproposedEBMs(Fonteijnetal.,2012;Huangand Alexan-der,2012;Youngetal.,2014),fitsaGaussianmixturemodel(GMM) toconstructthenormal/pre-eventprobabilitydensityfunction(PDF),

𝑝(𝑥⋅,𝑖|¬𝐸𝑖),andabnormal/post-eventPDF,𝑝(𝑥⋅,𝑖|𝐸𝑖).Event𝐸𝑖inthis

notationisusedtodenotethecorrespondingbiomarkerbecoming ab-normaland¬𝐸𝑖denotesthecorresponndingbiomarerbeingnormal.The aforementionedPDFscanbeexpressedas:

𝑝(𝑥⋅,𝑖|¬𝐸𝑖)=(𝜇𝑖,¬𝐸;𝜎𝑖,¬𝐸) (1)

𝑝(𝑥⋅,𝑖|𝐸𝑖)=(𝜇𝑖,𝐸;𝜎𝑖,𝐸) (2)

Where,(⨘ ,∫)isthenormaldistributionwithmean𝜇 andstandard deviation𝜎.

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Fig.2. OverviewofGMMoptimizationinDEBM.

For estimating these parameters robustly in the presence of biomarkerlabelnoise,thenormalandabnormalPDFestimatesarefirst initializedusingthemeanandstandarddeviationsaftertruncatingthe overlappingtailsoftheobserveddistributionsinCNandADsubjects. ThiscanbeobservedinFig.2,wheretheinitializationisperformedonly basedonthenon-overlappingpartsofgreenandredcurves,whilethe overlappingpartisleftouttoaccountforbiomarkerlabelnoise.Atthis stageofGMMinitialization,MCIsubjectsareleftoutaswell,because itisunsureaprioriwhethertheirbiomarkersarenormalorabnormal. TheresultinginitializedPDFsaredenotedaŝ𝑝(𝑥⋅,𝑖|¬𝐸𝑖))and̂𝑝(𝑥⋅,𝑖|𝐸𝑖).

Thisis followedbyanalternatingGMMmaximumlikelihood op-timizationschemeuntilboththeGaussian parametersaswellasthe mixingparametersconverge.Allthesubjects,includingMCI,areused forGMMoptimization.Afterconvergence,theseGaussiansareusedto representthePDFs𝑝(𝑥⋅,𝑖|¬𝐸𝑖)and𝑝(𝑥⋅,𝑖|𝐸𝑖).Themixingparameters(𝜃𝑖) areusedaspriorprobabilitiestoconvertthesePDFstoposterior prob-abilities𝑝𝐸𝑖|𝑥⋅,𝑖)and𝑝(𝐸𝑖|𝑥⋅,𝑖).Fig.2showsanoverviewofthis opti-mizationscheme.

Step2-Subject-specificOrderings:𝑝(𝐸𝑖|𝑥𝑗,𝑖)∀𝑖areusedtoestimate thesubject-specificorderings𝑠𝑗.𝑠𝑗isestablishedsuchthat:

𝑠𝑗𝑝 ( 𝐸𝑠𝑗(1)|||𝑥𝑗,𝑠𝑗(1) ) >...>𝑝(𝐸𝑠𝑗(𝑁)|||𝑥𝑗,𝑠𝑗(𝑁) ) (3)

Step3-CentralOrdering:DEBMcomputesthecentralevent order-ing𝑆 fromthesubject-specificestimates𝑠𝑗.Todescribethedistribution

of𝑠𝑗,ageneralizedMallowsmodelisused(FlignerandVerducci,1988). Thecentralorderingisdefinedastheorderingthatminimizesthesumof distancestoallsubject-specificorderings𝑠𝑗,withprobabilisticKendall’s

Taubeingthedistancemeasure(Venkatraghavanetal.,2019a).While𝑆

denotesthesequenceofbiomarkerevents,therelativepositionofthese events(event-centers)inanormalizedscaleof[0,1]isdenotedbythe vector𝜆.Thepair{𝑆,𝜆}togetherformsadiseaseprogressiontimeline.

Step4-PatientStaging:Oncethediseaseprogressiontimelineis created,subjectsinanindependenttestset(𝑇)canbe placedonthis timelinetoestimatediseaseseverity.Thisisachievedbyconvertingthe biomarkervaluesofthetestsubjectstoposteriorprobabilities𝑝(𝐸𝑖|𝑥𝑗,𝑖),𝑗𝑇.Thesecanbeusedtoestimatediseaseseveritiesintestsubjects byfirstestimating theconditionaldistribution 𝑝(𝑖|𝑆,𝑋𝑗),which esti-matestheprobability thatthefirst𝑖eventsof𝑆 haveoccurredfora test-subjectandtherestareyettooccur.

𝑝(𝑖|𝑆,𝑋𝑗)∝∏𝑖𝑙=1𝑝 ( 𝐸𝑆(𝑙)|||𝑥𝑗,𝑆(𝑙) ) × ∏𝑁 𝑙=𝑖+1 𝑝(‶𝐸𝑆(𝑙)|||𝑥𝑗,𝑆(𝑙) ) (4)

Thepatientstageofatestsubject(Υ𝑗)isdefinedastheexpectation of𝜆(𝑖)withrespecttotheconditionaldistribution𝑝(𝑖|𝑆,𝑋𝑗).

