400 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 34, NO. 2 ... · the dynamic character of 4-D PCI is exploited by Kim et al. [23], proposing a kt SPARSE SENSE approach reconstructing

400 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 34, NO. 2, FEBRUARY 2015

Multi-Dimensional Flow-Preserving CompressedSensing (MuFloCoS) for Time-ResolvedVelocity-Encoded Phase Contrast MRI

Jana Hutter*, Peter Schmitt, Marc Saake, Axel Stübinger, Robert Grimm, Christoph Forman, Andreas Greiser,Joachim Hornegger, and Andreas Maier

Abstract—4-D time-resolved velocity-encoded phase-contrastMRI (4-D PCI) is a fully non-invasive technique to assess hemo-dynamics in vivo with a broad range of potential applications inmultiple cardiovascular diseases. It is capable of providing quanti-tative flow values and anatomical information simultaneously. Thelong acquisition time, however, still inhibits its wider clinical use.Acceleration is achieved at present using parallel MRI (pMRI)techniques which can lead to substantial loss of image quality forhigher acceleration factors. Both the high-dimensionality and thesignificant degree of spatio-temporal correlation in 4-D PCI renderit ideally suited for recently proposed compressed sensing (CS)techniques. We propose the Multi-Dimensional Flow-preservingCompressed Sensing (MuFloCoS) method to exploit these prop-erties. A multi-dimensional iterative reconstruction is combinedwith an interleaved sampling pattern (I-VT), an adaptive maskedand weighted temporal regularization (TMW) and fully automati-cally obtained vessel-masks. The performance of the novel methodwas analyzed concerning image quality, feasibility of accelerationfactors up to 15, quantitative flow values and diagnostic accuracyin phantom experiments and an in vivo carotid study with 18volunteers. Comparison with iterative state-of-the-art methodsrevealed significant improvements using the new method, thetemporal normalized root mean square error of the peak velocitywas reduced by 45.32% for the novel MuFloCoS method withacceleration factor 9. The method was furthermore applied to twopatient cases with diagnosed high-grade stenosis of the ICA, whichconfirmed the performance of MuFloCoS to produce valuableresults in the presence of pathological findings in 56 s instead ofover 8 min (full sampling).

Index Terms—Compressed sensing, hemodynamics, magneticresonance imaging (MRI), phase contrast MRI.

Manuscript received July 23, 2014; accepted September 11, 2014. Date ofpublication September 19, 2014; date of current version January 30, 2015. As-terisk indicates corresponding author.*J. Hutter is with the Pattern Recognition Lab, Friedrich-Alexander-Univer-

sität Erlangen-Nürnberg, 91058 Erlangen, Germany, and also with the Schoolof Advanced Optical Technologies, 91058 Erlangen, Germany (e-mail: [email protected]).C. Forman, J. Hornegger, and A. Maier are with the Pattern Recognition

Lab, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen,Germany, and also with the School of Advanced Optical Technologies, 91058Erlangen, Germany.P. Schmitt and A. Greiser are with the MR Application and Workflow Devel-

opment, Siemens AG, Healthcare Sector, 91058 Erlangen, Germany.M. Saake is with the Department of Radiology, University Hospital Erlangen,

91058 Erlangen, Germany.A. Stübinger is with the Department of Vascular Surgery, University Hospital

Erlangen, 91058 Erlangen, Germany.R. Grimm is with the Pattern Recognition Lab, Friedrich-Alexander-Univer-

sität Erlangen-Nürnberg, 91058 Erlangen, Germany.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMI.2014.2359238

I. INTRODUCTION

T HE ASSESSMENT of hemodynamics is important inthe diagnosis of multiple cardiovascular diseases and is

widely used for therapy decisions and planning. One exampleis the bi-lateral assessment of carotid flow to evaluate conse-quences of stroke or stenosis or flow patterns in aneurysms toassess the probability of ruptures. A further example is the as-sessment of severe internal carotid artery (ICA) stenosis, whichcan be done by measuring retrograde flow or the visualizationof areas with very slow flow [1]. The requirements for clinicallyuseful hemodynamic information include a good temporal res-olution to observe the dynamics over the cardiac cycle and highspatial resolution in all three dimensions to assess even smallarteries and to detect turbulence. Furthermore, the accuracyof clinically relevant parameters such as wall shear stress [2]or volumetric flow is highly dependent on the knowledge ofanatomical vessel lumen and wall information.Acquisition techniques for hemodynamic information in-

clude computed tomography angiography (CTA), Dopplerultrasound (DUS), fractional flow reserve (FFR) [3], anddynamic contrast-enhanced MRI (DCE-MRI), but they allcome with significant drawbacks: CTA exposes the patient toionizing radiation; FFR is highly invasive; DCE-MRI requiresinjection of a gadolinium-based contrast agent; and ultrasoundis user-dependent and does not offer anatomical information[4]. In addition, Computational Fluid Dynamics is used tocreate simulations [5] based on patient-specific informationsuch as heart rate and geometry [6]. But those cannot replacea fully patient specific in vivo acquisition of the blood flowvelocities. 4-D PCI represents a fully noninvasive method toassess human blood flow. It provides furthermore intrinsicallyregistered—as simultaneously acquired—anatomical and ve-locity information, enabling the calculation of a wide range ofclinically relevant physiological parameters such as volumetricflow, peak velocity, and mean velocity [7].A general drawback of this technique—if considerable

3-D coverage and high temporal and spatial resolution is re-quired—is its long acquisition time, which originates from thesynchronization to the cardiac cycle, the required segmentedacquisition technique and the velocity encoding. The acqui-sition time for a full scan of the carotid bifurcation regionfor a reasonable parameter set offering both good spatial andtemporal resolution reaches up to 30 min.

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HUTTER et al.: MuFloCoS FOR TIME-RESOLVED VELOCITY-ENCODED PHASE CONTRAST MRI 401

Recently, approaches to reduce acquisition time for 4-D PCIhave been proposed: They rely either on optimized encodingstrategies [8], [9] or on specific reconstruction strategies ableto reconstruct the output volumes from highly undersampled-space data. Sampling below the Nyquist criteria leads to sig-nificant image artifacts which are addressed by specific recon-struction techniques. One group of methods consists of pMRIsuch as GRAPPA [10] and SENSE [11]. The spatially varyinginformation of multiple coils around the subject which acquire-space data simultaneously is used to reduce the scan time.The additional spatial sensitivity information of the coil setupis exploit to reconstruct images from undersampled -spacedata. pMRI techniques are employed for dynamic 4-D PCIoften in combination with temporal encoding schemes suchas kt-GRAPPA [12] or compartment-based kt-SENSE [13].Non-Cartesian trajectories such as spirals in combination withSENSE were introduced by Steeden et al. [14]. A differentgroup of methods, able to reconstruct even higher accelerateddata, are iterative reconstruction techniques and recently pro-posed compressed sensing (CS) techniques [15], [16] which arefrequently combined with pMRI methods. CS relies on gener-ating noise-like artifacts by using incoherent samplings, whichcan be separated from image information using adapted regu-larization. The basis for the employed regularizers are sparsityor transfer sparsity assumptions, The resulting optimizationproblem, combining the fidelity to the measured data with reg-ularization terms is solved using nonlinear reconstruction al-gorithms. This technique is promising for 4-D PCI as the fun-damental principles of incoherence and sparsity are applicable.Incoherence is achieved by choosing dedicated sampling trajec-tories for the acquisition in -space. Those are either Cartesianor non-Cartesian. Non-Cartesian trajectories such as radial orspiral sampling offer higher flexibility compared to Cartesianpatterns with regard to incoherence. But this benefit is associ-ated with the required additional gridding step [17] as majordrawback which significantly adds to the computational effortof the iterative reconstruction. It was shown that the theoret-ical best choice, randomized patterns, is less suitable for highlyaccelerated MRI. The central -space region, containing lowfrequencies requires denser sampling to achieve good results[18]. Furthermore, high acceleration randomized patterns canresult in significant gaps between the samples. Proposed al-ternatives include pseudo-random or Poisson-distributed sam-pling [19]. Sparsity is inserted into the CS formulation withspecific regularizers, varying between very general assump-tions about medical images such as a certain smoothness andthe possibility to be expressed by a low number of Waveletcoefficients and very acquisition specific constraints. These in-clude for 4-D PCI, complex difference sparsity as shown byKwak [20], divergence-free constraints described by Busch etal. [21] and specific phase regularization [22]. Furthermore,the dynamic character of 4-D PCI is exploited by Kim et al.[23], proposing a kt SPARSE SENSE approach reconstructingjointly reference and velocity-encoded data using a temporalFFT and PCA as sparsifying transform. Velikina et al. [24]include second order temporal differences. Spatially varyingtemporal constraint regularization has been applied to PC MRIby Hulet et al. [25]. Acceleration methods based on CS further

