Machine learning based white matter models with permeability: An experimental study in cuprizone treated in-vivo mouse model of axonal demyelination

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Machine learning based white matter models with

permeability: An experimental study in cuprizone

treated in-vivo mouse model of axonal demyelination

Ioana Hill, Marco Palombo, Mathieu Santin, Francesca Branzoli,

Anne-Charlotte Philippe, Demian Wassermann, Marie-Stéphane Aigrot,

Bruno Stankoff, Anne Baron-van Evercooren, Mehdi Felfli, et al.

To cite this version:

Ioana Hill, Marco Palombo, Mathieu Santin, Francesca Branzoli, Anne-Charlotte Philippe, et al..

Machine learning based white matter models with permeability: An experimental study in cuprizone

treated in-vivo mouse model of axonal demyelination. NeuroImage, Elsevier, 2021, 224, pp.117425.

�10.1016/j.neuroimage.2020.117425�. �hal-03164018�

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demyelination

Ioana

Hill

a,1

_,

_Marco

_Palombo

a,1,∗

_,

_Mathieu

_Santin

b,c

_,

_Francesca

_Branzoli

b,c

_,

Anne-Charlotte

Philippe

b

_,

_Demian

_Wassermann

d,e

_,

_{Marie-Stephane}

_Aigrot

b

_,

_Bruno

_Stankoﬀ

b,f

_,

Anne

Baron-Van

Evercooren

b

_,

_Mehdi

_Felﬂi

b

_,

_Dominique

_Langui

b

_,

_Hui

_Zhang

a

_,

Stephane

Lehericy

b,c

_,

_Alexandra

_Petiet

b,c

_,

_Daniel

_C.

_Alexander

a

_,

_Olga

_Ciccarelli

g

_,

Ivana

Drobnjak

a

a Centre for Medical Image Computing and Dept of Computer Science, University College London, London, UK

b Institut du Cerveau et de la Moelle épinière, ICM, Sorbonne Université, Inserm 1127, CNRS UMR 7225, F-75013, Paris, France c Institut du Cerveau et de la Moelle épinière, ICM, Centre de NeuroImagerie de Recherche, CENIR, Paris, France

d Université Côte d’Azur, Inria, Sophia-Antipolis, France e Parietal, CEA, Inria, Saclay, Île-de-France

f AP-HP, Hôpital Saint-Antoine, Paris, France

g Dept. of Neuroinﬂammation, University College London, Queen Square Institute of Neurology, University College London, London, UK

a

b

s

t

r

a

c

t

Theintra-axonalwaterexchangetime(𝜏i),aparameterassociatedwithaxonalpermeability,couldbeanimportantbiomarkerforunderstandingandtreating

demyelinatingpathologiessuchasMultipleSclerosis.Diffusion-WeightedMRI(DW-MRI)issensitivetochangesinpermeability;however,theparameterhasso farremainedelusiveduetothelackofgeneralbiophysicalmodelsthatincorporateit.Machinelearningbasedcomputationalmodelscanpotentiallybeusedto estimatesuchparameters.Recently,forthefirsttime,atheoreticalframeworkusingarandomforest(RF)regressorsuggeststhatthisisapromisingnewapproach forpermeabilityestimation.Inthisstudy,weadoptsuchanapproachandforthefirsttimeexperimentallyinvestigateitfordemyelinatingpathologiesthroughdirect comparisonwithhistology.

WeconstructacomputationalmodelusingMonteCarlosimulationsandanRFregressorinordertolearnamappingbetweenfeaturesderivedfromDW-MRIsignals andgroundtruthmicrostructureparameters.Wetestourmodelinsimulations,andﬁndstrongcorrelationsbetweenthepredictedandgroundtruthparameters (intra-axonalvolumefractionf:R2₌_0.99,_𝜏

i:R2=0.84,intrinsicdiﬀusivityd:R2=0.99).Wethenapplythemodelin-vivo,onacontrolledcuprizone(CPZ)mouse

modelofdemyelination,comparingtheresultsfromtwocohortsofmice,CPZ(N=8)andhealthyage-matchedwild-type(WT,N=8).WefindthattheRFmodel estimatessensiblemicrostructureparametersforbothgroups,matchingvaluesfoundinliterature.Furthermore,weperformhistologyforbothgroupsusingelectron microscopy(EM),measuringthethicknessofthemyelinsheathasasurrogateforexchangetime.HistologyresultsshowthatourRFmodelestimatesareverystrongly correlatedwiththeEMmeasurements(𝜌 =0.98forf,𝜌 =0.82for𝜏i).Finally,wefindastatisticallysignificantdecreasein𝜏iinallthreeregionsofthecorpus

callosum(splenium/genu/body)oftheCPZcohort(<𝜏i>=310ms/330ms/350ms)comparedtotheWTgroup(<𝜏i>=370ms/370ms/380ms).Thisisinlinewithour

expectationsthat𝜏iislowerinregionswherethemyelinsheathisdamaged,asaxonalmembranesbecomemorepermeable.Overall,theseresultsdemonstrate,for

theﬁrsttimeexperimentallyandinvivo,thatacomputationalmodellearnedfromsimulationscanreliablyestimatemicrostructureparameters,includingtheaxonal permeability.

1. Introduction

Theintra-axonalwaterexchangetime(𝜏i),aparameterassociated

withaxonalpermeability,isanimportantmicrostructuralpropertyof thetissue,whichhasbeenlinkedwithmyelinationinthecentral ner-voussystem(Nilssonetal.,2013a).Severalneurologicalconditionssuch as MultipleSclerosis(MS) cause a breakdownof themyelin sheath

∗_{Corresponding}_author.

E-mailaddress:marco.palombo@ucl.ac.uk(M.Palombo). 1 _These_authors_contributed_equally_to_this_work.

throughaprocessknownasdemyelination, whichmaylead toa de-creasein theexchange timeastheintra-axonal watermolecules en-counter lessbarriers. Changesin permeabilityhave alsobeenlinked withpathologiessuchasParkinson’sdisease(Vollesetal.,2001)and cancer Huetal.(2006),leadingtoawidespreadinterestin develop-ingpermeability-basedbiomarkers.Duetoitssensitivitytothemotion ofwatermoleculeswithintissue,modellingofDiﬀusion-WeightedMRI (DW-MRI)dataenablestheestimationof𝜏i.However,measuringithas

https://doi.org/10.1016/j.neuroimage.2020.117425

Received24July2019;Receivedinrevisedform29September2020;Accepted30September2020 Availableonline6October2020

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beenproblematicduetotheintractabilityofthemathematical expres-sionswhichaccuratelyincorporate𝜏iintoanalyticalmodels.

Sofar,thebiophysicalmodelsthatincorporatepermeabilityrelyon assumptionsthatareeithertoosimplistic(Callaghan,1997, Coddand Callaghan,1999, Vangelderenetal.,1994)ordonotholdinhuman tis-sue(Grebenkov etal.,2014, Kärgeretal., 1988).TheKärgermodel (Kärger et al., 1988) is themost widely used analyticalmodel that incorporatespermeability (Nilssonetal., 2010, Stanisz et al., 2005, Lättetal.,2009).However,itsassumptions(i.e.theindividualpools ofwaterarewellmixedandnotrestricted)donotholdinwhite mat-terandthemodelwasshowntofailwhenappliedtohighlypermeable tissue(Fieremansetal.,2010).Ameasurementtechniqueforaccessing exchangeistheapparentexchangerate(AXR)imaging,however,it re-quiresaspecialisedimagingprotocol(Lasič etal.,2011,Nilssonetal., 2013b).

Computational modelsbypass theneed foranalyticalexpressions andincorporatepermeabilitybycreatingamappingbetween simula-tionsoftheDW-MRIsignalandthegroundtruthmicrostructure param-eters. Nilssonetal.(2010)use MonteCarlosimulationswithknown groundtruthparametersincludingpermeabilitytogenerateasynthetic libraryofDW-MRIsignals.Givenapreviouslyunseensignal,they es-timatepermeabilityviaanearest-neighbouralgorithm.However,their approachrequiresnewlibrariestobegeneratedforeachacquisition pro-tocol-whichinsomecasesmayrepresentaproblem-andthe nearest-neighbouralgorithmingeneraldoesnothaveagoodgeneralisation ca-pacity.

Recently,Nedjati etal.(2017) applyforthefirsttimeamachine learningapproachusingarandomforest(RF)trainedonadatabaseof rotationallyinvariantfeaturesderivedfromtheDW-MRIsignals simu-latedusingsyntheticsubstratesofdenselypackedcylinders. Rotation-allyinvariantmetrics(e.g.MDandFAfromDTI)aremetricscalculated fromDW-MRIdatathatdonotdependontheparticularorientationof theunderlyingtissuewithrespecttothescannerreferenceframe,thus providingvaluablemetricsforinter-subjectandacross-platform anal-yses.ThemodelproposedbyNedjatietal.(2017)usesanRFinstead ofstandardmodel-fittingapproachesbasedon minimizationof (non-linear)least-squaresbecauseitismorecomputationallyefficient;itis lessprone tolocalminimumproblems;anditnaturallyencodeseven complex constraintson parametercombinations throughappropriate choiceof trainingdata, whileguaranteeinggood generalisation.The novelRFmodelisshowntooutperformtheKärger’smodelonsynthetic andin-vivohumandatabyprovidingmorereproducibleandrobust es-timatesof𝜏i(Nedjatietal., 2017).However,theirin-vivoapproachis

testedonlyqualitativelyonjusttwoMSpatients.Furthermore,Nedjati etal.(2017)hypothesisethat𝜏iislinkedwithdemyelinationinMS

le-sions,buttheydonotshowwhetherotherunderlyingprocessessuchas axonalswellingororientationdispersionaﬀecttheestimates.Here,we aimtoaddresstheselimitations.