Υ𝑗 = ∑𝑁 𝑖=1𝜆(𝑖)𝑝(𝑖|𝑆,𝑋𝑗) ∑𝑁 𝑖=1𝑝(𝑖|𝑆,𝑋𝑗) (5)

2.2. Group-specificandgroup-aspecificpartsofDEBM

WeproposeextensionsofDEBMforstratifiedpopulations,i.e.,when thedataset𝑋 canbe subdividedin groups𝑔∈ [1,𝐺],basedon,e.g., genotype orphenotypeof thesubjects.Since DEBMis adata-driven model,datastratificationintosmallergroupswouldleadtomore inac-curatemodels(Venkatraghavanetal.,2019a).ToobtainbetterDEBM accuraciesinsuchscenario,weproposetoco-trainDEBMforestimating diseasetimelines∀𝑔 bysplittingDEBMintogroup-aspecificand group-specificparts.Thegroup-aspecificpartsofDEBMareestimatedusing

Fig.3. OverviewofGMMoptimizationstrategiesinthedifferentapproaches

forDEBManalysisinstratifiedpopulations.(a)Thedefaultapproachinwhich

GMMineachgroupistrainedindependently.(b)GMMincoupledDEBM,where

thedifferentgroupssharetheGaussianparameters,butthemixingparameters

areestimatedindependently.(c)GMMinco-initDEBMinwhichthedifferent

groupsarejointlyinitializedbeforetheGMMoptimization,buttheoptimization

isdoneindependentlyforeachgroup.

theentiredatasetandgroup-specificpartsareestimatedforeachgroup independently.

We firstdiscussthedefaultway ofindependentlytrainingDEBM inthedifferentgroupsandthenproposetwodifferentapproachesfor splittingDEBMintogroup-aspecificandgroup-specificparts.

Approach1:IndependentDEBM

Inthisdefaultapproach,eachgroupisconsideredasanindependent datasetandthediseaseprogressiontimelineineachgroupisestimated independently.GMMinsuchascenarioisillustratedinFig.3a.

Approach2:CoupledDEBM

DEBM→ {

𝑝(𝑥⋅,𝑖|¬𝐸𝑖),𝑝(𝑥⋅,𝑖|𝐸𝑖) group-aspecific

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Inthisapproach,weassumethatthedifferentgroupssharethenormal andabnormalPDFs,buttheorderinginwhichthesebiomarkersbecome abnormalaredifferent.Themixingparameters(𝜃𝑖,𝑔)areconsideredas

group-specificpartof theDEBMalgorithmbecausetheproportionof subjectswithnormalandabnormalbiomarkervaluesineachgroup𝑔

iscorrelatedwiththepositionofthebiomarkeralongtheordering𝑆𝑔,

whichweexpecttobedifferentineachgroup.

Hence, in our approach, we modify the alternating GMM opti-mizationschemetojointlyoptimizetheGMMparametersofmultiple groups.First,theGMMalgorithmisinitializedwithoutconsideringthe groups,asexplainedinSection2.1.Secondly,aswiththedefaultDEBM, Gaussianparametersandmixingparametersarealternatelyoptimized. IncontrastincoupledDEBM,theGaussianparametersareestimated jointlyforallgroups,whilemixingparametersareestimatedseparately foreachgroup.ThishasbeenillustratedinFigure3b.

OncetheGMMoptimizationhasbeenperformed,𝑆𝑔and𝜆𝑔are

es-timatedineachgroup.Patientstaging(Υ𝑗)ofthetest-subjectsingroup

𝑔 arecomputedbasedonthediseaseprogressiontimeline{𝑆𝑔,𝜆𝑔}.

Approach3:Co-initDEBM

DEBM→ ⎧ ⎪ ⎨ ⎪ ⎩ ̂𝑝(𝑥⋅,𝑖|¬𝐸𝑖),̂𝑝(𝑥⋅,𝑖|𝐸𝑖) group-aspecific 𝑝𝑔(𝑥⋅,𝑖|¬𝐸𝑖),𝑝𝑔(𝑥⋅,𝑖|𝐸𝑖) group-specific 𝜃𝑖,𝑔,{𝑆𝑔,𝜆𝑔} group-specific (7)

Inthisapproach,weassumethatthedifferentgroupsdonotshare thenormalandabnormalPDFs,butthattheyareclosetoeachother. Hence, in co-init DEBM, we relax the constraint on 𝑝(𝑥⋅,𝑖|¬𝐸𝑖) and

𝑝(𝑥⋅,𝑖|𝐸𝑖)andinsteadconsidertheinitializedvaluesofnormaland abnor-malPDFs(̂𝑝(𝑥⋅,𝑖|¬𝐸𝑖)and̂𝑝(𝑥⋅,𝑖|𝐸𝑖))tobegroup-aspecificpartofDEBM. Weestimate𝑝𝑔(𝑥⋅,𝑖|¬𝐸𝑖)and𝑝𝑔(𝑥⋅,𝑖|𝐸𝑖)independentlyforeachgroup. ThisisillustratedinFig.3c.

Aswiththepreviousapproach,𝑆𝑔,𝜆𝑔andthepatientstagingofthe

test-subjectsingroup𝑔 arecomputedindependentlyforeachgroup.

3. Experiments

Section 3.1 describes the experiments to evaluate the proposed DEBMapproachesonastratifiedpopulation.Sinceground-truth order-ingsareunknown inrealclinicaldata,weusesimulateddatasetsfor evaluatingthemethods.Afterevaluatingtheproposedapproaches,we selectthebestapproachforanalyzingtheeffectofAPOEonAD progres-sionusingsubjectsfromtheAlzheimer’sDiseaseNeuroimaging Initia-tive(ADNI)database.Section3.2descibesthedetailsofthese experi-ments.

