spurs using graph cuts

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 34, NO. 2, FEBRUARY 2015 531

Simultaneous Phase Unwrapping and Removalof Chemical Shift (SPURS) Using Graph Cuts:Application in Quantitative Susceptibility MappingJianwu Dong, Tian Liu*, Feng Chen*, Dong Zhou, Alexey Dimov, Ashish Raj, Qiang Cheng,

Pascal Spincemaille, and Yi Wang

Abstract—Quantitative susceptibility mapping (QSM) is a mag-netic resonance imaging technique that reveals tissue magneticsusceptibility. It relies on having a high quality field map, typi-cally acquired with a relatively long echo spacing and long finalTE. Applications of QSM outside the brain require the removalof fat contributions to the total signal phase. However, currentwater/fat separation methods applied on typical data acquired forQSM suffer from three issues: inadequacy when using large echospacing, over-smoothing of the field maps and high computationalcost. In this paper, the general phase wrap and chemical shiftproblem is formulated using a single species fitting and is solvedusing graph cuts with conditional jump moves. This method isreferred as simultaneous phase unwrapping and removal of chem-ical shift (SPURS). The result from SPURS is then used as theinitial guess for a voxel-wise iterative decomposition of water andfat with echo asymmetric and least-squares estimation (IDEAL).The estimated 3-D field maps are used to compute QSM in bodyregions outside of the brain, such as the liver. Experimental resultsshow substantial improvements in field map estimation, water/fatseparation and reconstructed QSM compared to two existingwater/fat separation methods on 1.5T and 3T magnetic resonancehuman data with long echo spacing and rapid field map variation.

Index Terms—Field map estimation, magnetic resonanceimaging (MRI), quantitative susceptibility mapping, water/fatseparation.

Manuscript received August 21, 2014; revised September 25, 2014; acceptedSeptember 29, 2014. Date of publication October 08, 2014; date of current ver-sion January 30, 2015. This work was supported in part by the National NaturalScience Foundation of China under Grant 61271388 and Grant 61327902, inpart by the Beijing Natural Science Foundation under Grant 4122040, in part bythe Research Project of TsinghuaUniversity under Grant 2012Z01011, in part bythe Specialized Research Fund for the Doctoral Program of Higher Education,and in part by the U.S. National Institute of Health under Grant R43EB015293,R01EB013443, and Grant R01CA178007. Asterisk indicates corresponding au-thor.J. Dong and Q. Cheng are with the Department of Automation, Tsinghua Uni-

versity, Beijing 100084, China.*T. Liu is with Medimagemetric LLC, New York, NY 10044 USA (e-mail:

[email protected]).*F. Chen is with the Department of Automation, Tsinghua University, Beijing

100084, China (e-mail: [email protected]).D. Zhou, A. Raj, and P. Spincemaille are with the Department of Radiology,

Weill Cornell Medical College, New York, NY 10021 USA.A. Dimov is with the Department of Radiology, Weill Cornell Medical Col-

lege, New York, NY 10021 USA, and also with the Department of BiomedicalEngineering, Cornell University, Ithaca, NY 14853 USA.Y. Wang is with the Department of Radiology, Weill Cornell Medical Col-

lege, New York, NY 10021 USA, and also with the Department of BiomedicalEngineering, Cornell University, Ithaca, NY 14853 USA, and also with the De-partment of Biomedical Engineering, Kyung Hee University, Seoul 130-701,Korea.Digital Object Identifier 10.1109/TMI.2014.2361764

I. INTRODUCTION

Q UANTITATIVE susceptibility mapping (QSM) is amagnetic resonance imaging technique that revealstissue magnetic susceptibility by deconvolving a 3-D

field map with a dipole kernel [1]–[13]. Because the quality ofQSM hinges upon the quality of the field map, a multi-echoreadout with a long echo spacing ( ms) and a long last TE( ms at 3T) has generally been adopted to accrue sufficientphase signal. The multi-echo signal is then fitted with a singlespecies model to estimate the field map. This model is adequatefor typical QSM applications in the brain, as the contributionof fat to the MRI signal is minimal. In order to apply QSM toother body regions that contain fat, water/fat separation with anonlinear signal model is usually needed.The problem of field map estimation in the presence of both

water and fat is commonly solved as an energy mini-mization problem, with unknowns including a spatially varyingwater component , a fat component with a chemical shiftof , a susceptibility induced field inhomogeneity , and asignal decay rate necessary when TE is long [14]–[23]