Theaimofthisstudyistoexperimentallytestamachinelearning basedcomputationalmodelwithpermeabilityusingahighlycontrolled cuprizone-treated,in-vivomousemodelofdemyelination(CPZ),anda directcomparisontohistology.WeadopttheRFframeworkintroduced in Nedjatiet al(2017) toestimatetissue microstructureparameters. Priortoourin-vivoexperiments,weusesimulationsrepresentativefor ourmousedatatoinvestigatethesensitivityofthePGSEprotocolused toacquirethein-vivodatato𝜏i,andselectthemostinformativebshells

(i.e.bvaluesanddirections)withrespecttothisparameter.We addi-tionallyestablishabenchmarkperformanceforourmodelbytesting itsperformanceonsimulations.Totestthein-vivoperformanceofthe model,weusetwocohortsofmice:CPZandhealthyage-matched wild-type(WT),withDW-MRIscansandhistologydata.Ourdemyelination modelallowsustoinvestigatethedirectcorrelationbetweenthe esti-matedexchange timeandhistologicalmeasurementsofmyelin thick-ness.Furthermore,weinvestigatethepotentiallyconfoundingeﬀectsof dispersionandaxonalswellingtoeliminateanypotentialbiasinour

estimatesoftheexchangetime.Finally,weanalysethecorrelations be-tweentheestimationsofourmodelandhistologydata.

2. Methods

This sectionﬁrst describesthe imagingprotocol,in-vivo data ac-quisition,histologyanalysisandthemachinelearningmodelandthen outlinestheprincipalstepsofourexperimentalframework.Firstly,we investigateusingsyntheticdatathesensitivityofourimagingprotocolto changesin𝜏i.Secondly,weoptimiseourcomputationalmodelthrough

ashellselectionprocessandestablishabenchmarkperformanceforour modelinsimulations.Weﬁrstensurethereisagoodmatchbetweenthe syntheticandin-vivodataandweinvestigateanybiasinourmachine learningpredictionsof𝜏ibylookingattheeﬀectofpotential

confound-ingfactors. Finally,wetest thein-vivoperformanceof ourmachine learningmodelonacuprizonemousemodelofdemyelinationandwe analysethecorrelationsbetweenthepredictionsandtheex-vivo histo-logicalmeasurementsavailable.

2.1. Mousedata

2.1.1. In-vivodataacquisition

Weimagetwocohortsof8-weekoldC57BL/6Jfemalemice,CPZ (N=8)andWT(N=8),usingthesamescannerandacquisition proto-colaspresentedbelow. Allanimalexperiments areperformedin ac-cordance with the European Council Directive (88/609/EEC). Eight mice werefed0.2% cuprizonefor 6weeks,which corresponds toa demyelinationwithoutrecoveryphase,andeighthealthyage-matched wild-type(WT)miceofthesamebackgroundwerefedanormalchow diet andusedascontrols.AllmicearescannedonaBrukerBioSpec 11.7TscannerusingtheprotocoldescribedinSection2.1.2below.The WT data used in this study areavailable in the publicdomain and can be foundat https://zenodo.org/record/996889#.WgH5E9vMx24 (Wassermannetal.,2017).Theauthorsdonothavepermissiontoshare thedatausedinthisstudyfortheCPZtreatedmice.Allthecodeused fortheanalysisisavailableuponrequesttothecorrespondingauthors. Wepost-processtheimagesbycorrectingforeddycurrentsusing FSL-eddy(Smithetal., 2004).Nomotionartefactsareobserved.We restrict our analysisto whitematter voxelswithin the corpus callo-sum(CC).ToselecttheCCvoxels,wecomputemapsoflinearity(CL),

planarity(C_P)andsphericity(C_S)(Westinetal.,2002)fromthe diﬀu-siontensor(DT)ﬁttotheshellatb=1241s/mm2_._We_create_the_CC

mapsbyselectingthevoxelswithCL>0.3,CP<0.4,CS<0.5andfractional

anisotropy(FA)>0.40(valuechosentodistinguishWMformGMand CSFvoxelsalsointhecuprizonetreatedmice,whereFAvaluescanbe lowerthantheWTones).Followingthisprocedure,weobtainmasks ofthecorpuscallosumwhosethicknessvariesslightlyacrossallmice, randomlyandwithnostatisticallysignificantdifferences.Specifically, themean±s.d.of thenumberofvoxelscomprisingtheCCmaskin theWTgroupis161±13andintheCPZgroupis175±18.Followinga two-tailt-testwefindthisdifferencestatisticallyinsignificant(p>0.05). Previousstudies,suchas Wuetal.(2008),showedstatistically signif-icantincreaseinthevolumeofCCofCPZintoxicatedmicecompared toWT.However,wedonotmeasureastatisticallysignificantincrease andthefurtherinvestigationofthisobservationisoutofthescopeof thepresentstudy.

2.1.2. Diﬀusionimagingprotocol

WeusethesameDW-PGSEprotocolforsyntheticandin-vivodata, optimised tomaximisesignal reconstructionaccuracyunderrealistic timeconstraints(Filipiaketal.,2019).Our imagingprotocolhas25 shells,each withoneb=0measurementandadifferent combination ofdiffusiongradientstrengthGanddiffusiongradientseparationΔas summarisedinTable 1 .Theresultingprotocolhas345measurementsin total,diffusiongradientduration𝛿=5ms,|Gmax|=500mTm−1andshell

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Fig.1. Schematicpipelineofthestereologicalanalysistocomputeg_ratiosandaxonaldiametersinthecorpuscallosumofthemice.First,Tenequallyspacedslices arecutwithinthe1millimeterfromthemiddleofthecorpuscallosuminthesagittalsectiontowardstheedgeofthebrain(A).Then4slicesaresampledstarting fromarandomnumber.Inthiscase,therandomlychosenstartingnumberis1,andtheselectedslicesare#1,#4,#7and#10(B).Subsequently,theseslicesare usedtolocalisetheareasofinterest(e.g.,genu,bodyorspleniumasshowninC),andeachoneofthoseisslicedultra-thinly.Onarandomlychosenultra-thinslice foreachoftheROIs,30spotsarealsorandomlychosenovertheentireROIatsmallermagnificationtoassurethatimagesarenotintersected(Dshowsjust6ofthose)before acquiringthefinalEMimageat62Kmagnification.Eachofthe30randomspotsareselectedforstereologicalanalysisusingpointgridsof36regularlyspacedcrosses,each onerepresentinganareaof0.5𝜇m2₍_E_shows_two_of_those₃₀_spots,_one_for_WT_and_one_for_CPZ_)._These_point_grids_are_used_for_{quantification}_of_the_WT_and_CPZ_mice.

Table1

DW-PGSEparameters withthe corresponding nominal b-values in s/mm2_. 𝚫 (ms) 10.8 13.1 15.4 17.7 20 #grad dirs G (mT/m) 150 358 445 533 620 707 16 200 620 775 930 1086 1241 16 300 1384 1733 2083 2432 2781 8 400 2489 3110 3731 4352 4973 11 500 3892 4862 5833 6803 7773 13

b-valuesasshowninTable 1 .Additionalprotocoldetailsareasfollows: TE=33.6ms,TR=2s,FOV=16×16mm,matrixsize=160×160, num-berofslices=5,slicethickness=0.5mm.Totalacquisitiontime53min. 2.1.3. Histologysamples

TheWT(n=8)andCPZ(n=8)animalsaresacrificedbydeep anaes-thesiaandperfusedintracardiallywith1%paraformaldehydeand2.5% glutaraldehydeinphosphatebuffer0.12M,pH7.4attheendofthe 6-weekCPZtreatment.Theextractedbrainsarethenpost-fixedovernight at4°Cinthesamefixativeandrinsedinphosphatebuffer.Ten 100μm-thicksagittalsectionsarecutwithavibratome(ThermoScientific Mi-cromHM650VVibrationmicrotome)(Fig. 1 A).Theveryfirstsection closesttothebrainmidlineisconsideredas#1andsections#1,#4, #7,and#10 areselected (Fig. 1 B).Sections arepost-fixed with1% osmiumtetroxideinwaterfor1hatroomtemperature(RT°),rinsed

3×5minwithwaterandcontrasted“enbloc” for1hatRT° with2% aqueousuranylacetate.Afterrinsing,sectionsareprogressively dehy-dratedwith50%,70%,90%,and100%ethanolsolutionsfor2×5min each.Finaldehydration isachievedby immersingthesectionstwice in 100%acetonefor10min.Embeddingis performedinepoxy resin (Embed812,EMS,Euromedex,France)overnightin50%resin/50% acetoneat4°Cfollowedby2×2hinpureresinatRT°,and polymer-izationisachievedat56°Cfor48hinadryoven.Semi-thinsections (0.5𝜇m-thick)arecollectedwithanultramicrotomeUC7(Leica,Leica MicrosystèmesSAS,France)andstainedwith1%toluidinebluein1% boraxbuﬀer(Fig. 1 D).Ultra-thinsections(70nm-thick)arecontrasted withReynold’slead citrate(ReynoldES,1963),andobservedwitha transmissionelectronmicroscope(HITACHI120kVHT7700), operat-ingat70kV.Images(2048×2048pixels)areacquiredwithanAMT41B camera(pixelsize:7.4μmx7.4μm)(Fig. 1 E).

2.1.4. Post-mortemanalysis

Fromtheelectronmicroscopy(EM)samplesobtainedasoutlinedin

Section 2.1.3 ,weestimatethemeanandstandarddeviationofthegratio,

myelinthickness,axonaldiameterandtheintra-axonalvolumefraction oftheWTandCPZmice.Thestereologicalanalysisisperformedin iso-latedregionsoftheCC(genu,bodyandsplenium),where4random sec-tionswithuniformdistancearequantiﬁedperanimal(Fig. 1 B),with30 randomlylocatedimagesperregionandperanimalacquiredat62,000 magniﬁcation. Forvolumefraction(VF)weproceedaccordingtothe Delesseprinciple Mouton(2002):volumefractionsarecalculatedby

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di-vidingthetotal numberofpoints hittingthestructure(P(Y)) bythe totalnumberofpointshittingthereferencevolume(P(ref)),following theequation:𝑉𝐹(𝑌,𝑟𝑒𝑓)= ∑𝑚 𝑖=1𝑃(𝑌)𝑖 ∑𝑚 𝑖=1𝑃(𝑟𝑒𝑓)𝑖 .