3.1. Simulationexperiments

WeusedtheframeworkdevelopedbyYoungetal.(2015)for simu-latingcross-sectionaldataconsistingofscalarbiomarkervaluesforCN, MCIandADsubjectsintwogroups.Inthisframework,disease progres-sioninasubjectis modeledbyaseriesofbiomarkerchanges repre-sentingthetemporalcascadeofbiomarkerabnormalityasestimatedby anEBM.Individualbiomarkertrajectoriesarerepresentedbysigmoids varyingfromthebiomarker’snormalvaluetoitsabnormalvalue.To ac-countforinter-subjectvariability,thenormalandabnormalvaluesfor differentsubjectsaredrawnrandomlyfromGaussiandistributions.

The simulation dataset used in our experiments are based on a set of seven biomarkers as described in thesimulation experiments ofVenkatraghavanetal.(2019a).Thesimulateddatasetswere strati-fiedintotwogroups,witheachgrouphavingitsowndistinctdisease progressionpatterns.Therearetwowaysinwhichtheprogressionof diseaseinthegroupscandiffer:1.differenceinground-truthorderings

𝑆1 and𝑆2; 2.differencein theabnormalbiomarkerPDFsinthetwo

groupsi.e.𝑝1(𝑥⋅,𝑖|𝐸𝑖)and𝑝2(𝑥⋅,𝑖|𝐸𝑖).Eachofthesedifferencescould

af-fecttheaccuracyoftheproposedapproaches.Hence,weevaluatedthe proposedapproachesinthepresenceofeachofthesedifferences. Nor-malizedKendall’sTaudistancebetweentheestimatedordering(𝑆)and

theground-truthordering(𝑆𝑔𝑡)wasusedasanevaluationmeasurein theseexperiments: 𝜀𝑆=𝐾(𝑆,𝑆𝑔𝑡 ) (𝑁 2 ) (8)

where𝐾(𝐴,𝐵)is thenumberof swapsrequiredtoobtainorderingB fromorderingA.

Thenormalizationensuresthat𝜀𝑆fallsintherange[0,1],with0as

thedistancewhenthetwoorderingsarethesame,and1asthedistance whenthetwoorderingsarethereverseofeachother.

Experiment1:Thefirstsimulationexperimentstudiedtheeffectof thedifferenceinorderingbetweenthetwogroups.Theorderinginthe firstgroup(Group1)wasfixedandtheorderinginthesecondgroup (Group 2)wasselected randomlysuchthatthenormalizedKendall’s Taudistancebetweenthetwogroupswasafixednumber,say𝜀𝑂.𝜀𝑂

wasvariedfrom0to1instepsof0.2.ThenumberofsubjectsinGroup 2waskeptconstantat900.Thenumberofsubjectsin Group1was variedfrom100to900instepsof200,tostudyhowthedifferent ap-proachesperforminsmallaswellaslargegroups.Thenormaland ab-normalbiomarkerslevelsinthetwogroupsweresampledfromthesame Gaussiandistributionforthisexperiment.Wegenerated50random rep-etitionsofthesimulateddatasets,andreportedmeanandstandard de-viationof𝜀𝑆forindependentDEBM,coupledDEBM,andco-initDEBM ingroups1and2.

Experiment2:Thisexperimentstudiedtheperformanceofthe pro-posedapproacheswiththe𝜇𝑔,𝑖,𝐸parameterofthe𝑝𝑔(𝑥⋅,𝑖|𝐸𝑖)distribution beingdifferentinthetwogroups.𝜇1,𝑖,𝐸wasfixed,and𝜇2,𝑖,𝐸wasvaried

suchthatthedifference𝜇2,𝑖,𝐸𝜇1,𝑖,𝐸(𝜀𝐺)wasoneof{−0.2𝑑,0,+0.2𝑑}

where𝑑=𝜇1,𝑖,𝐸𝜇1,𝑖,¬𝐸.0isconsideredthereferencelevel,wherethe

abnormalGaussiansarethesameinthetwogroups.𝜇𝑔,𝑖,¬𝐸 werekept

thesame inthe twogroups.Hence, when𝜀𝐺=−0.2𝑑,theabnormal

biomarkerlevelsareclosertothenormalbiomarkerlevelsinGroup 2thaninGroup1.Thisresults inGroup2biomarkersbeingweaker thantheirGroup1counterpartswhen𝜀𝐺=−0.2𝑑 andstrongerwhen 𝜀𝐺=+0.2𝑑.ThenumberofsubjectsinGroup2waskeptaconstantat

900,whilethesubjectsinGroup1increasedfrom100to900.𝜀𝑂 be-tweenthetwogroupswasfixedat0.4.Weagaingenerated50random repetitionsofthesimulateddatasets,andreportedmeanandstandard deviationof𝜀𝑆forcoupledDEBM,co-initDEBMandDEBM.

Theseexperimentswereusedtoevaluatethedifferent approaches mentioned inSection2andselectthebestmethod foranalyzingthe effectofAPOEallelesinADprogression.

