PDF Medical Image Analysis - IBS

ContentslistsavailableatScienceDirect

Medical Image Analysis

journalhomepage:www.elsevier.com/locate/media

Model-Based High-Definition Dynamic Contrast Enhanced MRI for Concurrent Estimation of Perfusion and Microvascular Permeability

Joon Sik Park

^a

, Eunji Lim

^a

, Seung-Hong Choi

^d

, Chul-Ho Sohn

^d

, Joonyeol Lee

^a,c

, Jaeseok Park

^a,b,∗

aDepartment of Biomedical Engineering, Sungkyunkwan University, 2066, Seobu-Ro, Jangan-Gu, Suwon, Republic of Korea

bBiomedical Institute for Convergence, Sungkyunkwan University, Suwon, Republic of Korea

cCenter for Neuroscience Imaging Research, Institute for Basic Science, Suwon, Republic of Korea

dDepartment of Radiology, Seoul National University Hospital, Seoul, Republic of Korea

a rt i c l e i n f o

Article history:

Received 22 March 2019 Revised 20 September 2019 Accepted 26 September 2019 Available online 8 October 2019 Keywords:

Magnetic resonance imaging Dynamic contrast enhanced High definition

Microvascular permeability Brain cancer

a b s t r a c t

Thiswork introducesa model-based, high-definition dynamic contrast enhanced (DCE) MRI for con- currentestimation ofperfusion andmicrovascular permeabilityoverthewholebrain. Atimeseriesof reference-subtracted signalsis decomposedinto one component that reflectsmain contrast dynamics andtheotheronethatincludesresidualcontrastagents(CA)andbackgroundsignals.Theformerisde- scribedbylinearsuperpositionofafinitenumberofbasicvectorstrainedfromanaugmentedsetofdata thatconsistsoftracer-kineticmodeldrivensignalvectorsandpatient-specificmeasuredones.Contrast dynamicsisestimated bysolving aconstrained optimizationproblemthatincorporates the linearized signaldecompositionintothemeasurementmodelofDCE MRIandthencombiningthemain compo- nentwiththebackground-suppressed,residualCAsignals.Tothebestofourknowledge,thisisthefirst workthatprospectivelyenablesrapidtemporalsamplingwith1.5s(3∼4timeshigherthanclinicalrou- tines)whilesimultaneouslyachievinghighisotropicspatialresolutionwith1.0mm³ (4∼6timeshigher thanroutines),enhancingestimationofbothpatient-specificinputsandoutputsforquantificationofmi- crovascularfunctions.Simulationsandexperimentsareperformedtodemonstratetheeffectivenessofthe proposedmethodinpatientswithbraincancer.

© 2019ElsevierB.V.Allrightsreserved.

1. Introduction

Dynamiccontrastenhanced(DCE)magneticresonanceimaging (MRI) (Sourbronetal., 2009;Dutoitetal., 2013) hasbeenwidely used in a clinical routine to investigate vascular structures and functions (perfusion, microvascular permeability) forpathological tissuesinthepresenceofleakagesfromcapillaries.Particularlyin the brain, it hasbecome an essential tool to assess blood-brain- barrier (BBB) leakages in tumors (Roberts et al., 2000), multiple sclerosislesions(Ingrischetal., 2012),stroke(Meralietal.,2017), anddementia(VanDeHaaretal.,2016).InDCEMRI,atimeseries ofthree-dimensional(3D)T₁-weightedimagesistypicallyacquired todescribecontrastdynamics(wash-in,arisetoamaximumpeak, wash-out)afterintravenouslyadministeringT₁-shorteningcontrast agents (CA). The contrast dynamics in each voxel is reflected in time-varying MR signal intensities, which are then converted to concentration time coursesof CA by utilizing linear relation be-

∗ Corresponding author.

E-mail address: [email protected] (J. Park).

tween concentration of CA and change of T₁ relaxation rate. A tracer-kinetic model (Sourbron and Buckley, 2011; 2013), which employs a concentration time course in an arterial region asan input, is fitted to those in tissues of interest (output), revealing some ofthefollowing hiddenparameters relatedto tissueperfu- sionandmicrovascularpermeability:plasmaflow(Fp),plasmavol- umefraction(vp),permeability-surface area product(PS), andin- terstitialvolumefraction(ve).TheFpistheflowofplasmaintothe capillarybed,andthevpisthevolumefractioninsidethecapillary bed.PSindicatestheflowofmoleculesthroughthecapillarymem- branesina certain volumeoftissue andtheve isthe fractionof interstitialvolumethatcontainedinavolumeoftissue.Ingeneral, F_p andv_p are regardedas perfusion parameters, whilePS andv_e areknownaspermeabilityparameters.Ingeneral,Fpandvparere- gardedasperfusionparameters,whilePSandveareknownasper- meabilityparameters.Accurateestimationoftheperfusionparam- etersrequireshightemporalresolutionto capturerapidlyvarying signal intensities duringthefirst passof boluswhen CAremains mostlyintheblood stream.On theother hand,to preciselyesti- matethepermeabilityparameters,dataacquisitionistobelength- https://doi.org/10.1016/j.media.2019.101566

1361-8415/© 2019 Elsevier B.V. All rights reserved.

enedtoaccommodateaslowuptakeofCAintointerstitial spaces.

Thus,to improve estimationof both perfusion andmicrovascular permeability,DCEMRIrequiresasufficientlylongacquisitiontime withhightemporalresolution.