Agridof36regularlyspacedcrosses(Fig. 1 E)isgeneratedwithFiji, anopen-sourceplatformforbiologicalimageanalysis(Schindelinetal., 2012).Toidentifynon-perpendicularaxonsandremovethemfromthe analysis,wetakeintoaccounttheshapeoftheaxonsandthe micro-tubulesinsidethem.Perpendicularaxonshaveaminimallyelongated shapeandtheirmicrotubulesaresmallperfectlycircularstructures in-sidethem.Incontrast,non-perpendicular axonshavemoreelongated shapes(e.g.moreellipsoid-like)andtheirmicrotubulesappearlikelines, dependingontheangleofthesection.Stereologicalanalysisprovides MyelinVolumeFractions(MVF),Axon VolumeFractions(AVF),and thetotalAxon VolumeFractions(tAVF),whichincludesboth myeli-natedandunmyelinatedaxons.TotalAxonCount(tAxCount)is man-uallyquantiﬁed. Thegratio of myelinatedﬁbers isthencalculatedas

g_ratio=√ 𝐴𝑉𝐹

(𝑀𝑉𝐹 +𝐴𝑉𝐹) andthemeanaxondiameters(DAX)are

calcu-latedasDAX=2 × √

(𝑡𝐴𝑉𝐹 ×𝑠𝑢𝑟𝑓𝑎𝑐𝑒) (𝜋×𝑡𝐴𝑥𝐶𝑜𝑢𝑛𝑡) .

Theoutliersinducedbythenon-perpendicularaxonsintheimages arenottakenintoconsideration.FromthegratioandtheDAX,myelin

thicknessiscomputedas:myelinthickness= 𝐷𝐴𝑋

2𝑔𝑟𝑎𝑡𝑖𝑜(1−𝑔𝑟𝑎𝑡𝑖𝑜).

WecomparetheestimatesoftheRFwiththeEMmeasurementsby computingthegroup-wisemeanintheCCROIsofthemyelinthickness andintra-axonalvolumefraction(VF) andlookingatthecorrelation betweentheseandtheRFestimationsfor𝜏iandf.

2.2. Syntheticdata

Amachinelearningregressorcanbetrainedondiﬀerentdatabases. Inthiswork,weaimtocomparetheperformanceoftrainingdirectly onsimulatedsignalsversustrainingonfeaturesobtainedbymodelling thosesignals.Therefore,weconstructtwotrainingdatabases:one com-prisedofsyntheticDW-MRIsignalsandtheotherofrotationally invari-antfeaturesestimatedfromthosesignals.

Eachentryin thedatabasecorresponds toaunique digital phan-tomwhichmimicsthein-vivodataandforwhichthegroundtruth mi-crostructureparametersareknown.Eachsyntheticdatabaseisusedto traina machinelearningalgorithm,here anRF,tobuilda mapping betweenthesignalorfeaturesandthecorrespondinggroundtruth mi-crostructureparameters.Pleasenotethatinthiscontextwereferto “fea-tures” inamachinelearningsense:measurablepropertiesor character-isticsoftheDW-MRIsignal,andsomeofthefeaturesusedmaydepend onsomeoftheothers.

2.2.1. Syntheticsignalsdatabase

WeuseMonteCarlosimulationsoftheDW-MRIsignaltobuildour synthetictrainingdatabase.Thesignalsaregeneratedusingtheopen source Camino(Cooket al., 2006; http://camino.cs.ucl.ac.uk) simu-lationframework HallandAlexander(2009)togetherwiththe imag-ingprotocolinTable 1 .UsingtheCaminotoolbox,wegenerated syn-theticsignalsbyfirstsimulatingthediffusionofmanyspinsas three-dimensionalrandomwalkusingMonteCarlomethodsforeachsynthetic substratecomposedofrandomlypackedstraightcylinders.Then,from thesimulatedspinstrajectories,thediffusion-weightedsignalwas com-putedusingthephaseaccumulationapproach,accordingtothespecific diffusion-sensitisinggradientschemechosentomatchtheexperimental acquisitionprotocol.Thus,eachsimulatedsignalcorrespondstoa digi-talphantomwhichmimicsthein-vivomousebraindataintroducedin

Section 2.3 .Thedigitalphantomsarerepresentedbysyntheticsubstrates thatmodelwhitematterasacollectionof100,000non-abutting,parallel cylinderswithgamma-distributedradii,acommonchoiceinthebrain literature(Aboitizetal.,1992).Thecylindersarerandomlypackedin thesubstratesasdescribedinHallandAlexander(2009),withexample substratesshowninFig. 2 .Weconstructadatabaseof11,000unique

tissuesubstratesandtheircorrespondingDW-MRIsignalsbyrandomly samplingfromarangeofhistologicallyplausiblesubstrateparameters forwhitemattertissue(Aboitizetal.,1992, Barazanyetal.,2009).A whitemattersyntheticsubstrateisdeﬁnedthroughﬁveparameters:the mean𝜇R∈[0.2,1]𝜇mandthestandarddeviation𝜎R∈[min(0.1,𝜇R/5), 𝜇R/2]𝜇moftheaxonradiidistribution,theintra-axonalvolume

frac-tionf∈[0.4,0.7],theintra-axonalexchangetime𝜏i∈[2,1000]msand

theintrinsicdiﬀusivityd∈[0.8,2.2]𝜇m2_ms−1_._To_ensure_the

conver-genceandthehighprecisionofthesimulatedsignals,wegenerateour syntheticdatabaseusing100,000spinsand2,000timesteps(Halland Alexander,2009).TheMonteCarlosimulationsusedisplacements in continuousspace, withfixedstep sizein threedimensions𝑠=√6𝑑𝛿𝑡 Einstein(1905),with𝛿t=10𝜇s.Thepermeabilityofasubstrateis spec-ifiedwithintheCaminosimulationframeworkviatheprobability pa-rameter p.Thisparameter expressestheprobabilitythataspinsteps throughamembraneencounteredduringtherandomwalk(insteadof alwaysbeingreflectedbackwardsasitisthecaseforimpermeable sub-strates).Theprobabilitypisrelatedtothepermeabilitykthroughthe expression: 𝑝=2 3𝑘 √ 6𝛿𝑡 𝑑,

wheredistheintrinsicdiﬀusivityand𝛿tisthetemporalresolution.This expressionisobtainedbycombiningtheMonteCarlosteplength equa-tion𝑠=√6𝑑𝛿𝑡(HallandAlexander,2009)withthetransition proba-bilityequationasderivedin(ReganandKuchel,2000, Fieremansand Lee,2018).Here,wemeasurepermeabilitykviatheintra-axonalwater exchangetime𝜏i,whichisinverselyrelatedtokthroughtheexpression 𝑘= 𝑅

2𝜏𝑖 ,whereRistheaxonradius(Fieremansetal.,2010).

Tomaximisetheperformanceofourmachinelearningregressor,we aimtobuildatrainingdatabasethatresemblesascloselyaspossible thein-vivodata.Forthis,wegenerateanadditionalsetofsynthetic sig-nalstoaccountforthenoisepresentinthein-vivodata.WeaddRician noisewithastandarddeviation𝜎 correspondingtoanSNRof40,which reﬂectsthenoiseleveloftheb=0imageswiththelongest𝚫.

2.2.2. Syntheticfeaturesdatabase

Inordertomakethemethodgeneralizableacrossdiﬀerentscansand scanners,wetraintheRFregressorusingaconvenientdatabaseof fea-turesextractedfromtheDW-MRIsignalsthatareindependentof the speciﬁcorientationofthebrainwithinthescanner(i.e.rotationally in-variantfeatures)(Novikovetal.,2018, Reisertetal.,2017).Towards thisgoal,weobtainanequivalentrotationallyinvariantdatabaseby com-putingforeachofthesyntheticsignalsgeneratedinSection 2.2.1 asetof 15rotationallyinvariantfeatures(seeTableA.1),asdoneinNedjatietal. (2017).WecomputetheDTandthe4th_order_spherical_harmonic_(SH)

fitforeachbshellfromthesyntheticsignalsusingtheCaminotoolkit (Cooketal.,2006).Wethenderive15rotationallyinvariantfeatures foreachbshellandbuildanequivalentrotationallyinvariantsynthetic database.Thefirstfivesignal-derivedfeaturesarecalculatedfromthe DTfitandarethethreeeigenvalues𝜆1,𝜆2,𝜆3,themeandiffusivity(MD) andthefractionalanisotropy(FA).Theremainingtenfeaturesare de-rivedfromtheSHfit:themean,peak,anisotropy,skewnessandkurtosis oftheapparentdiffusioncoefficienttogetherwiththepeakdispersion (i.e.thestandarddeviationofthepeaksoftheSHfunctionsoverasetof evenlydistributedpointsinspace)andcombinationsofthefirst,second andfourthorderSH(Nedjatietal.,2017).SectionA.1intheAppendix presentsinmoredetailwhateachofthe15featuresrepresentsandhow itiscomputed.

2.3. Machinelearning 2.3.1. Randomforest(RF)

Due to theirinterpretability, robustnessto noiseand easiness of tuning (Criminisi etal., 2011),RFsarewidely usedasregressionor classiﬁcationtechniquesin themedicalﬁeld(Alexanderetal.,2017,

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Fig.2. ExamplesofthesynthetictissueusedforourMonteCarlosimulations.FromtwogivenexemplarGammadistributionsofaxondiameter(ﬁrstrow)four exemplardigitalsubstratesaregeneratedbypackingstraightnon-overlappingcylindersuptotwodiﬀerentintra-axonalvolumefractions:0.4(secondrow)and0.7 (thirdrow).

Geremiaetal.,2011, Nedjati-Gilanietal.,2017).AnRFisanensemble technique,builtofacollectionofdecisiontrees,calledweaklearners.An RFregressormakesestimatesbyaveragingtheanswersofallitsdecision trees,whichareindividuallytrainedthroughatechniquecalledbagging. Thistechniqueensuresthediversityofthetreesbytrainingeachtreeon adiﬀerentrandomtrainingsubset.Therandomnessanddiversityofthe treesensuretheirrobustnesstonoiseandgoodgeneralisation, result-ingintheRFactingasastronglearner Breiman(2001).Here,webuild anRFregressorthatlearnsamappingbetweenthesynthetictraining databaseofDW-MRIsignals/featuresandthegroundtruth microstruc-tureparametersofthecorrespondingsubstrates.Themappingislearnt throughagreedysplittingprocessoftheinputspace(thesynthetic sig-nals/features)guidedbytheassociatedtissueparametersprovidedas labelsduringtraining.