3.2. StudyingtheeffectofAPOE

Weconsideredthebaselinemeasurementsfrom417CN, 235MCI convertersand342ADsubjectsinADNI1,ADNIGOandADNI2studies.2

TheMCIconvertersaresubjectswhohadMCIatbaselinebutconverted toADwithin3yearsofbaselinemeasurement.Weexcludedsubjects withsignificantmemoryconcerns(withoutadiagnosisofADorMCI) andMCInon-convertersinourexperimentstoselectamore phenotyp-icallyhomogeneousgroupofsubjectswithprevalentorincidentAD.In eachoftheexperiments,thedatasetwasdividedintothreegroups(𝜀2 carriers,homozygous𝜀3carriers,and𝜀4carriers)basedonthesubject’s

APOEcarriership(vanderLeeetal.,2018).SubjectswithAPOE𝜀2,4 (n=34)werenot includedineithergroupbecauseof thepresenceof both𝜀2and𝜀4alleles

2TheADNIwaslaunchedin2003asapublic-privatepartnership,ledby

Prin-cipalInvestigatorMichaelW.Weiner,MD.TheprimarygoalofADNIhasbeen

totestwhetherserialmagneticresonanceimaging(MRI),positronemission

to-mography(PET),otherbiologicalmarkers,andclinicalandneuropsychological

assessmentcanbecombinedtomeasuretheprogressionofmildcognitive

im-pairment(MCI)andearlyAlzheimersdisease(AD).Forup-to-dateinformation,

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Table1

Demographicsfortheusedpopulation.2representsthesubjectswithAPOE

alleles𝜀2,2and𝜀2,3.33representsthesubjectswithreferenceAPOEallele𝜀3,3.

4representsthesubjectswithAPOEalleles𝜀3,4and𝜀4,4.Subjectswithboth

𝜀2and𝜀4alleleswereexcludedfromthisstudy(n=34).Edu.isanabbreviation usedforEducation.

Demographics Diagnosis CN MCIc AD 𝑛 417 235 342 APOE 2 ⋆ /33/ ⋆ 4 57/244/110 6/66/156 12/101/219 Sex M/F 209∕208 145∕90 189∕153 Age [yrs.] ( 𝜇 ± 𝜎) 74 . 8 ± 5 . 7 73 . 7 ± 7 . 0 75 . 0 ± 7 . 8 Edu [yrs.] ( 𝜇 ± 𝜎) 16 . 3 ± 2 . 7 15 . 9 ± 2 . 7 15 . 2 ± 3 . 0 Table2

BiomarkeravailabilityinnumberofsubjectsintheAPOEbasedgroupsof𝜀2

carriers,homozygous𝜀3carriers,and𝜀4carriers.

Biomarker availability Biomarker 𝜀 2 carriers ( 𝑁 = 75) Homozygous 𝜀 3 carriers ( 𝑁 = 411) 𝜀 4 carriers ( 𝑁 = 485) Imaging 74 408 481 ABETA 57 301 357 PTAU 57 301 357 TAU 57 299 348 NG 21 113 131 NFL 23 118 137 MMSE 75 411 485 ADAS 74 410 477

SubjectdemographicsandtheirAPOEcarriershipsaresummarized inTable1.Themodalitiesconsideredwerestructuralimaging biomark-ers,biomarkers extractedfrom cerebrospinal fluid(CSF),and cogni-tivebiomarkers.Structuralimagingbiomarkerswereobtainedfrom T1-weightedMRI acquiredat1.5Tor3T. DetailsoftheMRIacquisition protocolsofADNIcanbefoundinJackJr.etal.,2008,2015.

Imagingbiomarkers wereestimated fromT1-weighted MRI scans analysedwithFreeSurfersoftwarev6.0cross-sectionalstreamand out-putswerevisuallychecked.Weassumedasymmetricpatternof atro-phyinADandaveragedimagingbiomarkersbetweentheleftandright hemisphere.

Experiment3:Forthisexperiment,theselectedimagingbiomarkers were:hippocampalvolume,volumeoftheentorhinalcortex,fusiform gyrusvolume,middle-temporalgyrusvolume,precuneusvolume, to-getherwithwholebrainvolumeandvolumeoftheventricles(Archetti et al., 2019; Frisoni et al., 2010; Vemuri and Jack, 2010). The se-lected CSF based biomarkers were: CSF concentrationsof

Amyloid-𝛽42(ABETA),totalTau(TAU)andphosphorylatedTau181(PTAU)

pro-teins (Blennow and Hampel, 2003; Blennow et al., 2010), Neuro-granin(Thorselletal.,2010)andNeurofilamentlightchain(Jinetal., 2019;deWolfetal.,2020).Minimentalstateexamination(MMSE)and Alzheimer’sDiseaseAssessmentScale-Cognitive(13items)(ADAS13) wereusedascognitivebiomarkers.Theavailabilityofthesemultimodal biomarkersintheADNIdatabaseissummarizedinTable2.

Wedownloaded theCSFmeasurements fromthe ADNIdatabase. Themeasurementsof ABETA,TAU andPTAUhadbeenmade using themicrobead-based multiplex immunoassay, theINNO-BIAAlzBio3 RUO(Olssonetal.,2005).ThemeasurementofNFLhadbeenmadewith enzyme-linkedimmunosorbentassayNF-lightELISAkit(Mattssonetal., 2017).NGhadbeenmeasuredbyelectrochemiluminescence technol-ogy(MesoScaleDiscovery) usingamonoclonalantibodyspecificfor NG(Ng7)forcoatingtogetherwithadetectorantibodypolyclonal neu-rograninanti-rabbit(ab23570,Upstate)(Porteliusetal.,2015).As de-scribedpreviouslyinVenkatraghavanetal.(2019a),theTAUandPTAU measurementsweretransformedtologarithmicscalestomakethe dis-tributionslessskewedandmoresuitableforDEBManalysis.