(1)

where is the signal acquired at echo time , andis the number of echoes. Here, all the unknowns are scalar

fields with the spatial index omitted. There are multiple un-knowns for each voxel: two complex-valued unknowns and

and two real unknowns and , so this is a high-dimen-sional optimization problem. One solution is a gradient descentbased method, such as -Iterative Decomposition of waterand fat with Echo Asymmetric and Least-squares estimation( -IDEAL) proposed in [14]. However, this method is sus-ceptible to finding local minima close to the initial guess insteadof identifying the global minimum [17]. Fundamentally, currentsolutions to (1) suffer from the three issues: inadequacy whenusing large echo spacing, over-smoothing of the field maps, andhigh computational cost.1) Large Echo Spacing Challenge: It has been demon-

strated that the optimal echo spacingis for minimizing noise propagation [24]. For aminimum at 3T, the longest optimal echo spacing

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

is 0.75 ms, which would require either a very high receiverbandwidth consequently leading to poor SNR, or multiplescans that increase total acquisition time. In most existing QSMprotocols, multiple TEs are sampled within the same repetitiontime (TR) using an echo spacing of around 5 ms. This long echospacing poses a challenge to field map estimation: the phaseaccrual in one echo spacing from both the chemical shift andthe susceptibility inhomogeneity field may exceed , whichis beyond the range where signal phase can unambiguouslyrepresent the magnetic field [25].2) Over-Smoothing: In current methods to enforce the con-

tinuity of the field map [20], [26], [27], a regularization termis typically added to the data fitting term to penalize abruptchanges in the field map [17], [20], [28]. This regularizationformulation gives a maximum a posteriori estimation and hasbeen successfully applied in numerous applications in medicalimaging. However, the choice of regularization parameter is stillan open question and is usually determined heuristically, andthe regularization may introduce over-smoothing on the fieldmap that causes errors in the subsequent QSM [29]. Another ap-proach is to formulate the continuity condition as a combinato-rial optimization problem. Region growing (RG) based methodshave been proposed to efficiently connect the local minima iden-tified from VARiable PROjection (VARPRO), as in [25]. How-ever, RG is a greedy algorithm that does not guarantee a globaloptimum and its result is dependent on the RG path, i.e., an errorin a single step may propagate to the whole image. Loopy be-lief propagation (LBP) algorithm with additional swap moves[27] has also been suggested for solving this combinatorial op-timization problem. However, field maps obtained with LBP areprone to having piecewise smooth segments with abrupt discon-tinuities. This may be caused by the fact that LBP is suited for atree-structure Markov random field (MRF) but not a grid-struc-ture MRF [27].3) Computational Cost: Current attempts to find the

minimum in a global sense start with candidate generation.VARPRO is widely used in various methods [16], [17], [20],[25], [27], [28], [30]. VARPRO uniformly discretizes field mapvalues in a given range, scans through all possible values andidentifies the local minima as candidates. This step is time con-suming for a 3-D dataset, especially when a fine discretizationis used for accuracy or when is also subject to optimiza-tion. The candidate solutions may also be found through agolden-section search, as implemented in Multi-Resolutionfield map estimation using Golden Section search method(MRGS) [25], but this method does not account for the termin (1) that is necessary for long echo data in QSM.We propose here to formulate the removal of the echo-spacing

induced phase discontinuities and fat chemical shifts as adiscrete problem using an approximate signal model for mul-tiple species (fat and water). We demonstrate that the discreteproblem can be efficiently and effectively solved using graphcuts with conditional jump moves. This Simultaneous PhaseUnwrapping and Removal of chemical Shift (SPURS) methodcan provide a faithful initial guess without any smoothing forIDEAL to obtain the water and fat maps [14]. In this work,we show that the SPURS method outperforms two existingmethods on several 1.5T and 3T human datasets with long echo

Fig. 1. Flowchart of the SPURS method.

spacing and rapid field map variation, and we show promisingQSM results in liver imaging.