Duringthelearningphase,thetrainingdataispassedthroughthe de-cisiontree,startingattherootnodetowardstheterminalnodes.Ateach node,thedecisiontreesearchesforapartitionoftheincomingdatasuch thathavingseparatepartitionsoneithersideofthenodeimprovesthe estimation.Ifsuchapartitionexists,thenodeissplitandtwochildnodes areaddedonthelevelbelow.Thisprocedureisrepeatedforeverychild nodeuntilsplittingthedataintosmallerpartitionsdoesnotimprovethe estimationanymore.Ifnobetterpartitionisfound,thenodebecomes aterminalnode.Mathematically,thetrainingprocessisguidedbythe

optimisationofacostfunction,whichisusedtodeterminethebestsplit ateachnode.Theoptimisationsearchesforthefeature-thresholdpairs (fi,tﬁ)thatproducethebestsplit.Here,weusetheClassiﬁcationand

RegressionTree(CART)algorithmcostfunctionJ,deﬁnedas: 𝐽(𝑓𝑖,𝑡𝑓𝑖)=

𝑚𝑙𝑒𝑓𝑡

𝑚 𝑀𝑆𝐸𝑙𝑒𝑓𝑡+ 𝑚𝑟𝑖𝑔ℎ𝑡

𝑚 𝑀𝑆𝐸𝑟𝑖𝑔ℎ𝑡,

wherem_left/rightisthenumberoftraininginstancesintheleft/right sub-setand‘MSE’standsforthe’mean-squared-error’betweenthe ground-truthmicrostructureparameters(i.e.d,fand𝜏i)knownbydesignand

thepredictedones.

There aretwoimportantparametersthatneedtobeoptimisedto improvethelearningperformanceofanRF:thenumberoftreesandthe maximumtreedepth.Thenumberoftreesdeterminesthesmoothness of thedecisionboundary,andthetree depthparameterspecifiesthe maximumlevelsthateachdecisiontreecanhave.Toolargeavaluecan leadtooverfittingwhiletoolowavalueleadstounderfitting,depending onthecomplexityofthedata.Here,werunpreliminaryexperiments andoptimisethesetwoparametersforourtaskin ordertomaximise theperformanceofourmodel.

2.3.2. Trainingandtesting

WeimplementanRF regressorusingthescikit-learnopensource Pythontoolkit(Pedregosaetal.,2011).Followingpreliminary

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experi-ments,webuildanRFwith200treesofmaximumdepth20andbagging, asthesettingthatmaximisestheperformanceofthemodel.More gen-eralimplementationdetailscanbefoundathttp://scikit-learn.org/.We traintheRFforamulti-parameterregressiontask:weestimatethe intra-axonalexchangetime𝜏itogetherwiththeintra-axonalvolumefraction fandtheintrinsicdiﬀusivityd.UnliketheapproachinNedjatietal. (2017),wedonotﬁttheaxonradiusindex(Alexanderetal.,2010)due tothelackofsensitivityofthesignaltothisparameterforourimaging protocol(Burcawetal.,2015,Drobnjaketal.,2016).

Thedimensionalityofoursyntheticdatabasesis11,000by345for thesignaldatabaseand11,000by375forthefeaturedatabase.Weset thesizeoftrainingsetto11,000aswedidnotﬁndanyimprovementsin performanceabovethisnumber.Thelengthofeachsynthetictraining sampleisreducedfurtherduringtrainingaccordingtothenumberof bshellsselectedineachtrainingscenario.WetrainandtesttheRFon thesyntheticdatabasesusingtheassociatedgroundtruthparametersas labelsforthesupervisedregressiontask.Whenpredictingtheparameter mapsforthein-vivodata,wetraintheRFusingthenoisydatabasesas theyareamoreaccuraterepresentationofthein-vivodata.Wesplit oursyntheticdatabaseintoatrainingsetof9,500randomlyselected signal/featurevectorsandatestsetformedoftheremainingpreviously unseen1,500signal/featurevectors.AsshowninNedjatietal.(2017), theRFis notbiasedbytherandomselectionofthetrainingdataas longasthereissuﬃcientcoverageoftheparameterrange,whichwe alsoensure.Tobuildthetrainingset(tobe doneonlyonce) ittook approximately3days,using50nodesonourhigh-computingcluster ofCPUs.Thetrainingofthemachinelearningmodel(tobedoneonly once)took~1minandthepredictionofthemodelparametersfor~104

exemplarvoxelstook~1min,ona1.6GHzdual-coreIntelCorei5.Note thatthese times arejust indicative,andtheydependonthespeciﬁc hardwareused.

Inthisworkweexploretwopossiblewaysofusingmachine learn-ingformicrostructureestimation:usinga)signalsorb)featuresofthe signaltocreatethetrainingdatabase.The“signalstrainingdatabase” consistsofstandardDW-MRIsignalintensities(normalizedbytheb=0) forarangeofbvaluesandgradientdirections.The“Featurestraining database” iscreatedbyreplacingeachsignalatagivenbvalueinthe “signaltrainingdatabase” with15features,e.g.DTIandSHmetrics, cal-culatedusingallthegradientdirectionsforthatvoxelatthatbvalue, asdescribedinSection 2.2.2 .Whilethefirstapproachbuildsadirect mappingbetweentherawsignalsandthegroundtruthmicrostructure parameters,thesecondapproachintroducesanadditionalstepofmodel fittingandconstructsamappingbetweenDTandSHfeaturesoftheraw signalsandthemicrostructureparametersofinterest.Becausewechose touserotationallyinvariantfeatures,thesecondapproachis generaliz-ableacrossdifferentscansandscanners.

2.4. Experiments 2.4.1. Sensitivityanalysis

Firstly,weassessherethatintheanalyseddatathereissuﬃcient in-formationaboutthetargetedmicrostructuralparameters,inparticular 𝜏i.Toensurethatthereisenoughinformationinthedata,we

investi-gatethesensitivityofourPGSEprotocoltotheintra-axonalexchange timebylookingattherangeof𝜏i valuesforwhich theDW-MRI

sig-nalcan be distinguishedfrom thatof animpermeablesubstrate.For this,weconsidertwosyntheticsubstratesrepresentativeofmousewhite mattertissue,withthefollowingproperties:themeanaxonaldiameter 𝜇D=0.4𝜇mand𝜇D=2𝜇m,mimickingsmallandlargeaxonsintheCC,

theintra-axonalvolumefractionf=0.7(Barazanyetal.,2009),andthe intrinsicdiﬀusivity=1.2𝜇m2_ms−1₍_Wu_et_al.,₂₀₀₈_)._These_substrates

areagoodrepresentationofourin-vivomicedata,asshownbythe his-tologicalmeasurementsof𝜇DinSection 3.5 ,allwithintherangeofthe

gamma-distributionsabove.Notethatthechoiceofﬁxingthediﬀusivity to1.2𝜇m2_/ms_is_only_made_for_the_purpose_of_the_sensitivity_analysis_to

assessthesuitabilityoftheprotocol.Foralltheothersimulationsinthe

machinelearninganalysis,thediﬀusivityisvariedintheinterval[0.8, 2.2]𝜇m2_/ms,_as_done_in_Nedjati_et_al.,₂₀₁₇_,_and_as_shown_in_Wu_et_al.,

2008and Barazanyetal.,2009appropriateforrodents’brain.For ap-plicationonhumanbrain,higherdiﬀusivityof~2.2𝜇m2_/ms_should_be

usedforthesensitivityanalysis,accordingtorecentestimatesof intra-axonalaxialdiffusivityin-vivointhehumanbrain(Dhitaletal.2019). UsingtheCaminotoolbox,wegeneratesyntheticsignalsforeach substrateanddifferentvaluesof𝛿,ΔandG,correspondingtothebshells inourPGSEprotocol.Thediffusiongradientsaresetperpendicularto thecylindersinthesubstratetomaximisesensitivityto𝜏i.We

inves-tigatewhetherexchangetimeeﬀectscanbedetectedinthesignalby lookingatthediﬀerenceinthenormalisedDW-MRIsignalbetween im-permeable(𝜏i=∞)andpermeablesubstrates.Moreover,weanalysethe

eﬀectofnoisebylookingatarangeofdiﬀerentSNRs:SNR=∞,SNR=40 andSNR=20,whereSNR=40correspondstothelevelofnoisepresent inourin-vivodata.Byusingsyntheticsubstratesrepresentativeofour in-vivodataandthesameimagingprotocol,weexpecttheanalysisin thissectiontoprovideanindicativerangeofexchangetimevaluesfor whichthereisreasonablesensitivityinourin-vivodata.

2.4.2. Shellselection

Asourimagingprotocolusesanexplorativerangeofimaging pa-rameters,weselectthebshellsthatmaximisetheperformanceofour RFmodelwithrespectto𝜏i.Forthis,weevaluatetheperformanceof

ourRFmodelforeverypossiblecombinationof4,9and16shellsoutof the25inourprotocol.Wefirstevaluatecombinationsof4shellsusing asabenchmarkthe4-shellSTEAMprotocol(Nedjatietal., 2017) opti-mised Alexander(2008)foratwo-compartmentmodelwithexchange andbiophysicallyplausibletissueparameters.Asthereare12650 pos-sible combinationsof4shells,wetraintheRF12650times,onceon eachdifferentshellcombination.Then,foreachtrainingscenario corre-spondingtoauniquecombinationofshells,wecomputethecorrelation coefficientR2_for_f,_𝜏

ianddbetweenthegroundtruthandtheestimated

valuesinthetestset.Finally,wesortthediﬀerentshellcombinations accordingtotheirR2_score_for_𝜏

iandchoosethecombinationwiththe

highestscoreastheonethatmaximisestheperformanceofthemodel. Furthermore,weinvestigatetheeffectofincreasingthenumberof shellsusedfortraining.Forthis,wealsolookatcombinationsof9shells, astheminimumnumberofshellsrequiredtosampleindependently ev-eryuniqueGand𝚫 valueinourPGSEprotocol.Additionally,welook atcombinationsof16shellsasamiddlevaluebetweenthe9-shelland thefullprotocolscenario.Forthisanalysis,weusethesynthetic feature-baseddatasetdescribedinSection 2.2.2 .Finally,weinvestigatetheeffect ofnoiseontheperformanceofourmodel.Forthis,welookatarange ofdifferentSNRs:SNR=∞,SNR=40andSNR=20.