Thevolumesoftheselectedregionswereregressedwithage,sexand intra-cranialvolume(ICV)andtheeffectsofthesefactorswere subse-quentlycorrectedfor,beforebeingusedasbiomarkers.Theeffectsof ageandsexwereregressedoutofCSFfeatures,whereaseffectsofage, sexandeducationwereregressedoutofcognitivescores.

Forthe12selectedbiomarkers,weestimatedthediseasetimelinesin thethreeaforementionedgroupsusingthemethodselectedafter simu-lationexperiments.Westudiedthepositionalvarianceoftheestimated orderingsbycreating100bootstrappedsamplesofthedata.Inorder toevaluateiftheestimatedorderingsinthethreegroupswere signif-icantly differentfrom oneanother, weusedpermutation testing and estimatedthedistributionoftheKendall’sTaudistanceunderthenull hypothesis.Tocomputethisdistribution,wegenerated10,000random permutationsofthethreegroups.Wethencomputedtheone-sided𝑝 -valuesfortheactualKendall’sTaudistancesbetweentheorderingsof thethreegroups,calculatedastheproportionofsampledpermutations wherethedistancewasgreaterthanorequaltotheactualdistance,and usingBonferronicorrectiontoaccountformultipletesting.

Experiment 4:Inthisexperiment,wevalidatedthediseasestage (Υ𝑗)bycomputingitscorrelationwiththesubjects’MMSEandADAS13 values.Weuseda10-foldcrossvalidation,wherethetrainingsetwas usedtoestimatethediseasetimelineintheaforementionedgroupsand thetestsubjects’diseasestagewasevaluatedbyplacingthemonthis diseasetimeline.Weusedthevolume-basedandCSF-basedbiomarkers fromExperiment3,butexcludedMMSEandADAS13scoresfromthe model.

4. Results

4.1. Simulations

Experiment1:Fig.4(a)and(b)showtheorderingerrors(𝜀𝑆)in Group1ofthesimulationdatasetsforDEBM,coupledDEBMand co-init DEBMasa functionofnumberof subjectsinGroup1,when𝜀𝑂

betweenthetwogroupschangesfrom0to1.Fig.4(c)–(e)show𝜀𝑆in Group2ofthesimulationdatasetsfortheaforementionedmethods,asa functionofnumberofsubjectsinGroup1.Inourexperiments,Group1 datasetremainsthesamewhileGroup2datasetchangesas𝜀𝑂increase. HenceDEBMresultsdonotchangewithchangein𝜀𝑂inFig.4(a)and

(b),whereasinFig.4(c),DEBMresultsdonotchangewithincreasein numberofsubjectsinGroup1.

Itcanbeseenthatbothcoupled-trainingmethods(i.e.,co-initDEBM andcoupledDEBM) outperformthedefaultmethodof independently trainingDEBMmodels.Itcanalsobeobservedthatinbothco-initDEBM andcoupledDEBMtheordering errorsdecrease as𝜀𝑂 increasesand

thatco-initDEBMoutperformscoupledDEBMforlowervaluesof𝜀𝑂,

whereastheperformanceisonparwithcoupledDEBMforhighervalues of𝜀𝑂.

Experiment2:Fig.5(a)and(b)show𝜀𝑆inGroup1andFig.5(c)–

(e)showthesameinGroup2,whenvarying𝜀𝐺.Evenwith𝜀𝐺≠ 0,

cou-pledtraining(i.e.,co-initDEBMandcoupledDEBM)outperformedthe defaultmethodofindependentlytrainingDEBMmodels.Co-initDEBM showednegligiblechangeintheerrorswhen𝜀𝐺≠ 0.Theperformance

ofcoupledDEBMinGroup1worsenedfor𝜀𝐺=+0.2𝑑 (Fig.5(a))and inGroup2for𝜀𝐺=−0.2𝑑 (Fig.5(d)).

4.2. StudyingtheeffectofAPOE

Theresultsin Experiments1and2showthattheperformanceof co-initDEBMismoreaccurateandrobustthancoupledDEBMinmost scenarios.WehenceanalyzedExperiments3and4usingco-initDEBM.

Experiment3:Fig.6showsorderingsofCSF,globalcognitionand volumetricbiomarkersintheAPOEbasedgroupsof𝜀2carriers, homozy-gous𝜀3carriers,and𝜀4carriersalongwiththeiruncertaintyestimates. Itcan beseen thattheuncertaintyof theordering inthe𝜀2carriers groupwashigh.Despitethisuncertainty,somebiomarkers(i.e.MMSE,

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Fig.4. Experiment1:Theeffectof𝜀𝑂(thedifferenceingroundtrutheventorderingsinthetwogroups)ontheperformanceoftheproposedmethods.Theshaded regionintheseplotsrepresentsstandarddeviationoftheerrorinestimationoftheproposedmethodsin50randomiterationsofsimulations.Theplotsin(a)and(b)

showtheorderingerrorsinGroup1usingCoupledDEBMandCo-initDEBMwithindependentDEBMshowninboth(a)and(b),asafunctionofnumberofsubjects

inGroup1.Theplotsin(c),(d)and(e)showtheorderingerrorsinGroup2usingindependentDEBM,CoupledDEBMandCo-initDEBMrespectivelyasafunction

ofnumberofsubjectsinGroup1.