include work by Joseph et al., who showed good results ap-plying nonlinear inverse reconstruction techniques [26] and therecent work by Santelli et al. extending the L1-spirit methodto PCI data [27]. The only studies applying CS methods toin vivo carotid PCI data were presented by Tao et al. [28]introducing temporal Fourier transform in combination withuniformly random sampling and by Hutter et al. [29] usinga Low-Rank Sparsity assumption. Carotid PCI data is chal-lenging due to the small vessel diameter, the highly pulsatileflow and the complex flow behaviour around the bifurcation.For carotid PCI, the dynamic changes originate mainly fromblood flow effects, either directly or indirectly by vessel wallmotion due to its pulsatile nature. In addition, their spatial ex-tent is limited to the vessel proximity. Further possible originsof movement, patient-, cardiac-, and breathing motion can beneglected for this application. The temporal resolution of de-rived physiological parameters with clinical relevance, such asvolumetric flow, peak velocity, or wall shear stress depends to ahigh degree on the temporal fidelity of the flow reconstruction.A well-suited temporal regularization should thus exploit theanatomical correlation in the static tissue parts to offer volumeswith clinically accepted image quality while maintaining thetemporal fidelity in vessel proximity. Two basic requirementsexist for the usage of this prior knowledge: The first require-ment is a joint reconstruction algorithm which reconstructs allvolumes for all time steps as well as velocity encodings simul-taneously to allow for exploiting correlations spanning overdifferent images along all dimensions. Secondly, a stable dy-namic sub-division of the image volume into vessels and statictissue is required. The necessary information for this subdi-vision is intrinsically available in 4-D PCI with the anatom-ical reconstruction. The Multi-dimensional Flow-adapted Com-pressed Sensing algorithm (MDFCS) used this information asprior in combination with a dedicated pattern and iterative re-construction [30]. In this work, we extend this approach toa fully interleaved and incoherent sampling (I-VT) strategy,enabling higher acceleration through fully internally sharedcalculation of the coil profiles. The adaptive TMW regular-ization strategy is refined by adding additional weighting andmasking with a static and dynamic mask for a stable and au-tomatic differentiation into static and nonstatic tissue duringthe reconstruction. The resulting Multi-Dimensional Flow-pre-serving Compressed Sensing (MuFloCoS) algorithm exploitsthe significant spatio-temporal correlation in the dynamic ac-quisition while preserving the temporal flow resolution. Theadaptive masking strategy relies on inherent 4-D PCI features,which are made accessible by an interleaved sampling schemein the temporal and velocity encodings (I-VT). In summary,in contrast to other methods, not only the spatial and temporaldimension, but also the 4-D PCI inherent velocity encodingdimension is included in all steps of the algorithm. The TMWregularization penalizes nonsimilarity to neighboring temporalphases but differentiates between static and flow affected areasusing adaptive vessel masks in order not to introduce temporalblurring. The paper is structured as follows. The acquisition andreconstruction basics as well as the postprocessing of the imagedata for PCI are introduced, the novel MuFloCoS method isdescribed as well as the experimental setup consisting of a


Fig. 1. Static magnetization and moving magnetization under the influ-ence of the bipolar gradient in the acquisition time .

phantom experiment, a larger carotid volunteer study with 18subjects and two patient cases. Image based quantitative mea-sures as well as physiological parameters were calculated andused to evaluate the accuracy of the proposed method. Thenovel method is compared against the fully sampled referenceas well as different CS methods such as L1 regularized SENSE,kt-SPARSE SENSE once with temporal Fourier transform andonce with temporal principal component analysis (PCA), themethod proposed by Tao et al. 2013 [28] and MDFCS. Anotherexperiment shows the stability and acceleration capacities ofMuFloCoS. Finally, the results of the two patient cases of se-vere stenosis of the ICA, accelerated and reconstructed withMuFloCoS, are presented.

II. 4-D PCI THEORY

The magnetic field in an MRI measurement consists of a ho-mogeneous magnetic field and a gradient field , gener-ated from gradients in all spatial dimensions which are switchedduring the measurement sequence. Phase Contrast MRI relieson the linear dependency of the signal phase from velocity inpresence of special bipolar gradients consisting of twolobes with the same amplitude but opposite polarity. Thetotal field considering a gradient in -direction equals to

. The accumulatedphase at the end of an acquisition interval is calculated byintegrating the frequency over the time and using the Larmorfrequency as

(1)

The mechanism to obtain velocity information utilizing thesignal phase is illustrated observing two spins with differentvelocities (see Fig. 1). If a magnetization is static,

and magnetization is moving along the direction of thegradient with constant velocity , its time-dependent positionequals .During the first slope for , the transverse magneti-

zation of both spins is de-phased. The second slope re-phases thestatic magnetization, leading to a net phase change of 0. Mag-netization , however, during its movement experiences dif-ferent magnetic field conditions and is not completely re-phasedduring the second slope of the gradient. As a result, a term in-

cluding the first gradient moment and providing a linear depen-dency on the velocity is obtained

To obtain a linear relation between the velocity and the mea-sured phase as well as to compensate for phase effects unre-lated to flow such as field inhomogeneties an additional ve-locity-compensated reference acquisition is required. For 3-Dvelocity information, most 4-D PCI sequences use a 4-point ve-locity encoding scheme [31], which requires one flow-compen-sated reference scan and three velocity sensitive scans [32] inthrough-plane, anterior-posterior and right-left direction. Dy-namic information over the cardiac cycle is obtained by trig-gering the acquisition to the R-wave in the cardiac cycle of thepatient using electrocardiogram (ECG) or pulse triggering.time steps are defined starting from the R-wave, within whichsubsets of -space data are collected over multiple heart beatsin a segmented acquisition scheme.

A. Reconstruction

For the following, will be the column wise written notationfor a matrix . The variables and denote the numberof temporal phases and velocity encodings, rep-resents the problem size. is the number of-space samples, the number of parallel receive coils and

the image volume size.For each temporal phase , each velocity encoding , and each

channel , one -space vector is acquired. Theset of all -space acquisitions corresponding to the sametemporal phase and velocity encoding are represented as theall-channel vector . For each time step, vectors

are created, which formthe total raw data vector .The reconstruction leads to the image space vectors. The image space vector is formed analogously.Transformation between image space and -space is done via

the system matrix assembled of matrices, such that and

. . .. . .

(2)


For conventional fully sampled single-channel MRI, thesystem matrix consists of Fourier coefficients. For pMRI,the matrices include the coil sensitivity profiles andthe Fourier coefficients [11]. Direct solution of isimpossible in the highly accelerated case as the system isheavily under determined for .

B. Extraction of Flow-Related and Anatomical Information

The reconstructed data sets are further processed to ob-tain both anatomical information and quantitative 3-D velocityinformation. For the case of three-directional velocities,equals four. The complex difference between velocity-com-pensated and velocity-encoded acquisitions highlights theregions with moving magnetization by subtracting out the staticregions, corresponding to moving protons. It can therefore beused to obtain angiographic images which highlight the vesselanatomy. Respective complex differences are calculated forall velocity encoding directions. The sum of their magnitudesresults in the angiographic image , providing highcontrast for areas with flow irrespective of the direction. Thequantitative information is obtained via the signal phase dif-ference between velocity-compensated and encoded image foreach flow encoding direction as .Calculations of and are performed as

(3)The phase in is encoded to values within the interval

, which can lead to wrapping artifacts and misinterpre-tation of flow exceeding this range as slow flow in the oppositedirection. In order to avoid those artifacts, the velocity encodingparameter, also called venc, , is chosen as the maximum ex-pected velocity in each flow encoding direction. Finally, 3-Dvelocity fields are obtained as .