Simplified tracer-kinetic models such as the extended Tofts model (Tofts and Kermode, 1991) and the Patlak model (Hacksteinetal.,2005)havebeenwidelyusedforvariousclinical applications, in which the former enables estimation of vp, the volumetransferconstantK^trans(amixtureofFpandPS),andveby assumingF_p=∞(neglectingplasmatransittime)whilethelatter is a special case of the former and produces vp and K^trans with thefollowingconstraints:Fp=∞andnobackfluxfrominterstitial spaces. Despite the partial success of describing an uptake of CA, a capillary transit time is neglected, which may not reflect underlying tissue physiology in a real setting. Furthermore, the strong constraints preclude concurrent estimation of Fp and PS, leading to mixed information of perfusion and permeability. To addressthisissue,complexmodelswithfourunknownparameters (Fp, vp, PS’, ve), which include the two-compartment exchange model(2CXM) (Brix et al., 2004), the tissue homogeneity model (TH)(Simpson etal.,1999), theadiabaticapproximationtothetis- suehomogeneitymodel(AATH)(St.LawrenceandTing-Yim,1998), and the distributed parameter model (DP) (Garpebring et al., 2009), can be employed but require a highly defined uptake curve of CA with high temporal resolution during data acqui- sition (O’Connor et al., 2007). It was shown in (Kershaw and Cheng,2010) thatatemporalresolutionofatleast1.5sisneeded to reliably estimate all perfusion and permeability parameters withminimalbias.

In all tracer-kinetic models of DCE MRI, two distinct time- resolved signals need to be measured in an arterial region (in- put)andtissuesof interest(output), respectively.The arterial in- putfunction (AIF),whichestimatestheconcentration ofCAinan arteryfeeding the corresponding tissueof interest,exhibits rapid contrastdynamics with time, while the output function changes slowly due to a gradual uptake of CA in capillary plasma and interstitial spaces. Thus, it is very important to precisely cap- turethe two distinct contrastdynamics: rapidly varying AIF and slowlyvarying output signals.Giventheabove signalcharacteris- tics,dual resolution approachesin DCEMRI (Jelescu etal., 2011;

Li et al., 2012) were recently introduced, in which the AIF was measured using high temporal resolution and low spatial reso- lution while the output function was sampled using low tem- poral resolution and high spatial resolution. However, the mea- sured AIF, which is sampled with low spatial resolution, poten- tiallysuffers from partial volume effects (vanOsch et al., 2005), corrupting the measurement ofthe rapidly-rising upslope during thefirstpass ofbolusandthereby resultinginerroneousparam- eterestimates. Additionally, contrastdynamics in tissues is typi- cally quite slow, butis known to be substantially fast in tumors duetohighvasculatureandrapidtransferbetweencapillariesand interstitialspaces (Choyke etal., 2003;Jansen etal., 2008).Thus, toestimate thetwo distinct contrastdynamicsin arterialregions and tissues of interest, high-definition DCE MRI with high spa- tialandtemporalresolution isneeded.Toaddressthisissue,DCE MRIwascombinedwithparallelimagingandcompressedsensing forbrain(Lebel etal., 2014; Otazoet al., 2015;Guo et al., 2016;

2017), liver (Feng et al., 2014), and prostate (Rosenkrantz et al., 2015) usinghighundersamplingink−t spaceandnonlinear re- construction withsparsitypriors. Among them, onlya few stud- ies (Lebel etal., 2014;Guo et al., 2016; 2017) demonstrated the feasibility of the whole-brain DCE MRI for tumor characteriza- tion with prospectively enhanced spatial and temporal resolu- tion:0.94×0.94×1.9mm³,4.1sin(Lebeletal.,2014;Guoetal., 2016); 0.9×0.9×1.9mm³,5.0s in (Guo etal., 2017). Neverthe-

less,assuggestedin(KershawandCheng, 2010),temporalresolu- tionneedstobefurtherincreased.

Given the above considerations, in this work we develop a model-based, high-definition DCE MRI to enable concurrent es- timation of perfusion and microvascular permeability over the whole brain. A time-series of reference-subtracted signals is de- composed into: one component that reflects main contrast dy- namics and the other one that includes residual CA and back- ground signals. The former is described by linear superposition of a finite number of basic vectors trained from an augmented setofdatathatconsistsoftracer-kineticmodeldrivensignal vec- tors andpatient-specificmeasured ones. Contrastdynamicsises- timatedbysolvingaconstrainedoptimizationproblemthatincor- poratesthelinearizedsignaldecompositionmodelintothehighly undersampledmeasurementmodelofDCEMRIwithsparsitypri- orswhilecombiningtheresultingmaincomponentwiththeresid- ual background-suppressedCAsignals. Tothebest ofour knowl- edge, thisis thefirst work that prospectivelyenables rapidtem- poral sampling with1.5s (3 ∼ 4times higherthan clinical rou- tines)whilesimultaneouslyachievinghighisotropicspatialresolu- tionwith1.0mm³,(4 ∼ 6timeshigherthanclinicalroutines),en- hancingestimationofbothpatient-specificinputsandoutputsfor quantificationof microvascularfunctions. Simulations andinvivo experimentsareperformedtovalidatetheeffectivenessofthepro- posedmethodinbraincancer.

Therestofthepaperisorganizedasfollows:InSection 2,we introducethenotationsusedthroughoutthepaper,andpresenta measurementmodelinDCEMRI,alinearizedsignaldecomposition modelspecific to DCEMRI, anda detailedalgorithm on estimat- ingcontrastdynamicsfromincompletemeasurementsforquantifi- cationofmicrovascularfunctionparameters. InSection3,simula- tions andexperimentalstudies aredescribed. Lastly,in Section4, discussionandconclusionarepresented.