NGandPTAU)seemtooccurearlierthantheotherbiomarkersinthis group.

Inthehomozygous𝜀3carriergroup,ABETAwasveryprominently the earliest biomarker, followed by cognitive scores of MMSE and ADAS13.AmongtheCSFbiomarkers,PTAUfollowedimmediately af-terABETA,whichwasinturnfollowedbyTAU.NFLandNGwerelate biomarkers.Amongthestructuralbiomarkers,volumesoffusiformand middle-temporalgyriwerethefirsttobecomeabnormal,followedby ventricularvolumeandwholebrainvolume.Hippocampus,precuneus andentorhinalvolumeswerelatebiomarkersinthisgroup.

Inthe𝜀4carriergroup,theCSFbiomarkersfollowedapatternthat wassimilartothatofthehomozygous𝜀3carriergroup.Thecognitive biomarkerswereearlybiomarkersinthisgroupaswell.Howeverthe orderinginstructuralbiomarkerswasverydifferent fromthatinthe homozygous 𝜀3carrier group. Hippocampus andentorhinalvolumes

wereearlybiomarkersinthisgroup,followedbymiddle-temporaland fusiformgyrivolumes.Wholebrain,ventricularandprecuneusvolumes werelatebiomarkers.

Theorderingofthe𝜀2carriergroupwassignificantlydifferentfrom thatofthehomozygous𝜀3carriergroup(𝑝=0.0156,afterBonferroni correctionformultipletesting).Similarly,theorderingsfortheother twogroupsweresignificantaswell:𝑝=0.0147forthedifference be-tween 𝜀2 carrier group and𝜀4 carrier group and𝑝=0.0003 for the difference between the homozygous 𝜀3carrier group and 𝜀4 carrier group.

Experiment4:ThevariationofMMSEandADAS13scoreswith re-specttotheestimateddiseasestageshasbeenplottedinFig.7,forall threegroups.Thepatientstagesshowedasignificantcorrelationwith bothMMSEandADAS13scores.Thecorrelationcoefficientswerealso comparableinthethreegroups.

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Fig.5. Experiment2:Theeffectof𝜀𝐺(differenceinabnormalbiomarkerlevelsinthetwogroups),ontheperformanceoftheproposedmethods.Theshadedregion

representsstandarddeviationoftheerrorin50randomiterations.Theplotsin(a)and(b)showtheorderingerrorsinGroup1usingCoupledDEBMandCo-init

DEBMwithindependentDEBMshowninboth(a)and(b),asafunctionofnumberofsubjectsinGroup1.Theplotsin(c),(d)and(e)showtheorderingerrorsin

Group2usingindependentDEBM,CoupledDEBMandCo-initDEBMrespectivelyasafunctionofnumberofsubjectsinGroup1.

5. Discussion

DEBMmodelshavebeenshowntobeeffectiveindeterminingthe temporal cascadeof biomarkerabnormality as AD progresses, from cross-sectionaldata.Inthiswork,weintroducedanovelconceptof split-tingthedifferentstepsofDEBMintogroup-specificandgroup-aspecific partsforcoupledtraininginstratifiedpopulation.Weconsideredtwo novelvariationstosplitthestepsofDEBMinthismannerandthrough thoroughexperimentationinsimulationdatasetsweobservedthat co-initDEBMhelpsin obtainingmore accurateorderingsinastratified population.Usingthismethod,weestimatedthebiomarkercascadesin ADprogressionwith𝜀2alleles,homozygous𝜀3alleles,and𝜀4allelesof

APOE,basedoncross-sectionalADNIdata.Whilethefindingsinthe ho-mozygous𝜀3carrierand𝜀4carriergroupsfitthecurrentunderstanding ofprogressionofADwithhigh-confidence,thefindinginthe𝜀2carrier groupshowsevidenceforanalternativepathway(withrelativelylow confidence).Inthissection,wediscusstheinsightsprovidedbythe sim-ulationexperiments(Section5.1)usedformethodselectionaswellas theinsightsintotheADprogressionpathwaysprovidedbyour experi-mentsontheADNIdataset(Section5.2).

5.1. Choiceofthemethod

Coupled DEBM and co-init DEBM both split DEBM into group-specificandgroup-aspecific stepsfor coupledtrainingof anEBMin stratifiedpopulations.Experiment1and2showedthatcoupled

train-ingof thegroup-aspecific partsof DEBMandindependentlytraining thegroup-specificpartsofDEBMresultsinmoreaccurateorderingsin thegroupsbetterthanthedefaultapproachofindependentlytraininga DEBMmodelineachgroup.

WhilesplittingDEBMintogroup-specificandgroup-aspecificparts, westartedwiththeassumptionthatthelatenttruenormaland abnor-malbiomarkerdistributionsinthegroupsareeithersameorsimilar. Thedifferencebetweenco-initDEBMandcoupledDEBMisthat, co-initDEBMaccountsforslightdifferencesintheunderlyingbiomarker distributionsbetweenthegroupswhereascoupledDEBMdoesnot.