III. MUFLOCOS

The novel MuFloCoS approach is presented in detail here.Four main points are in the focus: the interleaved I-VT patternis explained in Section III-A; the multi-dimensional jointiterative Newton-based reconstruction in Section III-B; thevessel-masked and temporal weighted TMW regularization inSection III-C and finally the domain sub-division using theanatomical image detailed in Section III-D. Furthermore, thechoice of all parameters is given along with pseudo-code and aschematic flowchart.

A. The I-VT Sampling Strategy

The sampling for 4-D PCI can be varied in four dimensions,two intra-volume and two inter-volume directions. The intra-volume degrees of freedom are and direction. The acqui-sition in the read-out direction is substantially faster than thephase encoding directions and as no gradient switchesare required. The inter-volume directions are the time andthe flow encoding direction . While state-of-the-art approachestypically offer , and -direction variation, the freedom in

Fig. 2. (a) Schematic 2-D illustration of central -space variations. (b) Map-ping of time steps and velocity encodings for three different mapping func-tions : Temporal variations , regular , and random per-mutations .

flow encoding direction is rarely used. The MuFloCoS algo-rithm relies on a Cartesian pattern, which can be analyticallycalculated while offering both a substantial amount of incoher-ence and suitability for high acceleration factors. The omis-sion of a gridding step, possible through the use of a Carte-sian pattern, in every iteration reduces the computational ef-fort, and therefore the reconstruction time. The pattern is de-scribed by , with , it equals “1” for sam-pled points and “0” for omitted -space samples. The genera-tion of the pattern for all time steps and flow encodings isdetailed in the following. Intra-volume directions are exploitedby separating -space into a central region , which is regu-larly under-sampled and has the dimension , and aperipheral region, sampled irregularly with decreasing den-sity following an inverse root function [33]. The center is de-scribed by the spacing , fixed to for the 2-D version and

for 3-D, and the offset , obtained as re-sult of the mapping . Fig. 2(a) illustrates the center sam-pling patterns for with . Conventionaltemporal approaches without -variations can be modelled with

, where depends uniquely on . Two alterna-tives including -Variations for the mapping function are pro-posed, regular permutations and randomshifting , obeying two constraints: 1) two subsequenttime steps in the same velocity encoding never share the samecentral lines and 2) all -space central lines must be sampled ineach time step

(4)

(5)

(6)

The same offset is used for the start of -space peripheral in-verse root sampling, using a parametrizable quadratic function

to determine the distance between successive sam-pling points.The results for all three mentioned mapping functions are il-

lustrated in Fig. 2(b) for on the top, for in themiddle and for on the bottom. The corresponding pat-terns including peripheral inverse root sampling are shown inFig. 3 with their point spread functions. The higher peak-to-sidelobe-ratio in the I-VT variants in three illustrates the higher


Fig. 3. Exemplary patterns for 2-D sampling with for the threementioned mapping functions with the corresponding point spread functionsPSF using the same scaling illustrate the better peak-to-side loberatio for the random permutations.

incoherence of the under sampling artefacts, which is an impor-tant prerequisite for CS. The proposed I-VT sampling strategyexploits and direction to add incoherence, but also to enabletwo additional features.• The pattern, interleaved in and permuted in -direction,allows the calculation of approximate static and dynamicanatomical images , which are used later in the algorithmfor the anatomy-based sub-division (Section III-D).

• Instead of acquiring the coil profiles in a separate time-con-suming external scan, they can be obtained by combiningcentral -space lines over the velocity encodings. Thecombined fully sampled low resolution -space is filteredwith a Hanning window [11].

B. Multi-Dimensional Iterative Newton-Based Reconstruction

In contrast to the conventional case, where each volume isreconstructed individually, the approach combines the raw dataof all phases and velocity encodings in one reconstruction. Thisis favourable over individual reconstruction, as it enables the useof sparsifiying transforms spanning over different images alongall dimensions to exploit similarities during the reconstruction.The objective function for the iterative reconstruction consistsof a data fidelity term and the regularization term .The data fidelity term

(7)

includes the raw data and the pMRI reconstruction matrixincluding the Fourier coefficients,

the under sampling pattern and the coil profiles describedby

(8)

Fig. 4. Illustration and justification of the weighted and masked temporal reg-ularization strategy. (a) Magnitude of the flow-compensated and through-planeencoded images is shown for . (b) Finite differences over timebetween the images for and are depicted.

Thereby is the position of voxel in image space, isthe frequency. A more detailed description of the regularizationterm is found in Section III-C.

C. Vessel-Masked and Temporal Weighted Regularization

The MuFloCoS approach proposes an adaptive vessel-masked and weighted temporal regularization (TMW) whichexploits spatio-temporal correlation while maintaining the tem-poral flow fidelity using the anatomy-based sub-division. Thesignificant spatio-temporal correlations of PCI can be observedin the first five time steps for the velocity compensated (upperrow) and one of the velocity encoded scans (lower row). Thosecan be modelled using finite differences (FD) with differentstep length in time direction. For each temporal time pointand encoding , the FD to phase with

is calculated voxel wise as

(9)

Fig. 4(b) illustrates for andfor the velocity compensated and encodeddata scans. To assess the proposed sparsity assumption as wellquantitatively, the coefficients of the FD imagesfor are sorted by magnitude in Fig. 5 for ,

, , and . Three observations can be de-rived and will motivate the algorithmic choices detailed below.In general, it can be well observed, that the significant high con-tributions are concentrated within few pixels in the coefficientplot. This is illustrated in the difference images in Fig. 4(b),where the main contributions to the finite differences clearlyare concentrated at the vessels, which show the meaningful ve-locity changes over time, while the background has relativelylow contribution. There is a substantial difference between theflow compensated and encoded images.While the flow-encodedscans show the described enhancement of vessels, this is lessclearly observable in the compensated scans. The zooms into the


Fig. 5. Finite difference coefficients ordered by magnitude showthe sparsity of the used differences for and .

highest 0.1% of the coefficients show a more significant contri-bution of the velocity-encoded scans (dotted lines). The zoominto the lower coefficients reveals the inverse situation. In sum-mary, the coefficients for the flow-encoded scans are more com-pressed in the highest values. Finally, the concentration of con-tributions to the vessels decreases for larger time steps. How-ever, the general importance of the differentiation in vessels andbackground stays valid for a certain range of subsequent timesteps. Based on those observations, the MuFloCoS algorithmincludes three features into the temporal finite difference regu-larization:• a vessel-mask limiting the regularization to backgroundareas to preserve the dynamics within the vessels,

• different treatment for the flow compensated and velocityencoded scans, and finally

• a temporal weighting in time direction, attributing higherimportance to closer time steps.

The difference terms are arranged for all step sizes in a commonvector

(10)

and finally for all flow encodings to and all timesteps to

...... (11)

The weighting function is realized either witha box function or with a Gaussian kernel

, centred at the posi-tion with standard deviation , determining the extent of theinfluence of neighboring phases. Thereby the vectorequals . It is used to form theweighting matrix for time step as well asthe complete weighting matrix

(12)

Finally, the vessel masks are involved. Thosetheoretically equal to for voxels within a vessel and

for background voxels, for practical reasons nonbi-nary values are chosen. Their calculation is detailedin Section III-D. To limit the regularization to the backgroundareas, the masks are subtracted from the unity vector ,such that voxels within vessels are multiplied with “0” and donot contribute to the value of the regularization. The differen-tiation between compensated and encoded acquisitions is mod-elled by weighting all voxels of the compensated scan with “1,”meaning, that the regularization operates on them.The masks are arranged in the diagonal matrices

, and composed to form matrices. These are used in the assemblage of the final

entire mask matrix

. . .