2.4.3. Syntheticexperiments

ToassessthequalityoftheRFestimatesaftertrainingiscompleted, wecomputethePearsoncorrelationcoeﬃcientR2_between_the_ground

truthvaluesandtheRFestimatesoftheparametersinthepreviously unseentestset.Toevaluateanypotentialbiasintheestimates,weuse Bland-Altmanplotsshowingthemeanoftheestimatedandgroundtruth valuesagainsttheirdifference.Wefirstanalysetheperformanceofthe modelonthenoise-freesyntheticdatabasestoestablishabenchmark givenourdataandimagingprotocol.Next,weapplyourmachine learn-ingmodeltotheSNR=40databaseforamoreaccurateapproximation of theperformanceweexpect,giventhenoisepresentin ourin-vivo data.Foreachexperiment,weanalysebothtrainingscenariosoutlined inSection 2.4.2 (signal-basedandfeature-based)totestwhetherthere areanydifferencesinperformancebetweenthetwoapproaches. 2.4.4. In-vivoimagingexperiments

Beforegeneratingin-vivoparametermapsusingourtrainedmachine learningmodel,weﬁrstperformadataqualitymatchtocheckthatthe datasetusedtotrainourmachinelearningmodelrepresentswellthe characteristicsofthein-vivodataset..Inadditiontothis,weinvestigate

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Fig.3.DifferencesintheDW-MRnormalizedsignalbetweenimpermeable(𝜏i=∞)andtheequivalentpermeable(𝜏i∈[20,1000]ms)substratesatdifferentbvalues, fordifferentmeanaxonaldiameterandSNRsandintra-axonalvolumefractionf=0.7.A)resultsforasubstratewithmeanaxonaldiameter𝜇D=2𝜇m,representing largeaxonsinthemousebrain.B)resultsforasubstratewith𝜇D=0.4𝜇m,mimickingsmallaxonsinthebrain.Thelevelofsignaldetectabilityisdisplayedforthree SNRlevels(∞,40and20),representedbytheblackplanes,belowwhichanychangeinsignalisundetectable.

anypotentialbiasinourin-vivoestimatesof𝜏iduetochangesinthe

orientationdispersionbycomputingmapsoftheNODDIorientation dis-persionindex(ODI)(Zhangetal.,2012a)usingtheNODDIMatlab(The MathWorks,Inc,Natick,MA)Toolbox1_._Using_the_Camino_toolbox,_we additionallygenerateDTI mapsatb=1241s/mm2 _of_axial_diﬀusivity

(AD),fractionalanisotropy(FA)andradialdiﬀusivity(RD)asmeasures oftissuepropertiesthatcanbecomparedwithalreadypublishedworks incuprizonemodel(Boretiusetal.,2012,Songetal.,2005,Zhangetal., 2012b).

UsingtheRFtrainedonthenoisydatabase,wegenerateparameter mapsfortheCCsofthe16miceforthreeparametersofinterest:𝜏i,f

andd.Toinvestigatethedifferencebetweenthetwogroups(CPZand WT),wecomputebox-and-whiskerplotsofregion-specificcomparisons betweenWT(8mice)andCPZ(8mice)fortheDTIandNODDImetrics aswellasfortheRFestimates.Statisticalsignificanceisassessedbya two-tailedt-test,consideringp-values<0.05.Weruntheseexperiments usingthesignalsdatabase.TheCaminofeatureextractionofthein-vivo datadidnotproducehistologicallyplausibleresultsfortheshellswith veryhighgradientstrengths(G>300mT/m)inourprotocol,andwe thereforeexcludethistrainingapproachfromtheanalysisinthis sec-tion.Wediscussthepotentialexplanationsandtheimplicationsofthis inSection 4.1 .

3. Results

3.1. Sensitivityanalysis

Fig. 3 showstherangeofexchangetimevaluesforwhichtheDW-MRI signalS(𝜏i)canbedistinguishedfromthatofanimpermeablesubstrate

S(𝜏i=∞)inthepresenceofnoise.Forthis,wecalculatethechangein

sig-nal|S(𝜏i=∞)-S(𝜏i)|betweenanimpermeableandanequivalent

perme-ablesubstrate.Toillustratepracticallyachievablesensitivities,weplot thisdiﬀerenceagainstthreenoiselevels,denotedbytheblackplane: SNR=∞(1st_column),_SNR₌₄₀₍₂nd_column)_and_SNR₌₂₀₍₃rd_column).

Fig. 3 Aillustratestheresultsforasubstratemimickinglargeaxonsin thewhitematter(𝜇D=2𝜇m),whileFig. 3 Bcorrespondstoasubstrate

withsmalleraxons(𝜇D=0.4𝜇m).Thesecondcolumn showsthat,for

substrateswithlargeaxons(rowA)andanSNRof40,matchingthatof ourin-vivodata,itispossibletodistinguishexchangetimeeﬀectsfor valuesof𝜏i≤400ms.Forsubstrateswithsmallaxons(rowB),wecan

distinguishonlypermeablesubstrateswithexchangetimesupto𝜏i≤

250ms.Asexpected,whentheSNRdropsto20,itbecomesharderto distinguishbetweenimpermeableandpermeablesubstrates.Thistrend

1 _{http://mig.cs.ucl.ac.uk/index.php?n}₌_{Tutorial.NODDImatlab}_.

canbeobservedinthe3rd_column,_where_the_range_for_{distinguishable}

permeablesubstratesnarrowsfrom𝜏iϵ[0,400]msto𝜏iϵ[0,200]ms

forlargeaxonsandfrom𝜏iϵ[0,250]msto𝜏iϵ[0,140]msforsmall

axons.

3.2. Shellselection

Asouroriginal25-shellPGSEprotocolusesanexplorativerangeof imagingparameters,wechoosetheshellsmostsensitivetotheexchange time(seeSection2.5.2forfurtherdetails).InFig. 4 ,eachpointonthe x-axisrepresentsoneuniqueshellcombinationandthecorresponding y-axisvalueindicatestheR2_score_when_the_RF_is_trained_on_that

par-ticularshellcombination.Forexample,thex-axisinFig. 4 Awillhave 12650points,eachonecorrespondingtooneofthe12650unique 4-shellcombinations.Asweareinterestedintheperformanceofthemodel withrespectto𝜏i(1stcolumn),werearrangetheshellcombinationsin

increasingorder accordingtotheirR2_for_𝜏

i. Thisresultsin a

mono-tonicallyincreasingcurvefor𝜏i,asseenintheﬁrstcolumn.Forf(2nd

column)andd(3rd_column)_,_we_keep_the_x-axis_ordering_consistent_with

theresultsfor𝜏iinthe1stcolumn.

TheR2_scores_curves_in_the₁st_column_of_Fig.₄_show_that_only_a

lim-itednumberofshellcombinationshaveagoodcorrelationcoeﬃcient andareoptimalforestimating𝜏i,whiletheR2scoresinthe2ndand3rd

columnshowthatthemajorityofshellcombinationsprovidegood esti-matesoffandd.Forexample,inthenoisefree(bluecurves)4-shellcase inthetoprow,wenoticethatthediﬀerenceinR2_score_for_𝜏

ibetween

thebestandtheworstperformingshellcombinationsisapproximately 0.5.Incontrast,thisdiﬀerenceismuchnarrowerforfandd:≈0.02forf and≈0.01ford.WeobservethesametrendsforSNR=40(orange)and SNR=20(green).

BycomparingthebestR2_scores_on_the_blue_curves_in_{Fig. 4}_A_and

Fig. 4 B,wecanseethatthereisnodiﬀerenceinperformanceinthenoise freescenariobetweenusingthebestcombinationof4or9shells. How-ever,thischangeswiththeadditionofnoise.Forexample,forSNR=40 (orangecurves)theR2_score_of_the_best_9-shell_combination_is_0.67,_0.07

higherthanforthebest4shells.ThistrendissimilarforSNR=20(green curves),withadiﬀerenceof0.1between9and4shells.Forthe16-shell scenario,weﬁndnoimprovementinperformanceoverusing9shells.

Fig.4alsoshowstheeﬀectofnoiseontheestimationofeach param-eter.Asexpected,theadditionofnoiseresultsinlowerR2_scores,_a_trend

thatholdsforallparametersandacrossthe4and9-shellcase.However, theestimationof𝜏iisthemostaﬀectedbythepresenceofnoise:the

maximumcorrelationcoeﬃcientdropsfrom0.82inthenoisefreecase to0.67forSNR=40andevenfurtherto0.52forSNR=20.Forf(2nd

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Fig.4. PerformanceoftheRFmodelpredictionof𝜏i,fandd,trainedondiﬀerentcombinationsof4(A)and9(B)shells.EachcurveshowstheR2score(y-axis)of theRFtrainedonadiﬀerentcombinationofshells(x-axis).TheshellcombinationsaresortedinincreasingorderaccordingtotheirR2_score._We_show_the_results forthreelevelsofnoise:SNR=∞(bluecurve),SNR=40(orangecurve)andSNR=20(greencurve).TheR2_score_for_𝜏

iiscalculatedonlyforvalues≤400msasthis istherangeoverwhichwearesensitivetothisparameter(seeSection3.1).

SNR=∞to0.94forSNR=20.Theestimationoftheintrinsicdiffusivityd isveryrobusttonoise:thecorrelationcoefficientsremainingveryhigh (0.99)evenwhentrainingthemodelontheSNR=20dataset. Further-more,wefindthatallthetop100combinationscontainthetwohighest b-valueshells(6,803and7,773smm−2₎_with_the_two_longest_Δs.

Addi-tionally,weﬁndthathighb-valueshellsonlymaximisetheperformance oftheRFincombinationwithlowb-valueshells(775and930smm−2_).