II. METHODS

It has been shown in signal spectral analysis [31] that if asingle species model is used to fit a multiple species signal,the fitted frequency is close to the frequency of the dominantspecies. Thus if a single species fit is used for the signal ,the fitted frequency is close to if the dominant species iswater, or to if the dominant species is fat, allowingthe determination of the dominant species when there are mul-tiple species present in a single voxel [31]. Therefore, a fittedfrequency map has discontinuities due to 1) limited dynamicrange dictated by echo spacing (phase wraps) and 2) fat chem-ical shift. Characterizing the spatial continuity of the actual fieldmap by the norm of its gradient, the discontinuities can beaddressed by solving the following combinatorial optimizationproblem over integer scalar fields and a Boolean scalar fields

(2)

where denotes the gradient operator, corrects forthe field map wrap due to limited dynamic range with

, and corrects for the fat chemical shift in eachvoxel with . This joint optimization over andcan be efficiently solved using graph cuts with conditional jumpmoves.Fig. 1 illustrates the flowchart of the SPURS method in four

steps: 1) formulate phase wrap and chemical shift problemusing a single species fitting to obtain an approximate field mapestimate; 2) use graph cuts to solve the joint phase unwrappingand fat chemical shift removal problem; 3) refine the fitted fre-quency by performing a voxel-wise -IDEAL; 4) reconstructQSM using a modified version of the Morphology EnabledDipole Inversion method (MEDI) [6].

A. Formulation of Phase Wrap and Chemical Shift ProblemUsing Single Species Fitting

A vector representation of the signal is shown in Fig. 2. If, the complex signal generated by the two species

DONG et al.: SPURS USING GRAPH CUTS: APPLICATION IN QUANTITATIVE SUSCEPTIBILITY MAPPING 533

Fig. 2. Vector representation of the signal consisting of both water and fat.Geometric relationship between these vectors is used to provide an intuition ofthe error introduced by the single species fitting.

can be formulated as , where

is the magnitude of and

is the phase of .We use a single species model (frequency ) to fit the twospecies signal by minimizing the following nonlinear leastsquares fit [6]:

(3)

Equation (3) is solved using the Gauss-Newton method. Toanalyze model errors introduced by the single species fit, anapproximate analytical solution is obtained (detailed derivationin Appendix A)

(4)where

.Therefore, the true field map, , can be written as thesummation of the estimated field map, , the error term due tolimited dynamic range, the error term due to chemical shift,and the model error

(5)

where and model error is approx-imately

(6)

Note that larger echo spacing and more echoes tend to leadto a smaller approximate model error because the numerator isa summation of and the denominator is a summation ofmultiplied by echo spacing. In addition,

if or . That is, if the voxels consist of only wateror only fat, the model error of the single species fitting is zero.The true model error introduced in the single species fitting

will calculated numerically in the simulation section below.

B. Simultaneous Phase Unwrapping and Removal of ChemicalShift via Graph Cuts With Conditional Jump Moves

Equation (2) is solved in this step using a graph cuts algo-rithm. In the following, the field map is changed from Hz toradian by multiplying for notational convenience. It isassumed that the field map is continuous aftersumming , where is the effective fat chemicalshift derived by wrapping to the in-terval. Let , we have

(7)

Without loss of generality, we assume in the fol-lowing and denote

.A discrete energy function is generalized from (2)

(8)

where , subscript denotes a spatial location,represents any of the three Cartesian coordinate axes, isa weighting that is equal to the sum of the magnitude valueof the voxel and its neighbor along , and is the finitedifference operator along . The magnitude weighting termintroduced in the energy function serves as a quality map.Smaller weights are given to voxels with weaker signal inorder to enhance the robustness against phase noise [32]. Thesmaller weights also allow discontinuity in the field map wherethe corresponding magnitude has low SNR [33]. This energyfunction is defined on a 3-D first order MRF, in which everynode is locally dependent on its six neighboring nodes. Mini-mizing the above energy function is a multi-label optimizationproblem, which can be approximately solved by changing itto a sequence of binary optimization problem in which eachbinary problem is solved by graph cuts [34]–[36].We start with an initial labeling in which each voxel is as-