. . .(13)

The temporal differences are multiplied with the mask ma-trix and the weighting matrix

(14)

The objective function including the regularization weightequals

(15)

This is solved iteratively by

(16)

D. Anatomy-Based Sub-Division

During the iterative reconstruction the volume, defined by itsvoxel set with voxel indices , is divided into astatic part consisting of the voxel set and a part affected byflow motion with to allow guidance of thetemporal regularization to the static parts. This is important toavoid temporal blurring. The goal is therefore to obtain dynamicmasks with the theoretical property

ifif .

(17)

As the subdivision correlates mainly with the vessel anatomyin the chosen application, is referred to as vessel mask. Theselected differentiation feature is the occurrence of flow as it isinherent in the PCI technique through the anatomical imagesas explained in Section II-B. The proposed interleaved and in-coherent pattern leading to distributed incoherent artifacts inand direction allows an approximation of , which is used togenerate the masks . This information is then available for


Fig. 6. Generation of the vessel masks as a combination of the static and dy-namic anatomical images is illustrated. The (a) static images and (c) dynamicimages for the first three and the last iteration are depicted. (c) Evolution oftheir respective influence is shown depending on the parameter . (d) Finalvessel masks for are visualized.

the temporal regularization and directs it to the known loca-tions of correlation. Beside the approximation of the typical PCIanatomical map , as described in (3), a static approximationcalculated as the anatomical image over all phases is used

(18)

During the reconstruction, both are calculated based on the ac-tual image estimate for iteration . The firstnecessary estimate used during the first iteration, referred to bycorresponds to the conventionally reconstructed raw data

vector using the Sum-of-Squares [34] method: .By updating the masks in each iteration, the algorithm adapts tothe improving reconstruction quality. MuFloCoS uses a combi-nation of both static and dynamic image, which is crucial forthe stability and robustness of the algorithm particularly for thefirst iterations which are heavily influenced by aliasing artefacts.This is illustrated with a 9.0 times under-sampled data set inFig. 6. The obtained static masks for the first four iterations andthe last iteration are shown in Fig. 6(a), the dynamic imagesfor time step 2 in Fig. 6(c). The aliasing artefacts in the firstdynamic masks are visible, while the static images allow cleardepiction of the vessels from the first iteration on. This is pos-sible through the use of the I-VT pattern with variations in bothdirections, used combined to generate the anatomical images.The sub-division is entirely based on this intrinsically

obtained anatomical images and by applying a binarythreshold . The obtained binary masks and for voxelequal

ifif

ifif

(19)

The influence of the static and dynamic anatomical imageapproximation changes smoothly with the parameterdepending only on the iteration step with

(20)

The evolution of over iteration steps is visualized inFig. 6(b). Fig. 6(d) illustrates the final masks .

E. MuFloCoS Implementation Details

This section describes the details of the MuFloCoS algorithmimplementation. The algorithm was included into a C++ re-construction framework, offering both a linked version to theMR scanner and a standalone version for testing purposes. Thelinked version directly processes the raw data after acquisitionusing the inline data processing pipeline. The MuFloCoS algo-rithm seeks to find a solution for the problem as stated in (15).

Algorithm 1: MuFloCoS algorithm

Require:

Require:

1: Calculate combined coil profiles

2: Obtain direct reconstruction with

3: Calculate and

4: Initialize mask matrix

5: Assemble weighting matrix

6: Assemble encodingmatrix with

7: while do

8: Newton update evaluating the objective function asfollows:

9: Calculate the data fidelity term

10: Assemble FD vector

11: Calculate

12: Update

13: Update

14: Combine and to using

15: Update mask matrix

16: end while

17: return

The complete MuFloCoS algorithm is represented in the flowchart in Fig. 7 as well as in a detailed step overview in algorithm1. The preprocessing in steps 1–6 includes shared coil profilecalculation, initialization of the vessel mask, calculation of thetemporal weights and assembling of the encoding matrix. Then,the iterative process is started.Considering the size of the optimization problem with

unknowns, a limited-memory BFGS solver [35] is used, which


Fig. 7. Scheme of the MuFloCoS algorithm.

proved to be memory-efficient and stable. Replacement withfurther solvers such as the conjugate gradient method is possiblewithout introducing structural changes. For all gradient-basedsolvers, both the objective function and its gradient are requiredwithin each iteration. Finally, the vessel mask matrix is updated.The parameters for MuFloCoS were chosen equal for all

datasets and experiments as , and. The ensures computational effi-

ciency, as the differences are calculated only in the relevantkernel .

IV. EXPERIMENTS AND RESULTS

A. Phantom Experiment

Phantom data was acquired using an MR compatible pump(CardioFlow 5000MR, ShelleyMedical, Toronto, Canada) con-nected to a control unit outside the scanner room and a tubesystem filled with blood mimicking fluid. An inflow–outflowsetup was used for this study with two connected tubes of diam-eter 1.9 cm and a phantom bottle to simulate tissue contrast. Theimaging volume plane was chosen orthogonal to the tubes, suchthat each imaging plane contained a cross section of the bottleand both the in- and outflow tube. See Fig. 8(a) for a schematicrepresentation of the setup. A regulated laminar flow with 150ml/s was pumped through the tube system and imaged on a 3TMR scanner (MAGNETOM Skyra, Siemens Healthcare Sector,Erlangen, Germany). The imaging parameters were FOV 190130 mm, matrix 256 176 and a slice thickness of 3.1 mm,

, temporal resolution 49.6 ms and flipangle 20 . The controlled setup allows to verify the flow con-servation law between in- and outflow in each slice

, and between adjacent slices regarding the inflow andthe outflow

(21)

The fully sampled data sets were reconstructed with the con-ventional Sum-of-Squares (SoS) technique to obtain a referencevolume. The I-VT pattern with acceleration factor 9 was applied

Fig. 8. Setup and imaging planes for (a) the phantom experiments and (b) thein vivo carotid study.

TABLE IQUANTITATIVE EVALUATION OF THE PHANTOM DATA

retrospectively and the undersampled data was reconstructedusing our proposed MuFloCoS method. Table I illustrates thedeviation results for the reference reconstruction and for Mu-FloCoS in percent, which are all below 4%.

B. In Vivo Study

4-D PCI data was acquired on a clinical 3T MR scanner(MAGNETOM Skyra, Siemens Healthcare, Erlangen, Ger-many) from 18 healthy volunteers (18–72 years) using anECG-triggered PC sequence. The region of interest (ROI) waschosen as the region around the carotid artery bifurcation. Up to16 transverse slices were acquired, starting from the commoncarotid artery (CCA) 40 mm below the bifurcation up to theinternal and external carotid artery approximately 20 mm abovethe bifurcation. This setup is illustrated in Fig. 8(b). Imagingparameters were , temporal resolution49.76 ms, flip angle 20 , FOV 200 mm 200 mm, slicethickness 2–4 mm and imaging matrix 256 256, in-planeresolution of 0.78 . The velocity sensitivity rangeand phases were individually optimized. Between eleven and20 temporal phases were acquired and the was chosen be-tween 60 and 100 cm/s. The FOV was adapted if required. Thefully sampled data sets were used to have a reliable referenceespecially for the physiological values.Three experiments with different goals were performed with

the in vivo study data.Experiment I: Quantitative and physiological evaluation.Quantitative flow parameters obtained with the proposed re-

construction algorithm are compared to previously reported lit-erature values and to the results of different state-of-the-art iter-ative algorithms to ensure the validity of the proposed methodfor enhanced image quality and quantification purposes. Sevenreconstruction were therefore performed: the fully sampled dataset was reconstructed using the Sum-of-Squares technique toobtain a Reference. The remaining six reconstructions wereperformed using an acceleration factor of 9.0 for different it-erative techniques. Thereby, the same coil profiles and the sameformulation of the data fidelity term in (15) was used. The algo-rithms thus varied in the choice of the regularization, the samesampling was employed for all. All emerging optimization prob-lems were solved using the same lBFGSmethod which has been


shown to be very stable and especially suited for large opti-mization problems as the present. The number of iterations con-ducted for each algorithm was determined based on the relativedata fidelity term (7) reduction, calculated as

(22)

This resulted in for ISENSE, for ,for , for , for

MDFCS and for MuFloCoS.• Iterative SENSE algorithm ISENSE without regulariza-tion, corresponding to in (15).