2. Method 2.1. Notations

Notations, which are used throughoutthe paper, are listed in Table1.Matrices,vectors,andscalarsaredenotedbyboldfaceup- percaseletters,boldface lowercaseletters,anditalics,respectively.

CandRdenoteasetofcomplexandrealvalues,respectively.

2.2. MeasurementModelinDCEMRI

AfterintravenouslyinjectingCAintothebloodstream,aseries oftime-resolved,CA-inducedsignal modulation,whichrepresents contrastdynamicsofinterest,iscapturedvoxel-by-voxel.Thecor- respondingmeasurementmodelinDCEMRIisdescribedby:

y_l

(

^k,t

)

^{= c}l

(

^r

)

^x

(

^r,t

)

^{e−jk·rdr+n}l

(

^k,t

)

⁽¹⁾

wherey_l(k,t) isthemeasured DCEsignal attheFourierencoding indexkandthetimetforthelthreceivercoil;c_l(r)isthespatially varyingcoilsensitivityatthespatialpositionrforthelthcoilthat isassumedtoremaininvariantwithtime;x(r,t)isthetargetsignal inthespatialandtemporaldomain,representingadesired, time- varyingDCEsignalineachvoxel;n_l(k,t)isthemeasurementnoise forthelthcoilwithcomplexwhiteGaussiandistribution.

ADCEMRimage ata specific temporalphase (t) inEq.(1)is column-vectorizedbyx_t=

x(^r0,t)^x(^r1,t)...x(^rN_t−1,t)

T

.Theen- tireDCEMRimageinthex−tspace,X=

x₀x₁ ...x_Nt−1

,canbe

Table 1

Notations and their Descriptions

Notations Descriptions

k , x , t k-space, spatial, and temporal domains, respectivelyt

N , N t, N c, M Lengths of spatial, temporal, coil, and measured Fourier encoding dimensions, respectively Y ∈ C ^MNc×Nt Measured signal matrix in k-t space

X ∈ C ^NNc×Nt Target signal matrix in x-t space X 0∈ C ^NNc×1 Baseline signal matrix in x-t space X D∈ C ^NNc×Nt Main DCE signal matrix in x-t space X B ∈ C ^NNc×Nt Residual DCE signal matrix in x-t space

X B ∈ C ^NNc×Nt Background suppressed, residual signal matrix in x-t space U ∈ C ^T×r Spatial basis of X Dwith a rank of r

V r∈ C ^Nt×r Temporal basis of X Dwith a rank of r X P∈ R ^N×Np Residual pre-contrast background signal matrix X E∈ R ^N×Ne Residual augmented background signal matrix U B∈ R ^(Np+Ne)×r Spatial basis of residual background signal matrix F u: C ^NNc×Nt→ C ^MNc×Nt Undersampled Fourier encoding operator S : C ^N×Nt→ C ^NNc×Nt Coil sensitivity operator.

E : C ^N×Nt→ C ^MNc×Nt Sensitivity encoding operator.

writteninaCasoratimatrixformby:

X=

⎡

⎣

x

(

^r0,t0

)

^{... x}

(

^r0,tNt−1

)

..

. ... ... x

(

^rN,t0

)

... x

(

^rN,tNt−1

)

⎤

⎦

(2)

whereNisthetotalnumberofvoxelsineachtemporalphase,and Ntisthetotalnumberoftemporalphases.

Given the Casorati form of the spatiotemporal signal repre- sentation in x−t space, measured signals in a specific tem- poral phase (t) are correspondingly column-vectorized by y_l,t= [y_l(^k0,t)^yl(^k1,t))...y_l(^kM−1,t)^{]T where M is the total number} of Fourierencodingsin each temporal phase; y_l,t issubsequently stacked inthetemporaldimension column-wise,resultinginY_l=

y_l,0y_l,1...y_l,N

t−1

;Y_l isthenputinthecoildimension row-wise, yieldingY=

Y₀Y₁...Y_Nc−1

T

whereN_c isthetotalnumberofre- ceivercoils.WiththemeasuredmatrixY(∈C^MNc×Nt)andthede- siredtarget matrixX (∈C^NNc×Nt),themeasurementmodelofDCE MRIinEq.(1)isthenrecastinamatrixformby:

Y=FuS

(

^X

)

^+N

=E

(

^X

)

+N (3)

whereFu:C^NNc×Nt→C^MNc×Nt isthespatialFourierencodingop- erator with incomplete sampling (MN) in k−t space; S: C^N×Nt→C^NNc×Nt isthecoilsensitivitymappingoperatorthatper- formscolumn-vectorizationofcoilsensitivityprofiles ineachcoil byc_l=[c(^r0)^c(^r1). . .c(^rN−1)^]T,applies the Hadamard product be- tweenc_l andxtandstacksinthetemporaldimensioncolumn-wise by S_l=

c_lx0c_lx1...c_lxN_t−1

,andthen puts S_l inthe coil dimension row-wiseby S=

S₀S₁ ...S_Nc−1

T

;Nis thenoise ma- trix;E:C^N×Nt→C^MNc×Nt isthesensitivityencodingoperatorthat isequaltoFuS.

A spoiled gradient echo (GRE) MR pulsesequence, which en- codessteadystatesignalsbyrepeatingradio-frequencypulseswith a uniforminterval, is typically usedin acquiringa time seriesof DCEdatawhileintravenouslyinjecting CA(O’Connoretal.,2011).