signed label . In each iteration, every voxel's label canjump to or remain unchanged, where is the step size andthis jump move is called an -jump move. Every jump move it-eration decreases or preserves the function energy. In this study,was set equal to 2 (called the “2-jump move”) until no furtherdecrease in energy was observed. After the energy ceased to de-crease using , the iterative process was repeated with(called the “1-jump move”). Fig. 3 shows an example that illus-trates the evolution of labels in the jump move optimization.For 2-jump move optimization, the jump lengths are uni-

formly 2 for all voxels, so it is equivalent to traditional multi-label optimization problem with integer labels. However, theconstruction of the binary energy term for 1-jump move opti-mization requires a conditional jump move strategy due to theBoolean variable , in which a voxel's jump length depends onits current label. We use superscripts to denote the number of


Fig. 3. Example illustrating the evolution of voxel labels in the jump moveoptimizations. (a) 2-jumpmove optimization and (b) 1-jumpmove optimization.Bold labels indicate voxels that perform jump moves.

iterations, and subscripts to denote the spatial location. For ex-ample, denotes the label of voxel at iteration , and

(9)

where . is the jump length that depends onthe value of . If , the possible values of are

, which can jump to , respec-tively with step size 1, and the jump length is . If

, the possible values of are ,which can jump to , respectively with step size 1, andthe jump length is . So we have

(10)

In the th iteration, the label canmake a binary decisionto either jump to or remain unchanged

. We use graph cuts to optimize the binary variables. For every pair of neighboring voxels and , we aim to

minimize their difference in the following:

(11)

which is the binary energy term for graph cuts. For every binaryenergy term , the following inequality exists:

(12)

i.e., the energy function satisfies the submodularity conditionand can be efficiently solved by standard graph cuts method[37]. The proof of the submodularity condition is provided inAppendix B.After the following binary energy function

is obtained, a weighteddirected graph with source node and sink nodeis constructed by following Kolmogorov and Zabih's paper[37]. The edge weights of are assigned accordingto the energy function. Afterwards, the original energyminimization problem is changed to a min-cut/max-flowproblem in , which has been studied extensivelyand can be solved efficiently. The equivalence between the

minimization problem of the binary energy function and themin-cut/max-flow problem was proven in [37].

C. Voxel-Wise -IDEAL Water/Fat Separation

After the approximate field map is obtained in step 2, itis taken as the initial guess for the field inhomogeneity in-IDEAL, the objective function of which is identical to (1).

This step is a fine tuning step to remove the model error in thefield due to the single species assumption in the previous stepand to obtain the water and fat maps. Note that no additionalspatial smoothing is performed, as conventionally done in-IDEAL.

D. QSM Reconstruction

After the field map is fine-tuned by -IDEAL, it is takenas the input for QSM reconstruction using a modified versionof MEDI [6] that incorporates background field removal in thesusceptibility estimation [38]–[40]

(13)

where is a regularization parameter empirically determined tobe 1000, is a data weighting, is a matrix representing thedipole convolution, and is a binary weighting term derivedfrom the anatomical image, with 30% of its voxels equal to zeroto allow changes in and the rest equal to one [41]. This en-ergy minimization has a data fidelity term that is formulated inthe complex plane to account for Gaussian noise and an normregularization term to encode the prior. In contrast to the orig-inal nonlinearMEDI, this energyminimization has an additionalLaplacian in the data fidelity term. Since the background field isa harmonic function, the Laplacian eliminates any backgroundfields. The Laplacian is implemented as a convolution with akernel equal to the Kronecker delta functionminus a sphere withradius 5 mm and unit integral. For efficiency, this convolutionis evaluated in Fourier space.

III. MRI EXPERIMENTS

A. Single Species Fitting Error Simulations

Numerical phantom simulations were performed to computethe model error (6). Phantoms with differentpercentages of water and fat were generated. The fat fraction

was varied between 0 and 1 with a uniformspacing 0.01. was varied between 2 ms and 10 ms witha uniform spacing 0.1 ms. The true model error was computedas , where was the true fieldmap and was the result of SPURS. Itwas assumed in the simulation that the main magnetic field was3T, and the number of echoes was four, whichwas the same as used in our in vivo scans.