ForSNR=40(orangecurves),whichweusewhenpredictingonthe in-vivodata,weﬁndthattheoptimalcombinationof9shellssortedby b-valueis[620,775,930,1241,1384,2489,4973,6803,7773]smm−2

withanR2_score_of_0.67,_and_the_best_combination_of₄_shells_is_[775,

930,6803,7773]smm−2_with_an_R2_score_of_0.60._These_results_show

thattheoptimalb-valuesforboth4and9shellsareacombinationof lowandhighvalues,whichsamplebothshortandlong𝚫s.Similar re-sultswerealsoobtainedforthe“signalstrainingdatabase” (notshown). Sincewearelookingtooptimiseourframeworkforin-vivoestimation onthemousedata,werunthein-vivoexperimentsusingthebest9-shell combinationintheSNR=40scenario,asthenoiselevelwhichmatches ourin-vivodata.

3.3. Syntheticexperiments

Fig.5showstheRFresultsobtainedusingthefeature(toprow)and thesignal(bottomrow)noisefreedatabases.Toassessthequalityof ourfit,wedisplaytheresultsusingBland-Altmanplotsandcoloureach datapointaccordingtohowclosetheestimatesaretothegroundtruth values.Toaidvisualinterpretation,wecapthepercentageerrorat50%. Themeandifferencebetweenthegroundtruthandtheestimated val-uesisshownbytheblacklineandthe95%upperandlowerlimitsof agreementbythedashedlines.Forallthreeparametersofinterest,we observenooverallestimationbiasastheestimatesarespreadequally aroundthezero-differenceblackline.However,for𝜏i,theparameter

recoveryis notperfectandtheBland-Altmanplotsshow an overesti-mationbiasforsmallvaluesof𝜏iandanunderestimationbiasforlarge

values.TheR2_scores_show_a_strong_correlation_between_the_estimates_of

ourmodelandthegroundtruthparametervalues:R_𝜏i2₌_0.82/0.84

(fea-tures/signalsdatabase),Rf2=0.99(bothdatabases),andRd2=0.99(both

databases).Whenassessingthemodel’sperformancewithrespecttothe twotrainingdatabases(features/signals),weobservenosigniﬁcant dif-ferencebetweenthetwoapproaches.TheR2_scores_remain_unchanged

forfanddandshow onlyaminordiﬀerencefor𝜏i:R2features=0.82

/R2

signals=0.84.Theadvantagesofeachapproacharediscussed

fur-therinSection 4 .Thenoise-freeresultsinFig. 5 provideabenchmark performanceofthemodelgivenourdataandimagingprotocol.

Fig.6showstheequivalentresultsforSNR=40.Thepresenceofnoise resultsinwiderlimitsofagreementandaffectsdifferentlytheestimation ofeachparameter.Themeandifferencelinesforallthreeparameters remainatzero,showingnogeneralbiasintheestimates.Intra-axonal volumefractionanddiffusivitycontinuetobeverywellestimatedand theircorrelationcoefficientsareonlyverymildlyaffectedbythe pres-enceofnoise:Rf2=0.97andRd2=0.99,equalforbothtrainingdatabases.

Incontrasttothis,thepresenceofnoisehasastrongereﬀectonthe es-timationof 𝜏i,resulting inalowerR2score andamore pronounced

overestimation/underestimationbiasforsmallandlargevalues respec-tively.Despitethis,weﬁndthattheRFworkswellwithinthe sensitiv-ityrangecomputedinSection 3.1 ,withaverygoodcorrelation coeﬃ-cientbetweenthemodel’sestimationsandgroundtruthfor𝜏i≤400ms

(R2₌_0.68)._Outside_this_indicative_sensitivity_range,_the_correlation

co-eﬃcientisveryweak:R2₌_0.07_for_𝜏

i≥400ms.Inlinewiththenoise

freecase,wecontinuetoseenosigniﬁcantdiﬀerencebetweenthesignal andthefeatureapproach:R2

features=0.67/R2signals=0.68.

3.4. In-vivoimagingexperiments

Toshowthatourinvivodataiswellrepresentedbyoursynthetic trainingdatabase,weperformadataqualitymatch(Fig. 7 ).Weplotthe signalintensityasafunctionoftheanglebetweenthediffusion gradi-entsandthecylindricalfibres’axis𝜃 (indegrees),fordifferentdiffusion

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Fig.5.Bland-AltmanplotsfortheRFestimatesoff,𝜏ianddusingthefeatures(toprow)andsignals(bottomrow)noise-freesimulateddatabase.Toaidvisual interpretation,theplotsarecolor-codedwiththepercentageerrorcappedat±50%.

Fig.6. Bland-AltmanplotsfortheRFestimatesoff,𝜏ianddusingthefeatures(toprow)andsignals(bottomrow)simulateddatabasewithSNR=40,matchingthe noiselevelinourin-vivodata.Toaidvisualinterpretation,theplotsarecolor-codedwiththepercentageerrorcappedat±50%

gradientstrengths(G1-5=150–500mT/m)andforΔ={10.8,20.0}ms.

Fig. 7 providesacomparisonbetweenoneofoursimulatedsignals(at differentgradientstrengthsanddiffusiontimes)andtheexperimental signalsmeasuredfromavoxelinthecentreofthespleniumofaWT mouse.Wefindaverygoodmatchbetweenthesimulatedandin-vivo DW-MRIsignals,demonstratingthatourtrainingdatasetisagood rep-resentationofthein-vivomousedataset.Thisisanecessarycondition

foroursupervisedlearningapproachtobevalidandensuresthatduring thesupervisedlearningwelearnatrainingdatasetwhichissimilarto thetestdataset.However,pleasenotethatsimilarDW-MRIsignalsdo not necessarilyimply similarunderpinning microstructure.This very knownambiguity(Jelescuetal,2016b,Novikovetal,2019)isoneof themainchallengesinmicrostructureimaging,leadingtohigher uncer-taintyinthemodelparameterestimation.

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Fig.7. Comparisonbetweenthein-vivo(left)andsimulated(right)signalintensityasafunctionoftheanglebetweendiffusiongradientsandthecylindricalfibres’ axis𝜃 (indegrees),fordifferentdiffusiongradientstrengths(G1-5=150-500mT/m)andtwoΔs:10.8ms(bluelines)and20.0ms(greenlines).Thedashedblackline intheexperimentaldatarepresentsthenoisefloorlevel.

Fig.8. RepresentativeDW-MRIb=0imagesof:A)aWTmousescaninourcohortandB)aCPZmousescaninourcohort.C)ROIsoftheCCoverlaidonthezoomed inb=0imageoftheWTmousescan.ThethreeROIsaregenu(G-CC),body(B-CC)andsplenium(S-CC).TheyellowsquareindicatestheregioninwhichtheCCis found.

Fig.8showsexamplesofDW-MRIb=0imagesforaWT(Fig. 8 A) andforaCPZ(Fig. 8 B)mouse.Wecanobservetheappearanceofthe CCintheCPZscanisdifferentfromtheWT,showingtheeffectof de-myelination.Fig. 8 CshowsthethreeROIsoftheCCoverlaidontheb=0 imageoftheWTscan.WemanuallydefinethreeROIsontheCCmasks ofeachmousescan:splenium(S-CC),body(B-CC)andgenu(G-CC)by followingthedistributionoftheRDvaluestohelplocalizethecentral voxelsofthesethreemainregions:thegenuandspleniumofthecorpus callosumshowalowerRDthanthebody.Wethencalculatethemean parameterestimatesforNODDI(ODI),DTI(AD,RD,FA)andRF(f,𝜏i,

d)ineachROIforeverymouse,andstudythediﬀerencesbetweenthe WTandtheCPZgroups.Wepresenttheseresultsintheremainderof thissection.

Fig.9showsCCmapsforNODDIandDTIparametersforone ex-emplarhealthyWTmouse(ﬁrstcolumn)andoneexemplarCPZmouse (secondcolumn).AvisualinspectionoftheCCmapsrevealsno signif-icantchangesinODIandADbetween thetwomice,togetherwitha signiﬁcantincreaseinRDanddecreaseinFA.

Weobservethesametrendsin theDTIandNODDIparametersat grouplevel,asshowninFig. 9 B.Weillustratethediﬀerencebetween

theWT groupandtheCPZ groupthrough boxandwhiskerplotsin thethreeROIsoftheCC:genu(G-CC),body(B-CC)andsplenium (S-CC).We findtheestimates ofODI inthetwogroupstobe between 0.15and0.29,suggestingverylowdispersion,inlinewithrecently re-portedvaluesinliterature(Wangetal.,2019).Furthermore,wefind no statisticallysignificantdifferenceinNODDIODIbetween thetwo groupsinthethreeregionsoftheCC,afindingthatisalsoinlinewith Wangetal.(2019).TheDTIestimatesshownegligiblechangesinAD, asignificantincreaseinRDandasignificantdecreaseinFA.These re-sultsareconsistentwithalreadypublishedresults(Boretiusetal.,2012, Songetal.,2005, Zhangetal.,2012b).

Thein-vivoRFestimatesoff,𝜏ianddobtainedusingtherawsignal

databasearepresentedinFig. 10 .