Hence, due to the steady state signal acquisition, temporal sig- nal modulation in DCE MRI results only from time-varying con- centrationof CAthat subsequentlyproduces changes inT₁ relax- ation times fortissuesofinterest. Thus, the measurementmodel in Eq.(1) can then be recastby employing a steady state signal anditscorrespondingtemporalmodulationas:

y_l

(

^k,t

)

^{= c}l

(

^r

)

^xˆ

(

^r

)

^G

(

^T1

(

^r,t

))

^e−jk·rdr+n_l

(

^k,t

)

G

(

^T1

(

^r,t

))

^{= sin}

α

^{1−e−TR/T1(r,t)}

1−cos

α

^{e−TR/T1(r,t) (4)}

where xˆ(^r) ^{is the steady state signal in each voxel;} G(^T1(^r,t)) isthe nonlineartemporal signal modulation resultingfromtime- varyingT₁ relaxationtimes;

α

^{istheflip angleofRF pulse;TRis}

thetime intervalbetweenneighboringRF pulses.However,itmay bedifficulttofindanoptimalsolutionbyminimizingdataconsis- tencyerrorsduetothenonlinearitywithrespect toT₁(t)andthe potential scale mismatch between xˆ(^r) ^{and T}1(r, t) as shown in (Blocketal.,2007).

2.3.LinearizedSignalDecompositionModel

Priorto injectionof CAin DCEMRI,a set ofpre-contrast ref- erencecanbeconstructed byemployingeitherfull samplingina singlephase or interleaved samplingfollowed by averaging over multipletimephases.WithvaryingconcentrationofCA,atimese- riesofthereference-subtractedsignalvectorsisarrangedcolumn- wiseina Casoratimatrixform, whichcanthen beseparatedinto onematrixthatreflects contrastdynamicsinx−t spaceandthe other matrixthatcontainsresidual backgroundsignals.Giventhe aboveconsiderations,theproposedsignalmodelinDCEMRIisde- composedby:

RX=X−X0=XD+XB+N (5) whereR_Xisthereference-subtractedsignalmatrix;X₀istherefer- encesignalmatrixthatconsistsoftime-invariantbaselinevectors;

X_D isthe mainmatrixthat containsCA inducedsignal variations withtime voxel by voxel;X_B isthe additive matrix that denotes residualbackgroundsignalsotherthanCAinducedsignals.

WhileCAinthebloodstreamflowsintocapillaryplasmaspace, permeatesintointerstitialspace, andthenexchangesbetweenthe two spaces, signal intensities typically vary smoothly in a corre- latedfashion withtime. Thisimpliesthat the rowvectorsin the matrixX_D can be describedby linearlycombining a finiteset of therightsingularvectorsofthematrixX_Dcorresponding tolarge singularvaluesintermsofsingularvaluedecomposition(SVD).To find the temporal basis, a set of trainingdata is constructed, in whichtracer-kinetic modeldriven signals are row-vectorizedand then stacked row-wise with changing the following parameters:

F_p,v_p,PS,andv_e.Inthiswork,oneofthefour-parametermodels, 2CXM(Khalifaetal.,2014),inDCEMRIischosenasatracer-kinetic modeltoincludeconcurrent effects ofperfusionandmicrovascu- larpermeabilityontemporalsignalevolutions. Signalsinthecen- tralregionofk−t space, whichcontainmaincontrastdynamics, aretypicallyacquiredwithfull sampling.Measured signalsinthe corresponding central k−t space from each patient with brain cancerare augmented to thetracer-kinetic modeldriven training data row-wise to reflect actual temporal signal patterns. Lastly,

Fig. 1. A time series of the augmented training data in X T(a), a set of the corresponding basic vectors of the temporal basis in V r(b), and the corresponding singular values in r(c). Note that singular values in (c) decrease rapidly with increasing index.

thetracer-kineticmodeldrivensignalvectorsaretemporallyread- justedtoreflectanactualbolusdelayspecifictoeachpatientusing anintersectionaltimepointofthemostrapidlyvarying,measured signalvector.Thetemporalbasis isthencalculatedbyperforming SVDonthetrainingdatamatrixandtakingaportionofthecorre- spondingrightsingularvectorsby:

XT=

X_sim Xmeas

−→SVD Ur

rV^H_r (6)

whereX_T (∈R^T×Nt) is the trainingdata matrix consisting of the tracer-kineticmodeldrivensignalmatrix(X_sim)andthemeasured signalmatrixinthecentralportionofk−tspace(Xmeas);Tisthe totalnumberofrowvectorsinthetrainingdata;Ur(∈R^T×r)isthe unitarymatrixthatcontainstheleftsingularvectorscorresponding totherlargestsingularvalues;r(∈R^r×r)isthediagonalmatrix that containsthe rcorresponding singularvalues; Vr (∈R^Nt×r) is theunitary matrixthat containsthe corresponding rightsingular vectors,representingthedesiredtemporalbasis.X_T,V_r,andrare graphicallyillustratedinFig.1a–c.

Giventhelinearbasis,Vr,asapriori,theproposedsignalmodel inEq.(5)canbe rewrittenby decomposingX_Dintothe twoma- tricesthat contain spatialandtemporal information,respectively, as:

RX=XD+XB+N

XD =UV^H_r (7)

whereUcontainsthespatialinformationofX_D,whichiscomple- mentarytoVr,thetemporalinformationofX_D.Inthepresenceof the two distinct contrast dynamics, rapidly varying input signals inan arterialregion andslowlyvarying output signalsintissues, theproposedlowrankmodel,UV^H_r,inEq.(5)maynotcapturede- tailson theformersignal pattern, dependingon therank r. Asa result,theadditivematrixX_Bcontainsresidualcontrastdynamics thatismissinginX_Daswell asnon-contrastbackgroundartifacts.