• Regularized state-of-the art Compressed Sensing algo-rithm using the Daubechies 4Wavelet transform andTotal Variation with corner rounding parameter .

• The kt-SPARSE SENSE algorithm as previously proposed[23] using temporal Fourier transform and usingPCA [36] . The implementation followed theoriginal implementation provided by the authors in termsof objective function and gradients.

• The previously proposed MDFCS algorithm [30]. A boxfunction instead of the Gaussian weighting is used and thevessel mask is obtained only based on the dynamic images.

• The proposedMuFloCoS algorithm with .The sparsity weights for , , and were

optimized individually for each technique regarding NRMSEover the parameter space . The re-sulting values and for ,

and for andfor were used for all data sets.

Experiment II: Comparison against state-of-the art in CS forcarotid PCI.The aim of this experiment is the comparison of MuFloCoS

against the only known further state-of-the-art algorithm pro-posed for the same application, in vivo imaging of the carotidarteries, the method by Tao et al. [28]. The key algorithmic el-ements were as follows.• R: A pattern which is fully sampled in the central regionand randomly under sampled in the periphery. The samepercentage of central lines as in the study by Tao et al.(20/192) were used for the present data sets (26/256). Thepattern was not varied over encodings but includes randomvariations over time.

• : minimization in the yf-space for each encodingseparately was proposed as regularization by Tao et al.

Different reconstructions, evaluating these components againstthe corresponding MuFloCoS parts, the I-VT sampling and theTMW regularization were evaluated. First, the proposedmethodwas evaluated as proposed in the original paper combining arandom pattern with the minimization (TAO 3) and withthe same acceleration of 3 and regularization but using the I-VTpattern ( 3). Then, three reconstruction were per-formed with an acceleration factor of 9: The method by Tao etal. (TAO 9), the novel I-VT sampling combined with theregularization ( 9) and finally the proposed Mu-FloCoS method. To ensure fair comparison, the weights wereoptimized regarding NRMSE for each of these experiments, butkept fixed over all data sets.

Experiment III: Parameter and robustness.The influence, stability and robustness of different algo-

rithmic aspects and parameter choices are evaluated based onquantitative image measures and reconstruction parameters.For Experiment III, three elements were investigated: Theregularization weight , the sampling strategy and the useof the shared coil profile calculation. The parameter wasvaried between 0.0005 and 0.0125, the sampling pattern waschosen fixed for all phases and temporal phases or interleavedand permuted with regular permutations with length 1 orinterleaved and permuted as proposed by the I-VT samplingstrategy of MuFloCoS. The basis pattern parameters , , ,and were chosen identical for both. Furthermore, the influenceof coil sensitivity calculation was evaluated by using eitherthe shared version or by acquiring an external reference scan,both resulting in a total of 16 used reference lines. Feasibleacceleration factors for this application are determined, theused acceleration factor was therefore varied between 3, 6, 9,12, and 15. The reconstructions were performed with fixed

.1) Evaluation Strategy: The reconstructed volumes are

used to calculate the angiographic images and the phase con-trast images . For the derivation of flow parameters, the 3-Dvelocity fields and the vessel borders for the CCA, ICA,and ECA are required. The borders are obtained using an in-teractive up-sampling and segmentation tool. For the quantita-tive image evaluation, the normalized root mean square errorNRMSE and the structural similarity measure SSIM [37] offeran objective comparison between the reconstruction result

and reference volume . Thereby andare the mean and respectively the variance over . Further-more, to assess the image quality, the angiography specific con-trast-to-noise-ratio between vessel and tissue CNRVTwas used.Therefore, regions-of-interest (ROI) were chosen in the vesseland the tissue .Evaluated flow parameters include the volumetric flow rate, the peak velocity and the mean velocity .

They were measured in the 2-D slices and in three selectedvessel cross sections as illustrated in Fig. 8(b). The temporalnormalized root mean square error is calculated for a dynamicparameter , measured with a reference method

and with a test method . These values are com-puted as

and

The deviation is calculated for the volumetric flow and forthe peak velocity . Those results are illustrated with Bland-Altman diagrams for graphical illustration of the TNRMSE for


TABLE IIQUANTITATIVE IMAGE AND PHYSIOLOGY-BASED EVALUATION FOR THE INVIVO STUDY COMPARING MUFLOCOS TO THE REFERENCE AND FURTHER

ITERATIVE METHODS

all data sets and for each combination between MuFloCoS andstate-of-the-art methods. The results are ordered by the absolutevalues for the volumetric flow and the peak velocity of the datasets. This illustration aids to identify any systematic errors oroutliers.2) Results for Experiment I: Quantitative and Physiological

Evaluation: The quantitative results for the image based mea-sures of the study with 18 data sets are shown in Table II forMuFloCoS and the five comparison methods. The NRMSE ofMuFloCoS is significantly reduced compared to the iterativeSENSE reconstruction and all further considered methods. Sta-tistically, the NRMSE has been improved by 50.85 comparedto ISENSE, by 15.94 in comparison to and by 17.14compared to . The same is valuable for the structural sim-ilarity SSIM, which was improved by 3.72 compared to thebest comparison method . The contrast-to-noise ratio be-tween the vessel and the background could be improved for alldata sets, in the mean by at least 16.5 and up to 41.2. Table IIshows the mean deviation from the physiological parameters foreach reconstruction technique. One data set (P16) was excludedfrom the calculation of physiological parameters, as the overallimage quality even for the reference did not allow stable evalu-ation. A significant improvement was achieved for the peak ve-locity with 45.32% lower TN compared to and 53.04%lower TN compared to MDFCS. While performedbetter regarding the volumetric flow by 16.82%, MuFloCoSwas significantly better in the peak flow velocity measurementswith an improvement of 53.7%. Fig. 9 illustrates two represen-tative results for peak systole. Representative time curves areshown in Fig. 10, displaying the volumetric flow in the firstrow, the mean velocity in the second row and the peak ve-locity in the last row for all reconstruction results. The curvesof the reference and MuFloCoS are very similar, while the othermethods except tend to overestimate volumetric flow andmean velocity in both cases. The most significant difference isvisible in the peak velocity plots, which is heavily disturbedfor comparison methods but well preserved for MuFloCoS. Es-pecially the method, which produced a lower error involumetric flow significantly underestimated the peak velocity.The Bland-Altman diagrams for the best comparison methods

, , and are given in Figs. 11. No outlier

Fig. 9. Magnitude reconstruction results for the through-plane encoding forvolunteer P2 and P7 at peak systole. Shown for both in the top row: Reference,

, in the middle row: ISENSE, MDFCS and in the bottom row: ,MuFloCoS.

Fig. 10. Volumetric flow, mean velocity profile, and peak velocity profiles il-lustrated for volunteers P6 and P7.

or systematic bias is visible in the MuFloCoS result, whereasthe values for the other methods are spread for both volumetricflow and peak velocity. Fig. 12 illustrates the 3-D result for peaksystole and early diastole at three selected locations as depictedin Fig. 8(b).3) Results for Experiment II: Comparison Against Carotid

PCI State of the Art: With the original method proposed by Taoet al. [28], very good results were achieved for an accelerationfactor of 3, with a SSIM of 0.934 0.038, a NRMSE of 0.0640.023 and a TNRMSE of the volumetric flow of 0.123 0.104for the random pattern. The results using the same regularizationbut the proposed I-VT sampling further increase image qualityand the accuracy of the physiological values with a NRMSE of0.042 0.011 and TNRMSE of the peak velocity of 0.1610.060. For the higher factor of 9, however, the proposed Mu-FloCoS algorithm outperforms this method, particularly with


Fig. 11. Bland-Altman diagram for volumetric flow and peak velocity for thein vivo study for MuFloCoS and (a) , (c) , and (c) .