TheparametricCCmapsshowninFig. 10 Acorrespondtothesame WTmouse(firstcolumn)andCPZmouse(secondcolumn)inFig. 9 A. TheCCmapsshowastatisticallysignificantdecreaseinf(firstrow)and 𝜏i(secondrow),andnosignificantchangeind(thirdrow).Toprovidea

morequantitativeanalysis,weplottheboxandwhiskerplotsof region-speciﬁcparametercomparisonsbetweentheWTandtheCPZgroupover thethreeCCROIs(Fig. 10 B).ThetrendsobservedvisuallyinFig. 10 A

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Fig.9. A)ParametricmapsoftheCCinahealthyWTmouse(firstcolumn)andaCPZmouse(secondcolumn)obtainedfromconventionalDTIatb=1241s/mm2 andfromNODDIODI.B)Boxandwhiskerplotsofregion-specificcomparisonbetweenWT(N=8)andCPZ(N=8).DTImetrics(AD,RD,FA)areevaluatedwithin thegenu(G-CC),body(B-CC)andsplenium(S-CC)oftheCC.Statisticalsignificanceisassessedbyusinga2-tailedt-testwithequalvarianceandsignificancelevel: ∗₌_0.01,∗∗₌_0.005,∗∗∗₌_0.001._‘n.s.’_stands_for_{non-significant.}

Table2

MeanandstandarddeviationofRFestimatesforf,𝜏ianddinthethreeCCROIsfortheWTandCPZ group.CPZregionsthatarestatisticallydiﬀerentfromWTregionsaremarkedwith∗_for_p_<_0.01,∗∗ forp<0.005and∗∗∗_for_p_<_0.001.

f 𝜏i d

WT CPZ WT CPZ WT CPZ

S-CC 0.443 (0.005) 0.428(0.003) ∗∗∗ _{370 (7)} _{310 (15)}∗∗∗ _{1.12 (0.07)} _{1.18 (0.07)} B-CC 0.430 (0.002) 0.424(0.001) ∗∗∗ _{370 (9)} _{330 (10)}∗∗∗ _{1.10 (0.05)} _{1.15 (0.03)} G-CC 0.440 (0.006) 0.429(0.003) ∗∗ _{380 (14)} _{350 (12)}∗∗ _{1.15 (0.02)} _{1.11 (0.04)}

holdforthegroup-wisequantitativecomparison(WTversusCPZ):we observestatistically signiﬁcantdecreasesinfand𝜏i andanegligible

andstatisticallyinsigniﬁcantincreaseind.Thesetrendsareconsistent acrossallthreeregionsoftheCC.Themeanandstandarddeviationsof theRFparameterestimatesforeachROIarereportedinTable 2 . 3.5. Correlationwithpost-mortemanalysis

ThehistologicalEMmeasurementsinthesplenium,bodyandgenu oftheCCoverthecohortofWT(blue)andCPZ(black)micearereported inthehistogramsofFig. 11 .Ourhistologicaldatashowsnoaxonalsize changes(Fig. 11 C)andnosigniﬁcantaxonalloss(datanotshownhere) betweenthetwocohorts.TheaxonaldiametermeasurementsinFig. 11 C

do nottakeintoaccount thecommonlyaccepted shrinkagefactorof 30%(Barazanyetal.,2009,Innocentietal.,2015),afterwhichthe dif-ferencesbetweenthetwogroupscontinuetoremainstatistically non-significant. Wealsofindastatistically significantdecreasein myelin thickness(Fig. 11 B)correlatedwithanincreaseintheg_ratio(Fig. 11 A) andadecreaseintheintra-axonalvolumefraction(Fig. 11 D).Finally, wemeasureaweakbutnotstatisticallysignificantcorrelationbetween axonaldiameterandintra-axonalvolumefractionfromtheEManalysis (ϱ =0.34andp=0.51>0.05).

Next,westudythecorrelationbetweenthesechangesandthe esti-matesoftheRFmodelinFig. 12 .Weassessthestatisticalsigniﬁcance ofthelinearcorrelationbetween𝜏iandmyelinthicknessfromEMwith

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Fig.10. A)ParametricmapswiththeRFestimatesforf,𝜏ianddintheCCofahealthyWTmouse(firstcolumn)andaCPZmouse(secondcolumn).B)Boxand whiskerplotsofregion-specificcomparisonbetweenWT(N=8)andCPZ(N=8).RFestimatesforf,𝜏ianddarecomputedindependentlyforallvoxelswithinthe genu(G-CC),body(B-CC)andsplenium(S-CC)oftheCC.Statisticalsignificancewasassessedbyusinga2-tailedt-testwithequalvarianceandsignificancelevel: ∗₌_0.01,∗∗₌_0.005,∗∗∗₌_0.001._‘n.s.’_stands_for_{non-significant.}_The_difference_in_the_morphology_of_the_CC_between_the_WT_and_the_CPZ_mice_is_mostly_due_to_different masking,subjecttodifferentpartialvolumewithintheCSFofeachmouse.

Fig.11. Histologyresults.ThemeanandthestandarddeviationoftheEMmeasurementsinthesplenium,bodyandgenuoftheCCforthecohortofWT(blue)andCPZ (black)mice:thegratio(A),myelinthickness(B),meanaxonaldiameter(C)andintra-axonalvolumefraction(D).

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Fig.12. Statisticalsigniﬁcanceandcorrelationsbetween:A)theexchangetimefromDW-MRI(y-axis)andmyelinthicknessfromEM(x-axis)andB)theintra-axonal volumefractionfromDW-MRI(y-axis)andEM(x-axis).EachpointrepresentsthemeanoveroneregionoftheCCfortheWT(bluesquares)andCPZ(blackcircles) group.Errorbarsindicatethestandarddeviationovertheregion.

ofeachCCROIoftheWT(bluesquares)andCPZ(blackcircles)group (Fig. 12 A).WeﬁndaPearsonlinearcorrelationcoeﬃcient ϱ of0.82 andap-value<0.05for𝜏i,showingagoodcorrelationbetweentheRF

estimatesoftheexchangetimefromDW-MRI(y-axis)andhistological measurementsofmyelinthickness(x-axis).

Similarly, we investigate thestatistical significance of the linear correlationbetweenintra-axonalvolumefractionfasestimated from DW-MRI (y-axis) and from EM (x-axis) (Fig. 12 B). We find a Pear-soncorrelationcoefficientϱ of0.98andap-value<0.001,showinga strongcorrelationbetweentheRFestimatesandthehistological mea-surementsof theintra-axonalvolume fraction.Notethatthelowerϱ

valuefortheanalysisinFig. 12 Aislikelydue tothesensitivitylimit ofthecurrent experimentalprotocoltochangesin 𝜏i. Moreover,the

factthatEMmeasurementsofintra-axonalvolumefractionare consis-tentlyhigherthantheRFestimationfrominvivoDW-MRImaybedue tounaccountedshrinkageeﬀects, whichaﬀect mostlythe extracellu-larspaceandthuscanleadtoanincreaseintheintra-axonalvolume fraction.

Discussion

Inthiswork,wefocusontheexperimentalstudyofaRFbased com-putationalmodelforaxonalpermeabilityestimationusinganin-vivo cuprizonemousemodelofdemyelination.Becausenoanalyticalmodel isavailableforpermeabilitycharacterisationinthegeneralcaseof nei-therveryfastorveryslowexchange,here weusethecomputational approachproposedinNedjatietal.(2017).Speciﬁcally,weuseMonte Carlosimulationsof theDW-MRI signalandtrainourmodelto esti-matemicrostructureparameterswithafocusontheintra-axonal wa-terexchange time𝜏i,aparameterinverselyrelatedtoaxonal

perme-ability.Usingsyntheticsubstratesmimickingourin-vivodata,weshow thatourimagingprotocolhasgoodsensitivitytoexchangetimes𝜏i ≤

400msforlargeaxons(meandiameterof2𝜇m)andto𝜏i ≤250ms

forsmallaxons(meandiameterof0.4𝜇m)underthenoiseconditions ofourin-vivodata(SNR=40).Followingfromthis,weﬁndthattheRF modelwedevelopedworksverywellinthisrange:weﬁndagood cor-relationbetweenRF estimatesandthegroundtruthfor𝜏i ≤400ms

(R2₌_0.87_for_SNR₌_inf_and_R2₌_0.68_for_SNR₌_40),_and_a_weak

correla-tionfor𝜏i >400ms(R2=0.3forSNR=infandR2=0.07forSNR=40)

due tothelow sensitivityin our protocol forvalues above 400 ms. Inourin-vivoimagingexperiments,weﬁndthattheRFestimates of 𝜏iarewithinthesensitivityrangeandinlinewithliteraturevaluesof

theexchangetimereportedinhealthyratbraintissue(Prantner,2008, Quirketal.,2003).Furthermore,weﬁndthattheRFestimatesof𝜏iin

theCPZgrouparesigniﬁcantlylowerthanintheWTgroup,aﬁnding

thatonewouldintuitivelyexpecttoseeinamodelofdemyelination. Furthermore,weﬁndthatourintra-axonalvolumefractionestimates inCPZmicearealsosigniﬁcantlylowerthanincontrols.Theseresults arein strongagreement(ϱ_𝜏i = 0.82,ϱf = 0.98)withour EM

histol-ogy resultsof myelinthicknessandintra-axonalvolumefraction, re-spectively.Finally,weshowthatpotentiallyconfoundingfactorssuch asaxonalswellinganddispersionhaveanegligibleeffectonthe esti-mateddifferencesbetweentheWTandCPZgroup.Theseresults sug-gestforthefirsttime,quantitativelyandin-vivo,thatmachinelearning basedcomputationalmodelscouldactasasuitablebiomarkerto de-tectandtrackchangesindemyelinatingpathologies.Furthermore,they supporttheapplicationof𝜏i asmoresensitiveandspecificmarkerof

demyelination.

4.1. Simulations

Sensitivityanalysis.Oursensitivityanalysisshowsthatourimaging protocolhasgoodsensitivityforexchangetimesintherange𝜏i ∈[0,

400]msforsubstrateswithlargeaxons(meandiameter𝜇D=2𝜇m)and

in therange𝜏i ∈[0,250] msfor substrateswithsmallaxons(mean

diameter𝜇D=0.4𝜇m),undernoiseconditionsmatchingthatofour

in-vivodata(SNR=40).Generallyspeaking,thenoiseinthedataaﬀects thesensitivitydiﬀerently,dependingonthemeanaxondiameterinthe substrate. Forsubstrates withlargeaxons (𝜇D=2𝜇m), thesensitivity

halvesfrom𝜏i∈[0,400]msforSNR=40to𝜏i∈[0,200]msforSNR=20.

Forsubstrateswithsmalleraxons(𝜇D=0.4𝜇m),decreasingtheSNRfrom

40 to20 hasasmallereﬀectonthesensitivityrange,reducingitby 44%from𝜏i ∈[0,250]ms(SNR=40)to𝜏i ∈[0,140]ms(SNR=20).

Furthermore,weﬁndthatthelargertheaxonsinthesubstrate,thebetter thesensitivityrange.Substrateswith𝜇D=2𝜇mhaveasensitivityrange

widerby60%(forSNR=40)andby43%(forSNR=20)thansubstrates with𝜇D=0.4𝜇m.