B. Numerical Phantom Simulations

A numerical phantom with a mixture of water and fat wasconstructed to test the proposed method. The matrix size was128 128 64. The single peak fat model with fat chemical


shift ppm [42] was used. We assumed that the fat sus-ceptibility was 0.6 ppm with respect to water. The numericalphantom is a water phantom where most part is made up ofwater. In the center, there were six cylinders and two cuboidsof different size, where the cylinders and cuboids had differentfat fractions. The fat fractions in the cylinders and cuboids were23%, 35%, 47%, 60%, 73%, 86%, 94%, and 100%, respec-tively. The main magnetic field was 3T and a linear gradientbackground field with range Hz was added to themain magnetic field. We added complex Gaussian noise to pro-duce a noisy signal with signal-to-noise ratio (SNR) of 30 dB.The SNR was defined as the ratio of the power of signal tothe power of noise. The scan parameters were assumed to be

ms, ms, andmm .

C. In Vivo Validation

Among all the cases provided by the ISMRMFat-Water Sepa-ration workshop (http://ismrm.org/workshops/FatWater12/data.htm),we identifiedone3-Ddataset thatwasacquiredusingCarte-sian sampling and had a final TE larger than 15 ms. The imagingparameters were as follows: 1.5T, six echoes, ms,

ms, mm .Aunipolar readoutwas used in this experiment to avoid phase variations betweeneven and odd echoes. To establish a benchmark, we comparedthe Hernando et al.method [17] with the proposed method. TheHernando et al.method combined aVARPROformulationwith agraph cut solver to solve (1), so it was denoted as VARPRO-GC[16], [17]. Both methods gave virtually identical water and fatmaps when all six echoes are used. Thus, we took the six echoresults fromVARPRO-GC as the benchmark. To test the methodperformancewith longecho spacing,wediscardedall odd echoessuch that the effective ms. The field map error wasdefined as the norm of the absolute difference map dividedby the norm of the benchmark field map.

D. MR Acquisition

With Institutional Review Board approval and informed con-sent, a total of eleven healthy volunteers were recruited for liverMRI. Six of them were scanned on a 1.5T MRI system (GE Ex-cite HD, Milwaukee, WI, USA), two on a 3T MRI system (GEExcite HD, Milwaukee, WI, USA), and three at both 1.5T and3T. An 8-channel cardiac coil and a 4-echo spiral sequence wereused in 1.5T scans. The scan parameters were: 48 spiral leaves,

kHz, ms,ms, ms, mm . The scanswere finished in the time of a single breath hold (about 45 s).We used an 8-channel torso coil and a 4-echo 3-D spoiled

gradient echo sequence in 3T scans. The scan parameters were:kHz, ms,

ms, ms, mm . The scanswere finished in the time of a single breath hold (about 40 s).The matrix size for both 1.5T and 3T human data was 256 25626.

E. Image Processing and Analysis

The proposed method was compared with VARPRO-GC.Additionally, the Lu et al. Multi-Resolution field map esti-

mation using Golden Section search method (MRGS) wasalso compared [25]. The MRGS method identifies candidatesolutions from the VARPRO formulation and connects thecandidate solutions to form a continuous field map using regiongrowing. For VARPRO-GC andMRGS, the original codes fromthe ISMRM Fat-Water toolbox (http://www.ismrm.org/work-shops/FatWater12/) were used. Note that both VARPRO-GCand MRGS were designed for 2-D MR data, so VARPRO-GCand MRGS were run on every slice of the 3-D data. Weimplemented SPURS in both 2-D and 3-D. For numericalsimulations and in vivo validation, 2-D SPURS was comparedwith VARPRO-GC and MRGS to allow a fair comparison.For the acquired liver MR images, both 2-D SPURS and3-D SPURS were compared to the literature methods. Thesame single fat peak model was used for all methods. Thefield map range of VARPRO-GC was set toHz and all other parameters were set to the default values inthe code. The final in vivo field maps obtained by each of themethods were further processed using the modified versionof MEDI [6] to generate quantitative susceptibility maps.The tissue masks were created in an automated process usingmagnitude signal thresh-holding: voxels whose magnitudevalues were less than 5% of the maximum magnitude weremasked out. All the experiments were run on a personal com-puter with i7-2600 CPU and 8 GB memory. Running timefor water/fat separation was recorded. The performance ofthe water/fat separation for 14 abdomen cases was evaluatedby two experienced MR researchers on a 3-point scale, with