ThesimulationdatasetgeneratedinExperiment1hadthesametrue normalandabnormalbiomarkerdistributionsinthedifferentgroups, from whichthe simulatedsubjects wererandomlysampled,aligning wellwiththeassumptionofcoupledDEBM.However,thisdidnot re-sultinoverallbetteraccuraciesforcoupledDEBMthanthatofco-init DEBM.Co-initDEBMwasalsomorerobustthancoupledDEBMasits ac-curacywaslessdependenton𝜀𝑂,thedistancebetweentheground-truth

orderingsinthetwogroups.

Another observationin Experiment1, which wasrather counter-intuitive,wasthattheerrorsmadebytheco-initandcoupledDEBM modelsdecreasedasthedistancebetweentheground-truthorderings inthetwogroupsincreased.Whentheorderingsarefurtherapart,the combinedbiomarkerdistributionsinCNandADgroupshavealarger overlap.Thenon-overlappinginitialization(beforetheGMM optimiza-tion)thus resultsinthenormalandabnormaldistributionstobe fur-ther apart.Wehypothesizethatthis resultsin abetterestimationof

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Fig. 6. Experiment 3: Orderings of CSF, global cognition and volumetric

biomarkersintheAPOEbasedgroupsof𝜀2carriers,homozygous𝜀3carriers,

and𝜀4carriersalongwiththeiruncertaintyestimates.Uncertaintyinthe

esti-mationoftheorderingwasmeasuredby100repetitionsofbootstrapping,in

thethreeAPOEbasedgroups.Thecolor-mapisbasedonthenumberoftimes

abiomarkerisatapositionin100repetitionsofbootstrapping.Thenumberof

subjectsinthethreegroupswere75,411and485respectively.Theorderings

wereobtainedusingCo-initDEBM.

themixingparametersduringGMMoptimizationandin-turnresulted inmoreaccurateorderings,asmixing-parametersaredependentonthe biomarker’spositionintheordering.

InExperiment2, wecheckedthe performanceofour approaches whentheassumption(truenormalandabnormalbiomarker distribu-tionsbeingsameacrossgroups)isviolatedinthedataset.This experi-mentshowedthattheorderingsobtainedusingco-initDEBMaremore robust todifferencesbetween theabnormal Gaussians across groups thanthoseobtainedwithcoupledDEBM.WithcoupledDEBM,theerror increasedinthegroupwithweakerbiomarkersi.e.,Group1inthecase of𝜀𝐺=+0.2𝑑 andGroup2inthecaseof𝜀𝐺=−0.2𝑑.Thisshowsthat coupledDEBMintroducesasystematicbiasintheestimationofordering thatisdetrimentaltothegroupwithweakerbiomarkers.Co-initDEBM alsoshowedasimilarbias,buttoamuchlesserextent.

Wehenceselectedco-initDEBMasthepreferredapproachfor split-tingandperformedouranalysisonADNIdatasetusingthisapproach. We expect that this idea of splitting DEBM into group-specific and group-aspecific parts can be easily extended tothe EBMintroduced byFonteijnetal.(2012).

5.2. CascadeofbiomarkerchangesintheAPOEbasedgroups

Dividing the total population into groups based on APOE car-riership enabled us to create more phenotypically homogeneous groups(Weintraubetal.,2019),eachwithpotentiallyspecificdisease progression timeline. Inthis section,we discussour results in these

APOEcarriershipbasedgroups.

Our findings show that the three APOE-carriership based groups have significantlydifferenttemporalcascadesof diseaseprogression. Thissuggeststhattheunderlyingpathwaysofprogressionaredifferent forthethreegenotypes.AmongtheCSFbiomarkersinthehomozygous

𝜀3carrierandthe𝜀4carriergroups,ABETAabnormalityistheearliest biomarkereventfollowedbyPTAU.Thisfitscurrentunderstandingof ADprogression(Bloom,2014).Italsoconfirmstheneedforpreventing theaccumulationofABETAinhigh-riskpatients.NFLandNGarelate biomarkersinthehomozygous𝜀3carrierand𝜀4carriergroups,which suggeststhataxonal(Ashtonetal.,2019)andsynaptic(Thorselletal., 2010)degenerationdonotoccuruntilverylateinthediseaseprocessin thesegroups.NGbeingabnormalafterPTAUandTAUinthe homozy-gous𝜀3carrierand𝜀4carriergroupsisalsoconsistentwiththeprevious findingsthatTaumediatessynapticdamageinAD(Jadhavetal.,2015). Inthe𝜀2carriergroup,wefoundthattheabnormalNGandPTAU aretheearliestCSFevents,evenbeforeABETAbecomesabnormal.This couldhintattheexistenceofanalternativepathwayfortheformation of tautanglesinthebrainbeforeABETAaccumulation,assuggested inWeigandetal.(2019),butneedsmoreextensivevalidation.

Amongthevolumetricbiomarkers,Entorhinalcortexisoneofthe earlybiomarkersinthe𝜀4carriergroupwhichissupportedbythe find-ingsinHuijbersetal.(2014),butisoneofthelastbiomarkersto be-comeabnormal inthehomozygous𝜀3carriergroup.Ventricular vol-umeisalatebiomarkerinthe𝜀4carriergroupbutitbecomes abnor-malquiteearlyinthehomozygous𝜀3carriergroupasalsoobserved byNestoretal.(2008).Hippocampusvolumeistheearliestbiomarker in the𝜀4carrier group,butisa relativelylatebiomarkerinthe ho-mozygous𝜀3carrierand𝜀2carriergroups.Thissuggeststhatincidence ofhippocampalsparingAD(Ferreiraetal.,2017)couldcorrelatewith

APOEcarriership.