Thus, the proposed signal decomposition model inEq. (7)is ad- vantageousin that it provides a framework to recaptureresidual contrastdynamicswithoutapparentlossofinformationduringre- constructioninthe next sectionparticularlyin casethelow rank modelinX_DmaybroadenaprofileoftheAIF.

2.4.ReconstructionofContrastDynamicsfromIncomplete Measurements

In this section, we introduce a reconstruction framework of contrastdynamics that incorporates theproposed, linearized sig- naldecompositionmodel inEq.(7)into themeasurement model

in Eq.(3) while being formulated as a constrained minimization problemwith multiple sparsitypriors using the following objec- tivefunction:

J

(

^U,XD,XB

)

⁼

||

^Ft

(

^XD

) ||

¹⁺

λ

^U

|| ψ (

^U

) ||

¹⁺

λ

^B

||

^Ft

(

^XB

) ||

¹

s.t. RY=Y−Y0=E

(

^XD+XB

)

X_D=UV^H_r (8)

where Ft:C^N×Nt→C^N×Nt is the temporal Fouriertransform op- erator; Ft(·) ^{denotes the sparse} representation of the argument in x− f space;

ψ

^:C^N×r→C^N×r is the discrete wavelet trans- formoperator;

λ

U and

λ

B arethe balancingparameters thatcon- trol sparsities among Ft(^XD),

ψ

^{(U), and} Ft(^XB)^{; R}Y (∈C^MNc×Nt) istheresidualsignal matrixink−t spacethat isconstructedby subtractingthebaseline(pre-contrast)measuredsignalmatrix(Y₀) fromthemeasuredDCEsignalmatrix(Y).Coilsensitivitymapsin- cludedinthesensitivityencodingoperator(E)areestimatedusing thefullyacquiredbaselinedata,inwhicheachcoilk-spaceislow- pass filtered andinverse Fouriertransformed, and each resulting coilimageisthennormalizedbyroot-sum-of-squaredcombination ofallcoilimages.

TheobjectivefunctioninEq.(8)isreformulatedintoanuncon- strainedoptimizationproblemusingtheassociatedaugmentedLa- grangemultiplier(ALM)function(Boydetal.,2011)withadditional balancingparametersby:

L

(

^U,XD,XB,

1,

2

)

=J

(

^U,XD,XB

)

+

λ

¹

2

^RY−E

(

^XD+XB

)

⁺

λ

¹¹

¹

²

F

+

λ

²

2

^{XD−UVHr +}

λ

¹²

²

²

F

(9)

where1, and2 are theLagrange variables, and

λ

1 and

λ

2 are theadditionalbalancingparameters.Sincemultipleunknownvari- ablesareincludedinEq.(9),inthisworkaframeworkofthealter- natingdirectionmethodofmultiplier(ADMM)(Wangetal.,2008;

GoldsteinandOsher, 2009;Lingala etal., 2011)ischosen tomin- imizethecorrespondingALMfunctionwithrespecttoU,X_D,and X_B,inwhichU,X_D,andX_Bareestimatedinanalternatingfashion whileretainingthe other variablesobtained intheprevious step, andtheLagrangemultipliers(1and2)arethenupdatedateach iteration.Giventheaboveconsiderations,theADMMyieldsthefol- lowingseparablesubproblems andcorresponding solutionsforU, X_D,andX_Batthe(^k+1)^{thiterationstep:}

U^(k+1) =argmin

U L(^U,X⁽_D^k) ,X⁽_B^k) ,⁽1^k) ,⁽2^k) )

=argmin

U λUψ(^U)1+λ2

2

^XD^(k) −UV^H_r + 1 λ2⁽2^{k) 2}

F

=ψ^∗ SλUδU

ψ

U^(k) +δUλ2

X⁽_D^k) −U^(k) V^H_r + 1 λ2⁽2^k)

Vr

(10)

X⁽_D^k+1) =argmin

XD L(^U(k+1) ,X_D,X⁽_B^k) ,⁽1^k) ,⁽2^k) )

=argmin

XD Ft(^XD)1+λ1

2

^RY−E(^XD+X_B^(k) )+ 1 λ1⁽1^{k) 2}

F

+λ2

2

^XD−U^(k+1) V^H_r + 1 λ22^{(k) 2}

F

=Ft^∗

SδXD

Ft

X_D^(k) +δXDλ1E^∗

R_Y−E(^X(D^k) +X_B^(k) )+ 1 λ1⁽1^k)

+δXDλ2

X_D^(k) −U^(k+1) V^H_r + 1 λ2⁽2^k)

(11)

X⁽_B^k+1) =argmin

X_M L(^U(k+1) ,X⁽_D^k+1) ,X_B,⁽1^k) ,⁽2^k) )

=min

XM

λBFt(^XB)1+λ1

2^RY−E(^X(D^k+1) +X_B)+ 1 λ1⁽1^{k) 2}

F

=Ft^∗

SλBδXM

Ft

X⁽_B^k) +δXBλ1E^∗

R_Y−E(^X(D^k+1) +X⁽_B^k) ) +1

λ1⁽1^k)

(12)