no water/fat swap pockets of water/fat swap, andsubstantial water/fat swap. A performance score was

obtained by taking the mean of the grades assigned to eachof the axial slices for each method. Quantitative analysis wasperformed on the reconstructed QSM images from the 14liver cases. Regions of interest (ROIs) were chosen by placinga circle of radius 10 mm in the liver, a circle of radius 7.5mm in the back muscle, and a circle of radius 3 mm in thesubcutaneous fat regions in the central slice of the 3-D volume.Relative susceptibility values with respect to the back musclewere recorded for each of the subjects. Inter-subject mean andstandard deviation were calculated.

IV. RESULTS

The error due to single species fit with different fat fractionsand different TEs is shown in Fig. 4. It is noted that the modelerror peaked when there was an equal amount of water and fatin a voxel and reduced to zero when only a single species waspresent. The error is not a monotonic function of echo spacing,but the peak error across fat fraction decreases with increasingecho spacing, as can be seen from (6). For example, when theecho spacing is larger than 4.4 ms, all model errors are smallerthan 14.97 Hz. The average size of the error is 2.67 Hz, whichcorresponds to 0.02 ppm at 3T or 0.04 ppm at 1.5T, and it canbe further refined by running voxel-wise -IDEAL.Results of numerical validation are shown in Fig. 5 and

Table I. The field map errors shown in Table I are defined asthe ratio of norm of the difference map and the norm ofthe true map. As shown in the difference maps, 2-D SPURS


Fig. 4. Error of using single species model to fit the signal with different fatfractions and different TEs. If the voxel consists of only water or only fat, thefitting error is zero. Average size of fitting error for different parameter is 2.67HZ.

Fig. 5. Comparison of 2-D SPURS with VARPRO-GC and MRGS method ona numerical phantom with different fat fraction. First column is the true values,the second column is the difference maps of 2-D SPURS and the true value.Third column is the difference maps of VARPRO-GC and the true value. Fourthcolumn is the difference maps of MRGS and the true value. Arrows point outerrors in the estimation.

gives accurate field map estimation and water/fat separationresults across various fat fractions. The field maps of bothVARPRO-GC and MRGS have moderate error at the boundaryof cylinders and cuboids. The MRGS result still suffers fromwraps in which the field is rapidly varying. For water/fat sepa-ration, VARPRO-GC and MRGS also have moderate error atthe boundary of cylinders and cuboids.Results of the in vivo validation are shown in Fig. 6 and

Table I. As the figure shows, 2-D SPURS on the 6-echo data pro-duces a result that is virtually identical to the benchmark. Whenonly 3-echo data was used, all three methods showed notice-able differences with respect to the benchmark, but 2-D SPURSgave the lowest error compared to VARPRO-GC and MRGS.The results of VARPRO-GC and MRGS have artifactual dis-continuities in the estimated field map (arrows), which may bethe cause of the large errors measured. For VARPRO-GC andMRGS, pockets of water/fat swaps are observed near tissue-airinterfaces.

TABLE ICOMPARISON OF DIFFERENT METHODS

Fig. 6. Comparison of 2-D SPURS with VARPRO-GC and MRGS in the invivo validation. First column is the benchmark results. Second to fifth columnsare the diffrence map between various reconstructions and the benchmarks.Second column is from 2-D SPURS in 6-echo data. Third column is from2-D SPURS in 3-echo data. Fourth and fifth column are from the results ofVARPRO-GC and MRGS, respectively. For VARPRO-GC and MRGS, thereare discontinuities in field map and water fat swaps in water/fat map, whichare indicated by white arrows.