Shellselection.Tooptimisetheperformanceofthemachinelearning model,weexplorethewiderangeofparametersinourPGSEprotocol andselectthebestcombinationof4and9shells.Weshowthatforour in-vivodatawithSNR=40thenumberofshellsthatmaximisesthe per-formanceofthemodelis 9,withtheb-values[620,775,930,1241, 1384,2489,4973,6803,7773]smm−2_and_an_R2_score_of_0.67._When

analysingthebestcombinationsof4and9shells,weobservethatthey sampleeveryvalueof𝚫 inoursequence,resultinginacombinationof lowandhighb-valueshells.Thisﬁndingisinaccordancewiththe opti-misedSTEAMprotocolinNedjatietal.(2017),whichcontainstwolong

𝚫 andtwoshort𝚫 shells.Thissuggeststhattomaximisesensitivityto theintra-axonalexchangetime,itisnecessarytoincludeacombination ofshortandlong𝚫s.

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Weshowthatnoiseisanimportantfactorfortheperformanceofour model.Wefindthatinthenoisefreecase,itissufficienttouseonly4 shellsasintroducingmoreshellsdoesnotimproveperformance. How-ever,inthepresenceofnoise,wefindthatincreasingthenumberof shellsfrom4to9improvestheR2_score_between_the_estimated_and_the

groundtruth𝜏i.Apotentialexplanationforthisisthattheadditionof

noisecorruptstheinformationineachshell,andhavingmoreshellsto corroborateinformationfromhelpstheRFmodellearnbetter.Our anal-ysisalsorevealsthatincreasingthenumberofshellsabove9doesnot offeranyadditionalbenefitseveninthepresenceofnoise.Moreover,we showthatnoisehasastrongereffectontheestimationof𝜏i,forwhich

theR2_score_drops_from_0.84_in_the_noise_free_case_to_≈0.5_for_SNR₌_20.

Theestimationoffanddisconsiderablymorerobust:R2

noise-free=0.99

versusR2

SNR=20=0.94forfandnodropford.ThissuggeststhatSNR

playsanimportantroleinaprotocol’ssuitabilityforpermeability esti-mationusingourapproach.

Featureextraction. Whenextracting therotationally invariant fea-turesfromoursyntheticsignals,weobtainmeaningfulvaluesforallb shellsinthesyntheticdata.Whenweapplythesamemethodtoin-vivo data,thefeatureextractionbecomesdifficultanddoesnotgive mean-ingfulresultsforbshellswithhighgradientstrength(above300mT/m) andhighb-values.Webelievethatthisdifferenceismostlikelydueto theeffectoffibredispersion,presentinthein-vivodatabutnotincluded inoursimulations.Asthegradientstrengthincreases,thedispersed fi-breswouldcauselargerdropsinthesignal,ascanalsobeseenin(Fig. 7 ), wherewenoticethatthedropinthesignalintensityrelativetothe gra-dientdirection is lessprominentin thesyntheticsignalsthanin the in-vivodata.Moreover,wenotethattheoreticallythediffusiontensor featuresatbvalueshigherthan2000s/mm2_lose_their_physical

mean-ing.However,herewedonotinterpretthediffusiontensorfeaturesat highbvaluesintermsoftissuemicrostructure,butweratherusethem justasconvenientmetricstorepresentthesignal.Notethatweinclude allofthediffusiontensorfeaturesevenifsomeofthemarenotmutually independent.Weprefertoworkwithacomprehensivesetoffeaturesto ensurethatourmachinelearningalgorithmfindsthemostinformative splitcriteria.

Syntheticdataexperiments.TheRFmodelestimatesinthenoisefree casehaveverystrongcorrelationswiththegroundtruthvalues, pro-vidinganexcellentbenchmarkperformanceforourmodelandimaging protocol(f:R2₌_0.99,_𝜏

i:R2=0.84d:R2=0.99).Weshowthatthe

addi-tionofnoisewithSNR=40,matchingourin-vivodata,doesnotaﬀect muchtheestimationof fandd(f:R2₌_0.97,_d:_R2₌_0.99),_however,_it

hasastrongereﬀectontheestimationof𝜏i.Inlinewithoursensitivity

results,for𝜏i<400mstheeﬀectispresent,however,theperformanceis

stillsuﬃcientlygood(R2₌_0.68),_while_for_𝜏

i>400mstheperformance

ofthemodelisseverelyaﬀected(R2₌_0.07)._The_estimation_of_f_and_d

isconsiderablymorerobustthanthatof𝜏idue totheuseof arange

ofgradientstrengthsfrom50to300mT/m,whichhasbeenshownto improvethesensitivitytofandd(Huangetal,2015,Sepehrbandetal 2016).Moreover,therobustestimationoffanddisinagreementwith whathasbeenshownby Fieremansetal.(2011)abouttheestimation offanddinthecaseofparallelfibres.Indeed,thecaseofparallelfibers issolvableanalyticallyusingonlyfourestimatedparameters:diffusivity andkurtosisinthedirectionsparallelandperpendiculartothefibres. Sincealltheinformationforcomputingtheseparametersispresentin thedata,thisexplainsthehighfidelityoftheprediction.However,we notethatFieremansetal.’smodelisconfoundedbythefiberorientation dispersion,whichisknowntobepresentinwhitematter(Ronenetal., 2014)andtherefore,inthecaseofnon-negligiblefibredispersion,our parameterestimatesmaybebiased.

Inadditiontothis,wecompareforthefirsttimethesignaland fea-turetrainingapproachesandshowthatthereisnosignificantdifference intheRFperformanceaccordingtowhichdatabaseisusedfortraining. Thisisasignificantresultasitshowsthatwhenextractingthe rotation-allyinvariantfeaturesfromtherawsignalswedonotloseinformation thatisessentialfortrainingourmodel.Consequently,wecanusethe

featuresdatabasewithoutaffectingtheperformanceofourmodel.The advantage ofarotationallyinvariantfeature approachisthatitdoes notrequirethegenerationofanewlibraryforeverynewacquisition protocol aslong astheb-valuesandthe TEof theprotocolsmatch. Nevertheless,asdiscussedabove,cautionshouldbe appliedwiththis approach whentheacquisitionprotocoluses highgradientstrengths (G≥300mT/m)andtheSNRislow,suchasconditionsoftenfoundin thepre-clinicalsetting,andthenusingsignalsdatabasemightbe the preferablechoice.Ontheotherhand,intheclinicalsetting,imaging protocolshavemuchlowergradientstrengthsandsufficientSNRtofit theDTIandSHmodelparametersinthefeatureextractionapproach, andconsequently,weexpecttherotationallyinvariantfeatureapproach tobeabetterchoice(asusedinNedjatietal.(2017)).Irrespectiveof thetrainingapproach,weexpectourmodel’sperformancetobesimilar inboththeclinicalandpreclinicalsetting.

4.2. In-vivomousedataandcorrelationwithpost-mortemanalysis Ourdataqualitymatchshowsthatoursynthetictrainingdataisa goodrepresentation ofthein-vivodata.Our DTIresults showan in-creaseinRDandadecreaseinFAbetweenthetwogroups.Thiscould beexplainedbythebreakdownofthemyelinlayerwhichallows wa-tertodiffusemoreintheradialdirection,leavingADunchangedand having theoveralleffect ofreducing FA.These changesin DTI met-ricsareinagreementwiththosereportedinseveralstudiesoftheCPZ mousemodelofdemyelination(Boretiusetal.,2012,Songetal.,2005, Zhangetal.,2012b).Nevertheless,theDTImetricsarenotspecific be-causetheyprovideonlyindirectmeasuresoftheunderlying microstruc-turalchangesintheCPZmodel.Forinstance,theobservedincreasein RDmaybeduetotheincreasespacesbetweentheaxonsandnottothe higherpermeabilityoflessmyelinatedaxons.

Ontheotherhand,ourRFestimatesof𝜏iprovideamoredirectand

speciﬁcmeasureofpermeability.Infact,inourcomputationalmodel, diﬀusivity(viad)andpermeability(via𝜏i)aredecoupledand

individ-ually estimatedfrom thedata. Weﬁndthat ourestimationsof 𝜏i in

thehealthymicecomparewellwithliteraturevalues.Studieson sph-ingomyelinmembranesfoundinaxonalmembranessuggestvalues be-tween300msand600msforaxonswithradiibetween0.5and1𝜇m (Finkelstein,1976).Contrastagentandrelaxometrystudiesintherat brainestimatetheintracellularwaterexchangelifetimeintheratbrain tobebetween200ms(Prantner,2008)and550ms(Quirketal.,2003). Itisworthwhiletonotethatourexperimentalprotocoldoesnot pro-videenoughsensitivitytodetectexchangetimes >0.4s.Hence,our methodwouldestimate𝜏i~ 0.4sforanyactualexchangetime≥0.4s.

Nevertheless,wehavehighsensitivitytoreliablymeasureanychanges in 𝜏ioccurringbelow0.4sduetodemyelination.Asaccurate

histol-ogymeasurementsof𝜏iarenotavailableduetotissueﬁxationaltering

themembranepermeability,wecompareourestimatesof𝜏iwithEM

measurementsofmyelinthickness.Wecomputemyelinthicknessfrom myelinatedaxonsonly,anditincludesboththeeﬀectofdemyelination inducedbyCPZandsomeremyelinationthathappensspontaneouslyin theCPZmodel(MatsushimaandMorell,2001).Weﬁndastrong corre-lationbetweentheRFestimatesof𝜏iandmyelinthickness(ϱ𝜏i=0.82).

This isin verygoodagreementwitharecentlypublishedsimulation workinvestigatingthelinkbetweenexchangetimeandmyelinthickness (Brusinietal.,2019),furthersupportingtheﬁndingsthatmyelin wrap-pingcanmeaningfullycontributetothesignalinDW-MRIandimpact𝜏i.

Furthermore,ourRFestimatesofdlieintherange1–1.3𝜇m2_sm−1_,_an

expectedrangeforthemouseCC(Wuetal.,2008),andourestimates offcorrelateverystronglywiththeEMintra-axonalvolumefraction measurements(ϱf=0.98).

Whencomparingthetwogroups,weobservethefollowinggeneral trends: a statisticallysigniﬁcant decreasein the intra-axonalvolume fractionfandintheintra-axonalexchangetime𝜏i,togetherwitha

neg-ligibleandstatisticallyinsigniﬁcantincreaseintheintrinsicdiﬀusivity d.WeexpectftobelowerintheCPZgroupasthereisanincreasein