⁽₁^k+1)=

⁽₁^k)+

λ

^{1 RY−E}

(

^X(_D^k+1)+X⁽_B^k+1)

)

(13)

⁽2^k+1)=

⁽2^k)+

λ

^{2 X}D^(k+1)−U^(k+1)V^H_r

(14) where

δ

U,

δ

X_D, and

δ

X_B are thestep sizes withiterations; Ft^∗ is theadjointoperator ofFt;

ψ

^{∗ istheadjointoperatorof}

ψ

^;Sτ is the soft thresholding operator that is equivalent to the proximal operatorof ₁normby:

S_τ

(

^z

)

=

z

( |

^z

|

⁻

τ )

/

|

^z

|

^if

|

^z

|

^≥

τ

0 otherwise (15)

where

τ

^{is the} thresholding value for Sτ. Given U^(k+1), X_D^(k+1) and X⁽_B^k+1), the Lagrange variables are updated as shown in Eqs. (13) and (14). Iterations in the above ADMM method con- tinue withthefollowing stopping criterion:

X_D^(k+1)−X_D^(k)

²

F≤

δ

X_D^(k)

²

F ork>Kmaxwhere

δ

^{andKmax arethepredefinederror}

toleranceandthemaximumnumberofiterations,respectively.The proposed model-based reconstruction of contrast dynamics from incompletemeasurementissummarizedinAlgorithm1.

2.5.CapturingResidualContrastDynamicswithBackground Suppression

ThelowrankmodelinEq.(7)maynotbeabletodescribede- tailedinformationonrapidcontrastdynamicsparticularlyinarte- rialregions.Intheproposedreconstructionframework,bothresid- ualcontrastdynamicsandbackgroundsignalsarethencapturedin theadditivematrixX_B.Torestoreresidualcontrastdynamicswhile furthersuppressingbackgroundsignals,inthissectionafiniteset ofbasicvectors,whichcanrepresentonlybackgroundsignalsother than residual CA induced signals,is found underthe assumption that background signals mainlyresult frommotion-mismatch in- ducedsignaldifferencesbetweentimeframes.

Algorithm 1. Model-based reconstruction of contrast dynamics fromincompletemeasurementsinDCEMRI.

1.Task:FindX_D,X_B,andXbyminimizingEq.(9) 2.Initialization:

Iterationindex:k=1 Errortolerance:

Maximumiterationnumber:Kmax

Regularizationparameters:

λ

U,

λ

B,

λ

1,

λ

2

stepsize:

δ

U,

δ

X_D,

δ

X_B

Initialsolutions:U⁽⁰⁾=XV_r,

3.Reconstructionalgorithmforcontrastdynamics:

Step1:UpdateU,X_D,X_B,1,2usingEqs.(10–14) Step2:Increaseiterationindexk

Step3:Stopifk>Kmax or^{||XD(k+1)−XD(k)||F}

||X_D(k)||F ≤

4.Residualcontrastdynamicswithbackgroundsuppression: UpdateX_BbyX_B=(^I−U_MU^H_M)^XBusingEq.(17)

5.Output:X_D^(k+1)→X_D,X_B^(k+1)→X_B,X^(k+1)→X_D+X_B

Thecorrespondingsubspacecontainingonlynon-contrastback- groundsignals is learned fromthe following residual signal ma- trix, X_P=[x₁−x₀x₂−x₀ ... x_Tp−1−x₀], consistingofdifference vectorsbetweenthereferenceandthemeasured Tp dynamicim- ages before the arrival of CA columnby column. However, since theresidual signal matrix overTp phasesbefore the firstpass of bolus does not fully reflect motion-induced background changes intheentiredynamicphase,itisextendedtoaccommodatetime- varyinginformationbyemulatingtranslational,rotational,andran- dom motions. To this end, motion-emulated synthetic vectors, whichare generated by performing voxel shifting,voxel rotation, andrandomrearrangementofneighboringvoxelswithinareason- ablerange,aresubtractedfromthereferenceandthenaugmented tothe measured matrix X_P (∈C^N×Np) inthe columndirectionas shownin (Park etal., 2017).A basis of thebackground subspace isthen foundby performingthe SVDontheaugmented, residual signalmatrixby:

[X_P X_E]^SVD→U_B

BV^H_B (16)

where X_E (∈C^N×Ne) is the residual signal matrix consisting of motion-emulatedsyntheticvectors; U_B (∈C^N×r) is theleft singu- larmatrixthatcontainsthespatialbasisofbackgroundsignals;B

(∈R^r×r)isthediagonal,singularvaluematrix;V_B(∈C^(Np+Ne)×r)is therightsingularmatrix.

Given the spatial basis of the background subspace, residual contrastdynamicscanbeseparatedfromnon-contrastbackground signalsbyprojectingthematrixX_Bontothesubspacespannedby thebasicvectorsinU_Bandthensubtracting theprojectedmatrix fromXBas:

XB=

(

^I−UBU^H_B

)

^XB (17) where X_B is the matrix that contains background-suppressed, residualcontrastdynamics;U_BU^H_Bistheprojectionmatrixforback- groundsubspace.Complete contrastdynamicsisthenconstructed byX_D+X_B.Anoverallreconstructionofcontrastdynamicsissum- marizedinAlgorithm1.Sourcecodesanddatainthisworkwillbe availableonline(http://misl.skku.edu).