Fig. 12. 3-D velocity vector field from the left carotid bifurcation region il-lustrated with 3-D vectors at four different locations for peak systole and earlydiastole.

TABLE IIIQUANTITATIVE IMAGE-BASED EVALUATION FOR THE IN VIVO STUDYCOMPARING MUFLOCOS AGAINST THE RESULTS OF CAROTID PCI

STATE-OF-THE-ART

respect to the TNRMSE of the physiological values and theNRMSE. However, the method of Tao et al. shows some ad-vantages regarding the SSIM value and the CNR. Comparing

Fig. 13. Image and reconstruction parameter results for a representa-tive volunteer using different . (a) Image results and zooms tothe right ICA for, from left to right, the reference and MuFloCoS with

. (b), (c) Evolution of the datafidelity term (b) and TMW term (c) is depicted for 20 iteration steps.

TABLE IVQUANTITATIVE IMAGE BASED EVALUATION FOR DIFFERENT

MUFLOCOS VARIANTS

the results using the regularization in combination witheither the random or the I-VT pattern, the I-VT pattern deliversmore accurate flow results and better NRMSE, but lower SSIMand CNR.4) Results for Experiment III: Robustness and Acceleration:

Image results for different values are shown in Fig. 13(a)with a zoom to the right carotid artery. The data fidelity termas well as the TMW regularization term for selected choices of

are illustrated in Fig. 13(b) and (c). This reconstructionresulted in the values illustrated in Table IV. The respective im-ages for for early diastole and differenceimages to the complete MuFloCoS method are shown for MnPand MnC in Fig. 14 along with the corresponding differenceimages to the normal MuFloCoS in the lower row. Volumetricflow and mean velocity are illustrated in Fig. 15. Reconstruc-tion specific parameters including the data fidelity term and theregularization term over time are illustrated in the lower row ofFig. 15. Both the data fidelity and the norm show a similar be-havior for M andMnC, but higher values for the variant withoutthe interleaved and shifted pattern (MnP). Results of varying ac-celeration factors of 3, 6, 9, 12, and 15 and fixed can beseen for for the entire image and a zoom in Fig. 16(a).The corresponding curves for volumetric flow are illustrated inFig. 16(b), the peak velocity in Fig. 16(c).


Fig. 14. Image results for the through-plane encoding comparing the referenceandMuFloCoS (a) with the I-VT pattern and the shared coil profiles, (b) withoutthe I-VT pattern, and (c) with external coil profiles. Lower row illustrates thedifference to MuFloCoS scaled by a factor of 10.

Fig. 15. Physiological parameters (a), (b) and reconstruction results (c), (d)for MuFloCoS with the I-VT pattern and the shared coil profiles, without theI-VT pattern and with external coil profiles. Volumetric flow and meanvelocity are shown. Evolution of the data fidelity term and the evolutionof the TMW- term are depicted for 20 iteration steps.

C. Patient Cases

Two patients m/72y, m/63y, diagnosed with stenosis in theICA, were examined on a clinical 3T MR scanner (MAG-NETOM Verio, Siemens AG Healthcare Sector, Erlangen,Germany) using an ECG-triggered PC MRI sequence. Patient1 was diagnosed with a high-grade stenosis of the right ICA(NASCET 80%) and a low-grade stenosis of the left ICAon CTA and DUS as indicated by red arrows on the coronarCTA maximum-intensity projections in Fig. 17. For patient 2,DCE-MRI revealed the unilateral high-grade stenosis of theright ICA, as shown in Fig. 17. Both patients were scheduled foran endartectomy. Axial slices at three locations pre- and postthe respective stenosis, “pre,” “ste,” and “post,” were acquiredwith TE/TR 3.96/3.25(3.96 ms/6.51 for Patient 2) ms, temporalresolution 26.04 ms. flip angle 20 , FOV ,slice thickness 4 mm, imaging matrix 224 224, and in-plane

Fig. 16. Image results and physiological parameters for a representative vol-unteer for different acceleration factors , , , , and

. (a) Image results from the through-plane encoding. (b) Volumetricflow and (c) peak velocity .

Fig. 17. Diagnostic data from two patients with severe ICA stenosis. Patient1: (a) coronar MIP highlighting the plaque around the right carotid bifurcation.Patient 2: (b) DCE-MRI indicating the severe stenosis of the right ICA. Loca-tions of the chosen PCI slices are highlighted in green.

resolution of 0.89 , to evaluate the flow and velocityprofiles. The venc was chosen as (150 cm/s) andthe number of temporal phases as .1) Results: The under sampled data was reconstructed with

MuFloCoS and evaluated with a special focus on the pathologydetection by analyzing the peak velocities in the CCA and ICAat three positions in the CCA, in close proximity to the stenosisand post-stenotic in the ICA. For patient 1, the correspondingpeak velocity plots are illustrated in Fig. 18(a). A clear differ-ence is visible between the velocities in the pre-stenotic CCAand after the stenosis in the ICA. Especially on the right side,where the high-degree stenosis has been diagnosed, a sharp in-crease in peak-velocity is observable, which correlates well withthe reduced lumen at this position. The difference between thehigh-grade stenosis on the right side to the low-grade stenosis


Fig. 18. Peak velocity profiles for patient 1 and 2 from the PCI scan, acceleratedwith factor and reconstructed using MuFloCoS.

on the left side is well visible in the difference of the peak ve-locities. The results for patient 2 in Fig. 18(b) indicate well theunilateral stenosis diagnosed with DCE-MRI. The stenotic pro-file in the ICA is corrupted by aliasing due to the high velocites.No correction was applied to this.

V. DISCUSSION

Iterative reconstruction with the proposedMuFloCoSmethodwas successful for all phantom and in vivo experiments and pro-duced comparable consistent results. The high noise level bothin the background and the tissue of the image reconstructed withiterative SENSE was significantly reduced with the iterativemethods. While they all managed to recover the image struc-tures, the quality of the images as well as the accuracy of thephysiological parameters are measurable better for MuFloCoS.The image quality allowed in summary good visualization of theanatomy and corresponds to that of the fully sampled referenceimage.

A. Quantitative Evaluation

In the phantom experiment, all inter- and intra-slice devia-tions were under 4%, which illustrates the capability of Mu-FloCoS to preserve the flow values over the entire dataset. Con-cerning the in vivo data, the evaluation of the flow parametersshowed good preservation of flow parameters and had a sig-nificantly reduced deviation compared to the further methodsin both the phantom experiment and the volunteer study (seeTable I, II). With MuFloCoS, the volumetric flow , the

mean velocity and the peak velocity over time(Fig. 10) in the CCA were in good agreement with the corre-sponding values obtained with the fully sampled directly re-constructed data sets. A comparison to further state of the artmethods showed improvements in NRMSE, SSIM, and particu-larly in the peak velocity, which is an important diagnostic valuein the classification of stenosis. This result indicates that bothtemporal and spatial resolution are well preserved. Volumetricflow and mean velocity corresponded well to values for the vol-umetric flow rate as previously reported by Long et al. [38], andfor the mean velocity obtained by Ringgaard et al. [39]. The pa-tient cases and the respective peak velocity profiles shown in 18confirmed the CT and DCE-MRA findings and even alloweddifferentiating between high- and low-grade stenosis.The results compared to the PCA regularization as proposed

by Kim et al. [23] revealed a slight advantage for this regular-ization for the evaluation of the volumetric flow, which shows,that a combination of the global PCA method with the proposedtemporal masked and weighted MuFloCoS regularization is ofhuge interest. The method as shown in [23] was a two stepmethod, starting with Fourier Transform and than switching toPCA, which was not done in this analysis to make it compa-rable to the further methods. Including the proposed MuFloCoSalgorithm for the first step instead of the temporal Fourier trans-form, and adding the temporal PCA at a later stage could fur-ther add value. The improvement compared to MDFCS showsthe positive influence of combining static and dynamic anatom-ical images and of the Gaussian weighting. The comparisonagainst state-of-the art for CSmethods applied to carotid PCI in-dicated the improvements achieved with MuFloCoS regardingboth image- and physiology-based measures, particularly in forhigher acceleration factors.For carotid PCI accelerated by factor 9.0, the comparison

with the method by Tao et al. revealed significant advantages forthe novel MuFloCoS method, particularly regarding the phys-iological values. The proposed method by Tao et al. showed,however, good results for the SSIM and the CNR. Regardingthe desired hemodynamic information obtained from PCI ac-quisitions, the accuracy of the physiological values is of majorinterest. Further investigation could include a combination ofboth regularization terms in the objective function.