Thefindings relatedtothese orderingsof biomarkereventswere validatedbycorrelatingthepatientstagesderivedfromtheseorderings withMMSEandADAS13scores.Patientstagesofsubjectsinallthree groups,whenusedastest-subjectsinacross-validatedmanner,showeda significantcorrelation(𝑝<0.001)withthesescores.Thesecorrelations validateourfindingsandsuggestthatthesegenotype-specificdisease progressiontimelinescouldbeusedforpatientmonitoring.

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Fig. 7.Experiment 4: Correlation of

esti-mateddiseasestageswithMMSEandADAS

scoresintheAPOEbasedgroupsof𝜀2carriers, homozygous𝜀3carriers,and𝜀4carriers.The

plotontopofeachsubfigureshowsthe

prob-abilitydensityfunctionofthediseasestages,

andtheplotontherightof eachsubfigure

showstheprobabilitydensityfunctionofthe

cognitivescoreinthesubfigure.The2Dplot

ineachsubfigureshowsthejointdensity func-tionofthetwoaxes.Thelineineachsubfigure

showsthelinearregressionofMMSE/ADAS

scoreswiththeestimateddiseasestageandthe shadedareaaroundthelineshowsits95% con-fidenceinterval.Figures(a),(c)and(e)depict

correlationbetweenMMSEscoreandobtained

diseasestagesinthethreeAPOEbasedgroups. Figures(b),(d)and(f)depictcorrelation

be-tweenADAS13scoreandtheobtaineddisease

stagesinthethreeAPOEbasedgroups.

6. Conclusionandfuturework

Weconcludethatco-initDEBMprovidesthebestaccuracyand ro-bustness when estimating orderings in stratified populations. Future workonco-initDEBMcanfocusonextendingtheapproachfor high-dimensionalimagingbiomarkers(Venkatraghavanetal.,2019b).This

work also provides groundwork for extending the method towards hypothesis-free,data-drivenstratificationofphenotypes.

WegainednewinsightsintothediseaseprogressiontimelineofAD intheAPOEbasedgroupsof𝜀2carriers,homozygous𝜀3carriers,and

𝜀4carriers.Whileweobservedthattheestimateddiseaseprogression timelines in the𝜀4 carrierandthehomozygous𝜀3 carriergroupsfit

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thecurrentunderstandingofADprogressionwithhighconfidence,the estimatedtimelinesinthe𝜀2carriergroupmaysuggestanalternative pathwayfortheformationoftautanglesinthebrainbeforeamyloid

𝛽 accumulation,albeitwithrelativelylow condence.Weexpect that

thesegenotype-specificdiseaseprogressiontimelineswillbenefitpatient monitoringinthefuture,andmayhelpoptimizeselectionofeligible subjectsforclinicaltrials.

Acknowledgement

ThisworkispartoftheEuroPONDinitiative,whichisfundedbythe EuropeanUnion’sHorizon2020researchandinnovationprogramme undergrantagreementNo. 666992. E.E.Bronacknowledgessupport fromtheDutchHeartFoundation(PPPAllowance,2018B011),Medical DeltaDiagnostics3.0:DementiaandStroke,andtheNetherlands Cardio-VascularResearchInitiative(Heart-Brain Connection:CVON2012-06, CVON2018-28).Datacollectionandsharingforthisprojectwasfunded bytheAlzheimer’sDiseaseNeuroimagingInitiative(ADNI)(National InstitutesofHealthGrantU01AG024904)andDODADNI(Department ofDefenseawardnumberW81XWH-12-2-0012).ADNIisfundedbythe NationalInstituteonAging,theNationalInstituteofBiomedical Imag-ingandBioengineering,andthroughgenerouscontributionsfromthe following: AbbVie,AlzheimersAssociation;Alzheimer’sDrug Discov-eryFoundation;AraclonBiotech;BioClinica,Inc.;Biogen;Bristol-Myers SquibbCompany;CereSpir,Inc.;Cogstate;EisaiInc.;Elan Pharmaceu-ticals,Inc.;EliLillyandCompany;EuroImmun;F.Hoffmann-LaRoche LtdanditsaffiliatedcompanyGenentech, Inc.;Fujirebio;GE Health-care;IXICOLtd.;JanssenAlzheimerImmunotherapyResearch& Devel-opment,LLC.;Johnson&JohnsonPharmaceuticalResearch& Develop-mentLLC.;Lumosity;Lundbeck;Merck&Co.,Inc.;MesoScale Diagnos-tics,LLC.;NeuroRxResearch;NeurotrackTechnologies;Novartis Phar-maceuticalsCorporation;PfizerInc.;PiramalImaging;Servier;Takeda PharmaceuticalCompany;andTransitionTherapeutics.TheCanadian InstitutesofHealthResearchisprovidingfundstosupportADNI clin-icalsitesinCanada.Privatesectorcontributionsarefacilitatedbythe FoundationfortheNationalInstitutesofHealth(www.fnih.org).The granteeorganizationistheNorthernCalifornia InstituteforResearch andEducation,andthestudyiscoordinatedbytheAlzheimer’s Thera-peuticResearchInstituteattheUniversityofSouthernCalifornia.ADNI dataaredisseminatedbytheLaboratoryforNeuroImagingatthe Uni-versityofSouthernCalifornia.

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