3. ExperimentsandResults

To demonstrate the feasibility of the proposed, model-based high-definitionDCEMRI inenhancing estimationaccuracy ofmi- crovascular functions, simulations and experimental studies were performedinpatientswithbraincancerasfollows:(1)Retrospec- tive validation for high-definition contrast dynamics estimation,

(2)Quantitativecomparisonbetweenhigh-andlow-definitionDCE MRIswithvaryingspatialandtemporalresolutions,and(3)Statis- ticalanalysis onthe effect of spatialand temporal resolutionon estimation of microvascular function parameters from the retro- spectivestudies aswell ason comparisonbetweenthe proposed methodand conventional routine DCE MRI from the prospective studies.

3.1.RetrospectiveValidationforHigh-DefinitionContrastDynamics Estimation

A time series of 3D whole brain DCE data was acquired in 5patients diagnosed with brain cancer on a 3Twhole-body MR scanner(Skyra,SiemensHealthineers,Erlangen, Germany)usinga spoiledGREMRpulsesequence.Toachievehightemporalandspa- tial resolution (1.5s, 1.0 mm³) in DCE MRI, each set of 4D data ink−t space was vastly undersampled, in which phase encod- ingsin ky−kz spacewere performed ina pseudo-radial fashion andthen subsequentlyrotatedwitha goldenangle(111.25^◦) with varyingtimephasesasshownin(Parketal.,2017).Hence,alarge portionofphaseencodingsineachdatawasprospectivelyskipped inadifferentwaywithtime duringdataacquisition,leadingtoa very high reduction factor (R) ∼ 50. Phase encodings only in a rangeofintermediate to high frequencies were then sharedover three neighboring time phases to reduce signal leakages. Asym- metric sampling (70%) was applied in the readout direction to decrease T₂^∗ effects. Imaging parameters were: time-of-repetition (TR)/time-of-echo(TE)=3.18/1.1ms,flipangle=15^◦,matrixsize

=192 × 192 × 144,readoutbandwidth=650Hz/Px,numberof phases=170,andtotalimagingtime=4.5min.Additionally,vari- ableflip angleimaging (Deoni etal., 2005) was performedusing 2^◦,8^◦,and15^◦toestimateareferenceT₁₀map.Allimagingproto- colswereapprovedbytheInstitutionalReviewBoard,andwritten informedconsentswereobtainedfromallpatients.

ToinvestigatetheeffectivenessoftheproposedmethodinDCE MRI with error analysis, a set of DCE images without apparent artifacts,which is available from the proposed reconstruction, is chosen as a reference, and then Fourier transformed to produce a fully sampled data ink−t space. The reference image is uti- lizedtoevaluaterelativeperformance ofreconstruction strategies, whichhasbeenoftenusedinotherliteratures(Lingalaetal.,2011;

Zhaoetal., 2012;Parketal.,2017).ThefullysampledDCEdatain k−t space are retrospectivelyunder-sampled usinga new sam- plingmaskwhileretainingthesamespatialandtemporal resolu- tionasthereference,yieldinganewsetofDCEdata.

A set of DCE images was generated using zero-padded in- verse FT (Fig. 2a), image difference between the reference and the zero-filled reconstruction (Fig. 2b), main contrast dynamics (X_D) (Fig.2c), residual CA signals withbackground artifacts (X_B) (Fig. 2d), background-suppressedresidual contrast dynamics (X_B) (Fig. 2e),combined contrast dynamics (X_D+X_B) (Fig. 2f), a final reconstructedDCEimage (X₀+X_D+X_B) (Fig.2g),and(imagedif- ference betweenthe referenceand the final reconstructed image (Fig.2h). Thenormalized,root-mean-square-error (nRMSE)iscal- culatedby:

nRMSE= 1

max

(

^XRef

)

1 N

N

i=1

X_Ref,i−XRec,i

2

(18)

wherei isthepixelindex, X_Ref,i isthereferenceimage, andX_Rec,i isthereconstructed image.Incoherentaliasing artifactsinFig.2a andbaresubstantiallyreducedinFig.2g andh.ThenRMSEwith zero-filled reconstruction is 68% higher than that with the pro- posedmethod. Mostof CAsignals appear inFig. 2cwhile a low level of CA and background signals is observed in Fig. 2d. It is

notedthat backgroundsignalsinFig.2darefurthersuppressedin Fig.2e(dotted arrow)whileresidualcontrastdynamicsinFig.2d iskept inFig. 2e(solid arrow).Signal time coursesinan arterial region(Fig.3a)andatissueofinterest(Fig.3b)werecorrespond- inglyplotted. X_D+X_Bdescribesarterialpeaksignalsmoreclearly thanX_D(Fig.3a),whileX_D+X_BandX_Dappearnearlyidenticalin estimatingrelatively slowlyvaryingcontrast dynamicsina tissue ofinterest(Fig.3b).

Theeffectofthelowrankmodelonsignaltimecoursesinar- terial andtissue regions wasinvestigated withr = 3, 8, 15,and 30(Fig.4).IntheAIF(Fig.4a),signaltimecoursesare:largelyun- derestimated withr= 3, well describedwith r= 8 and15, and substantially noisywith r= 30.Likewise, in the output function (Fig. 4b), signal time coursesremain nearly identical withr= 8, 15,and30whilebeinglargelydeviantwithr=3.r=8wascho- seninthiswork.

Theeffectoftheregularizationparameters inEq.(8)onimage reconstruction wasinvestigatedby calculatingnRMSEswithvary- ing

λ

U and

λ

B ranging between 1 and 12. Reconstruction errors werethelowestwith

λ

U=5and

λ

B=7whilegraduallyincreasing as

λ

U and

λ

B deviatefromtheoptimalvalue (Fig.5a).