B. Evaluation of I-VT, shared coil profiles and TMW

The benefits of the masked temporal regularization and theI-VT pattern to exploit the inherent data correlation both in ve-locity encoding and temporal direction become evident with theresults of their respective influence. The shared coil profile cal-culation led to a very slight change in the results for NRMSEfrom 5.53 to 5.79 while it contributed to the final accelerationin substantial amount by allowing real undersampling by factorfour in the central -space. Its influence, illustrated in Fig. 14(b),was barely visible in the difference image where the main de-viations occur outside the object. The influence of the inter-leaved and shifted pattern was substantially higher, as shownin Table IV and in image 14(c). The norm plot in Fig. 15on the left illustrates that the higher similarity at the beginningdue to the same sampling, expressed through low differencevalues, increased with iterations and converged at higher values


for

where such thatelse.

(23)

for

where such that

for

where such thatotherwise.

(24)

than the I-VT pattern variant. This can be explained by the prop-erty of the I-VT pattern to allow a better minimization for thewhole objective function (15) including data fidelity term andregularization term. The NRMSE decreased from 5.90 to 5.53compared to the shifted and interleaved pattern.Experiment III illustrated furthermore that the proposed

TMW- regularization offers a very stable regularizationoption. Different choices of the weighting factor ledto comparable results both in terms of visual impression andin the physiological parameter estimation. Fig. 13(a) and (b)illustrated even a convergence for both the data fidelity term, aswell as for the TMW- term independent of . Forin the range of [0.0015, 0.0125], the term converged to thesame value after 20 iterations. No critical bound was obtained,the changes with growing are smooth in Fig. 13(a). Allreconstructions for the 18 volunteers were performed with afixed and produced comparable results, theparameter can therefore be assumed to be robust in a widerange and stable over different data sets.Experiment II illustrated furthermore the acceleration capac-

ities of MuFloCoS. Even for high factors, such as 15, resultingin using only around 6.7% of the data, good physiological re-sults could be achieved. This acceleration is significantly higherthan feasible with currently clinically used methods, which canachieve an acceleration of 2–4 (25%–50% of the data) in thisapplication. The performed study had the limitation that the ref-erence values are calculated based on the fully sampled PCMRIscans. This provided a reliable reference for the physiologicalvalues. Further prospective under sampled studies need to bedone in the future. A further extension could be direct compar-ison with a different modality such as a flow meter or DUS.

VI. CONCLUSION

Acceleration factors of in volunteers have been suc-cessfully applied to PCI, leading to a significant speed up of theacquisition. The acquisition time for good temporal and spatialresolution in the carotid artery region can be significantly re-duced. For the concrete example of the patient data acquisitionusing three three-directional 2-D slices pre- and post-stenosis inthe CCA and ICA, the acquisition time could be reduced from8 min 24 s to 56 s. This significant reduction can be an impor-tant step in the clinical acceptance of this technique. This savedtime could also be invested in a higher spatial and/or temporalresolution allowing to visualize especially pathological hemo-dynamic situations with better accuracy. The proposed method

TABLE V

is not limited to the introduced masked and weighted temporalregularization, but is easily expendable to different regularizerssuch as TV, Wavelet or more PCI specific constraints as diver-gence free flow fields or phase constraint. Only intrinsic proper-ties of the 4-D PCI acquisition were exploited which makes theproposed method ideally suited to be applied to different bodyregions imaged with PCI.

APPENDIXANNEX A: PATTERN CALCULATION

See (23) and (24) at top of the page.

ANNEX C: SYMBOL TABLE

See Table V.

ACKNOWLEDGMENT

The authors gratefully acknowledge funding of the ErlangenGraduate School in Advanced Optical Technologies (SAOT) bythe German Research Foundation (DFG) in the framework ofthe German excellence initiative. The authors further thank the


authors of the paper [23] for providing the code for the temporalFT and PCA regularization.

REFERENCES

[1] , R. G. González, J. A. Hirsch, M. H. Lev, P. W. Schaefer, and L. H.Schwamm, Eds., Acute Ischemic Stroke, Imaging and Intervention, 2nded. New York: Springer, Oct. 2011.

[2] A. F. Stalder, M. F. Russe, A. Frydrychowicz, J. Bock, J. Hennig,and M. Markl, “Quantitative 2-D and 3-D phase contrast MRI: Op-timized analysis of blood flow and vessel wall parameters,” Magn.Reson. Med., vol. 60, no. 5, pp. 1218–1231, Nov. 2008.

[3] N. H. J. Pijls, B. de Bruyne, K. Peels, P. H. van der Voort, H. J. R. M.Bonnier, J. Bartunek, and J. J. Koolen, “Measurement of fractional flowreserve to assess the functional severity of coronary-artery stenoses,”N. Eng. J. Med., vol. 334, no. 26, pp. 1703–1708, Jun. 1996.

[4] A. Harloff, T. Zech, F. Wegent, C. Strecker, C. Weiller, and M. Markl,“Comparison of blood flow velocity quantification by 4-D Flow MRimaging with ultrasound at the carotid bifurcation,” AJNR Am. J. Neu-roradiol., vol. 34, no. 7, pp. 1407–1413, Feb. 2014.

[5] I. Marshall, S. Zhao, P. Papathanasopoulou, P. Hoskins, and Y. Xu,“MRI and CFD studies of pulsatile flow in healthy and stenosed carotidbifurcation models,” J. Biomech., vol. 37, no. 5, pp. 679–687, May2004.

[6] J. Endres, T. Redel, M. Kowarschik, J. Hutter, J. Hornegger, and A.Dörfler, “Virtual angiography using CFD simulations based on patient-specific parameter optimization,” in Proc. 9th IEEE Int. Symp. Biomed.Imag., Barcelona, Spain, 2012, pp. 1200–1203.

[7] M. Markl, M. T. Draney, M. D. Hope, J. M. Levin, F. P. Chan, M. T.Alley, N. J. Pelc, and R. J. Herfkens, “Time-resolved 3-dimensionalvelocity mapping in the thoracic aorta: Visualization of 3-directionalblood flow patterns in healthy volunteers and patients,” J. Comput. As-sist. Tomogr., vol. 28, no. 4, pp. 459–468, Jul./Aug. 2004.

[8] F. Santini, S. G. Wetzel, J. Bock, M. Markl, and K. Scheffler, “Time-resolved three-dimensional (3-D) phase-contrast (PC) balanced steady-state free precession (bSSFP),”Magn. Reson. Med., vol. 62, no. 4, pp.966–974, Oct. 2009.

[9] R. Hausmann, J. S. Lewin, and G. Laub, “Phase-contrast MR angiog-raphy with reduced acquisition time: New concepts in sequence de-sign,” J. Magn. Reson. Imag., vol. 1, no. 4, pp. 415–422, Jul./Aug.1991.

[10] M. A. Griswold, P. N. Jakob, R. M. Heidemann, M. Nittka, V. Jellus,J. Wang, B. Kiefer, and A. Haase, “Generalized autocalibrating par-tially parallel acquisitions (GRAPPA),” Magn. Reson. Med., vol. 47,pp. 1202–1210, Jun. 2002.

[11] K. P. Pruessmann, M. Weiger, M. B. Scheidegger, and P. Boesiger,“SENSE: Sensitivity encoding for fast MRI,”Magn. Reson. Med., vol.42, no. 5, pp. 952–962, Nov. 1999.

[12] Z. Stankovic, J. Fink, M. Russe, M. Markl, and B. A. Jung, “On the useof K-T accelerated 4-D flow MRI in the portal system,” in Proc. 21stAnnu. Meet. ISMRM, Salt Lake City, UT, 2013, p. 4451.

[13] D. Giese, T. Schaeffter, and S. Kozerke, “Highly undersampledphase-contrast flow measurements using compartmentbased k-t prin-cipal component analysis,” Magn. Reson. Med., vol. 69, no. 2, pp.434–443, Apr. 2012.

[14] J. A. Steeden, D. Atkinson, M. S. Hansen, A. M. Taylor, and V. Muthu-rangu, “Rapid flow assessment of congenital heart disease with high-spatiotemporal-resolution gated spiral phase-contrast MR imaging,”Radiology, vol. 260, no. 1, pp. 79–87, Jul. 2011.

[15] M. Lustig and J. M. Pauly, “Spirit: Iterative self-consistent parallelimaging reconstruction from arbitrary k-space,” Magn. Reson. Med.,vol. 64, no. 2, pp. 457–471, Aug. 2010.

[16] E. J. Candès and M. B. Wakin, “An introduction to compressive sam-pling,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21–30, Mar.2008.

[17] J. I. Jackson, C. H. Meyer, D. G. Nishimura, and A. Macovski, “Selec-tion of a convolution function for Fourier inversion using gridding,”IEEE Trans. Med. Imag., vol. 10, no. 3, pp. 473–478, Sep. 1991.

[18] M. Doneva, H. Eggers, and P. Börnert, “CS-SENSE or denoised sense:The influence of irregular sampling in l1 regularized sense reconstruc-tion,” in Proc. 20th Annu. Meet. ISMRM, Melbourne, Australia, May2012, p. 2240.

[19] J. Yellott, “Spectral consequences of photoreceptor sampling in therhesus retina,” Science, vol. 221, no. 4608, pp. 382–385, Jul. 1983.

[20] Y. Kwak, S. Nam, K. V. Kissinger, B. Goddu, L. Goepfert, W. Man-ning, V. Tarokh, and R. Nezafat, “Accelerated phase contrast imagingusing compressed sensing with complex difference sparsity,” J. Car-diovasc. Magn. Reson., vol. 14, p. W24, Feb. 2012.

[21] J. Busch, D. Giese, L. Wissmann, and S. Kozerke, “Divergence-freereconstruction for accelerated 3-D phasecontrast flow measurements,”J. Cardiovasc. Magn. Reson., vol. 14, p. W38, Feb. 2012.

[22] F. Zhao, D. Noll, J. Nielsen, and F. Fessler, “Separate magnitudeand phase regularization via compressed sensing,” IEEE Trans. Med.Imag., vol. 31, no. 9, pp. 1713–1723, Sep. 2012.

[23] D. Kim, H. A. Dyvorne, R. Otazo, L. Feng, D. K. Sodickson, andV. S. Lee, “Accelerated phase-contrast cine MRI using k-t SPARSE-SENSE,” Magn. Reson. Med., vol. 67, pp. 1054–1064, Apr. 2012.

[24] J. V. Velikina, K. M. Johnson, W. F. Block, and A. A. Samsonov, “De-sign of temporally constrained compressed sensing methods for accel-erated dynamic MRI,” in Proc. Joint Annu. Meet. ISMRM—ESMRMB,Stockholm, Sweden, May 2010, p. 4865.

[25] J. P. Hulet, A. Greiser, J. K. Mendes, C. McGann, G. Treiman, and D.L. Parker, “Highly accelerated cardiac cine phasecontrast MRI usingan undersampled radial acquisition and temporally constrained recon-struction,” J. Magn. Reson. Imag., 2013.

[26] A. Joseph, K. Merboldt, D. Voit, S. Zhang, M. Uecker, J. Lotz, and J.Frahm, “Real-time phase-contrast MRI of cardiovascular blood flowusing undersampled radial fast low-angle shot and nonlinear inversereconstruction,” NMR Biomed., vol. 25, pp. 917–924, Jul. 2012.

[27] C. Santelli, T. Schaeffter, and S. Kozerke, “Radial k-t spirit: Autocal-ibrated parallel imaging for generalized phase-contrast MRI,” Magn.Reson. Med., Nov. 2013.

[28] Y. Tao, G. Rilling, M. Davies, and I. Marshall, “Carotid blood flowmeasurement accelerated by compressed sensing: Validation in healthyvolunteers,” Magn. Reson. Imag., vol. 31, no. 9, pp. 1485–1491, Nov.2013.

[29] J. Hutter, P. Schmitt, G. Aandal, A. Greiser, C. Forman, R. Grimm,J. Hornegger, and A. Maier, “Low-rank and sparse matrix decomposi-tion for compressed sensing reconstruction of magnetic resonance 4-Dphase contrast blood flow imaging (LoSDeCoS 4-D-PCI),” in 16th In-ternational Conference for Medical Image Computing and Computer-Assisted Intervention (MICCAI 2013), K. Mori, I. Sakuma, Y. Sato, C.Barillot, and N. Navab, Eds. New York: Springer, 2013, vol. 8149,Lecture Notes in Computer Science, pp. 558–565.

[30] J. Hutter, A. Greiser, R. Grimm, C. Forman, J. Hornegger, andP. Schmitt, “Multi-dimensional flow-adapted compressed sensing(MDFCS) for time-resolved velocity-encoded phase contrast MRA,”in Proc. 10th IEEE Int. Symp. Biomed. Imag., San Francisco, CA,2013, pp. 13–16.

[31] N. J. Pelc, M. A. Bernstein, A. Shimakawa, and G. H. Glover, “En-coding strategies for three-direction phase-contrast MR imaging offlow,” J. Magn. Reson. Imag., vol. 1, no. 4, pp. 405–413, Jul./Aug.1991.

[32] P. Chai and R. Mohiaddin, “How we perform cardiovascular magneticresonance flow assessment using phase-contrast velocity mapping,” J.Cardiovasc. Magn. Reson., vol. 7, no. 4, pp. 705–716, Jan./Dec. 2005.

[33] J. Hutter, R. Grimm, C. Forman, J. Hornegger, and P. Schmitt, “In-verse root sampling pattern for iterative reconstruction in non-CE MRangiography,” in Proc. Annu. Conf. ESMRMB, Leipzig, Germany, Oct.6–8, 2011.

[34] P. B. Roemer, W. A. Edelstein, C. E. Hayes, S. P. Souza, and O. M.Mueller, “The NMR phased array,”Magn. Reson. Med., vol. 16, no. 2,pp. 192–225, Nov. 1990.

[35] F. Nocedal, “Updating quasi-newton matrices with limited storage,”Math. Computat., vol. 35, no. 151, pp. 773–782, Jul. 1980.

[36] L. Feng, R. Otazo, H. Jung, J. Jensen, J. Ye, D. Sodickson, and D.Kim, “Accelerated cardiac t2 mapping using breathhold multiecho fastspin-echo pulse sequence with k-t focuss,”Magn. Reson. Med., vol. 65,no. 6, pp. 1661–1669, 2011.

[37] Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality as-sessment: From error visibility to structural similarity,” IEEE Trans.Image Process., vol. 13, no. 4, pp. 600–612, Apr. 2004.

[38] Q. Long, X. Y. Xu, K. V. Ramnarine, and P. Hoskins, “Numerical in-vestigation of physiologically realistic pulsatile flow through arterialstenosis,” J. Biomech., vol. 34, no. 10, pp. 1229–1242, Oct. 2001.

[39] S. Ringgaard, S. Oyre, and E. Pedersen, “Arterial MR imaging phase-contrast flow measurement: Improvements with varying velocity sen-sitivity during cardiac cycle,” Radiology, vol. 232, no. 1, pp. 289–294,Jul. 2004.

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400 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 34, NO. 2 ... · the dynamic character of 4-D PCI is exploited by Kim et al. [23], proposing a kt SPARSE SENSE approach reconstructing