λ

U and

λ

B

were setto5and7,respectively. Theother parameterswere em- piricallysetasfollows:

λ

1=1,

λ

2=2,

δ

U=1.0e⁻⁷,

δ

X_D=1.0e⁻⁶, and

δ

X_B=1.0e⁻⁷.Additionally,nRMSEswere calculatedwithsev- eralsetsof

δ

U,

δ

^XD,and

δ

^XB withincreasing iterationsto demon- strate convergencebehaviorofthe proposedmethod.nRMSEwas convergedafter roughly 20∼30iterationswiththe abovechosen values of

δ

U,

δ

X_D, and

δ

X_B (Fig. 5b). A typical image reconstruc- tiontimewasroughly12hforeachsetofdata(XeonCPU3.2GHz, 32GBmemory).

To demonstrate the effectiveness of the proposed method in estimating the contrast dynamics, a set of DCEimages were re- constructedfromtheretrospectivelyundersampleddatausingthe proposed method.To compare with the proposed method, k−t RPCA (Otazo et al., 2015), and k−t SPARSE-SENSE (Feng et al., 2013)wereimplementedoff-lineusingtheMATLABprogramming environmentwithpubliclyavailablesoftware.Comparedwithk− tRPCAandk−t SPARSE-SENSE,theproposed methodyieldsbet- tersuppressionofbackgroundnoiseandartifactswithsuperiorde- pictionofstructures (enlargedfigures intheyellowdottedboxes) andthelowestnRMSEamongallcomparisonmethodswithoutap- parentnoiseandartifactsinFig.6aandb.Correspondingconcen- trationtimecoursesofCAinanarterialregionandatissueofin- terestwereshowninFig.6candd,respectively.Arterialpeaksig- nals are underestimated in k−t RPCA and k−t SPARSE-SENSE while the proposed methods capture the signal evolution more better comparedwith thereference(Fig.6c). Moreover,a satura- tion level ofconcentration time coursesin a tissue ofinterest is underestimatedink−t RPCAandk−t SPARSE-SENSE whilethe proposed methodis estimatedrelativelyaccurate ascompared to thereference(Fig.6d).

Quantitative maps of perfusion and microvascular permeabil- ity(Fp,PS,vp,ve),which resultfromthe proposed method,k−t RPCA, and k−t SPARSE-SENSE, were then generated (Fig. 7). To thisend,signaltime courseestimatesinbotharterialregions (in- put) and brain tissues(output), which reflect contrast dynamics, were converted to concentration time courses in corresponding regions by exploiting the linear relation between concentration of CA and changeof T₁ relaxationrate withthe pre-definedT₁₀ (O’Connor etal., 2011). The AIF wasmanually measured ina re- gion of the internal carotid artery. Given both the AIF and out- putfunctions,perfusionandpermeabilitymapswereestimatedus- ing nonlinear least squares fittingwith the2CXM mode (Ahearn etal., 2005; Kargar etal., 2018). Comparedwith k−t RPCA and k−tSPARSE-SENSE,theproposedmethodestimatesapparentbet-

Fig. 2. DCE images reconstructed using: zero-filled inverse Fourier transformation (a), the difference image between the reference and the zero-filled image (b), the proposed method with main CA signals (c), residual CA signals (solid arrow) and backgrounds (dotted arrow) (d), residual CA signals with background suppression (e), combined CA signals (f), a final reconstructed DCE image (g), and the difference image between the reference and the final reconstructied image (h). Note that arterial CA signals in (f) are better estimated with higher signal intensity than those in (c). Additionally, background signals in (d) are well suppressed in (e).

Fig. 3. Signal time courses in DCE MRI for arterial input function (a) and output function (b). Note that X D+ X Bestimates a rapidly varying arterial concentrations better than X Dalone (brown solid arrow) while both X D+ X Band X Dare nearly identical in describing slowly varying output function.

Fig. 4. The effect of the low rank model on signal time courses in an arterial region (a) and a tissue of interest (b) with varying r . Note that signal time courses with r = 3 are largely deviant from those with r = 8, 15, 30. Additionally, an arterial signal time course appears relatively noisy with r = 30.

ter perfusionandpermeabilityvaluesinbraincancertissues(yel- lowdottedboxes)inpartduetorelativeaccuratedepictionofboth theinputandoutputfunctionsascomparedtoreferenceinFig.6c andd.AndthenRMSEwiththeproposedmethodremainsthelow- estlevelcomparedwithk−tRPCAandk−t SPARSE-SENSE.

3.2.QuantitativeComparisonbetweenHigh-andLow-DefinitionDCE MRI

Since the high-definition reference data (temporal resolution:

1.5s,spatialresolution:1.0mm³)isavailable,multiplesetsoflow-

Fig. 5. The parameter sensitivity of the proposed method to λUand λBwith increasing λU= [1,12] and λB= [1,12], respectively (a) and the convergence behaviors (b). It is noted that the nRMSE of the optimal parameters λU= 5 and λB= 7 leads to the lowest nRMSE. And the nRMSE was converged after roughly 20 ∼30 iterations with the parameters δU= 1 . 0 e ⁻⁷, δXD= 1 . 0 e ⁻⁶, and δXB= 1 . 0 e ⁻⁷.

Fig. 6. Comparison of DCE images (a, b) and concentration time courses (c: AIF, d: tissue output function) reconstructed using the proposed method, k −tRPCA, and k −t SPARSE-SENSE. Note that the proposed method (Input region: nRMSE