Fig. 7 shows the comparison results on a 3T human data.Both VARPRO-GC and MRGS have water/fat swaps in subcu-taneous fat regions. The field maps generated by VARPRO-GCand MRGS had large artifactual discontinuities within tissue(white arrows). These resulted in discontinuities and artifactsin the QSM results (indicated by white arrow). The field mapby 2-D SPURS had a small discontinuity (indicated by thewhite arrow). This was no longer present in 3-D SPURS,which takes the field map continuity along the slice selec-tion direction into account. Since SPURS does not performany additional smoothing on the field map, the tissue de-tails are better delineated in the reconstructed QSM result.The susceptibility values of the liver were ppm

and the susceptibility values ofthe fat region were ppm.The average score and running time of the three methods

in 14 abdomen datasets are summarized in Table I. The 3-DSPURS had the best water/fat separation quality on average andhad the lowest standard deviations. 3-D SPURS outperformed2-D SPURS in water/fat separation. 2-D SPURS performedbetter than VARPRO-GC and MRGS. The running time of 2-DSPURS was significantly faster than that of VARPRO-GC.


Fig. 7. Comparison of SPURS with VARPRO-GC and MRGS method. Firstcolumn is the results of 3-D SPURS. Second column is the results of 2-DSPURS. Third column is the results of VARPRO-GC. Fourth column is theresults of MRGS. From top to bottom are: field map, water map, fat map, andQSM. For VARPRO-GC and MRGS, there are discontinuities in field map,water fat swaps in water/fat map, and artifacts in QSM, which are indicated bywhite arrows.

V. DISCUSSION

In this paper, a 3-D method for Simultaneous Phase Unwrap-ping and Removal of chemical Shift (SPURS) is described toestimate the susceptibility inhomogeneity generated field mapfrom tissues in which fat is present. The solver is implementedusing graph cuts with conditional jump moves. Experimentalresults on several datasets with large field inhomogeneity andlarge echo spacing acquired at both 1.5T and 3T show that theSPURS method outperformed two existing methods.In most current QSM protocols, multiple TEs are sampled in

one repetition time (TR)with a long last echo to order to enhancethe SNR on the estimated field map. Due to receiver bandwidthand other hardware limits, the echo spacing in such acquisitionsis about 5 ms. This is much longer than the optimal TE spacingpreferred by most existing water/fat separation methods. In thispaper, the field map is first estimated using a single species fit-ting, and long echo spacing actually reduces the model bias inthis simplified model. Compared with multi-species fitting thatinvolves solving for six real unknown variables, single speciesfitting is more robust because it only has a single unknown beingthe off-resonance frequency, and is not an unknown subjectto optimization [6]. With the single species fitting, the water/fatambiguity is reduced to a problem parallel to phase unwrapping,which has been carefully studied in literature.The spatial continuity of the field map is enforced using

graph cuts algorithm with conditional jump moves to judi-ciously connect the candidate solutions. Graph cuts algorithm iswell-suited for combinatorial optimization problems satisfyingthe submodular condition, such as the problem formulated in(8). An additional fine-tuning of the field map is performedusing a voxel-wise -IDEAL algorithm without spatial reg-ularization. The sole purpose of this step is to correct themodel error introduced in the single species fitting and toobtain water and fat maps. Because the expected distancebetween the initial guess and the truth is the model error, and

the model error is small from the theoretical and numericalanalysis, the -IDEAL step converged to the global minimumin most cases. It should be noted that no additional spatialsmoothing is performed, so SPURS does not have the problemof over-smoothing that VARPRO-GC has. In some cases, the-IDEAL fine tuning step may be optional for QSM applica-

tions, as the employed long TE in QSM applications effectivelyreduces the model error introduced in the single species fitting.TheproposedSPURSmethod is also faster thanVARPRO-GC

andMRGS, partly because the candidate generation step is moreefficient. There are some other efficient water/fat separationmethods that save time in the candidate connection stage, suchas proposed in [20], [28]. However, wraps often remain in theestimated field map in these methods, as field map continuity isnot strictly enforced. In comparison, SPURS results in a spatiallycontinuous field map without phase wraps, which is importantfor optimal QSM reconstruction.Because the magnetic dipole kernel is a long-range field in

nature, discontinuities in one location will propagate to other lo-cations and other slices. QSM images from VARPRO-GC andMRGS still had discontinuous artifacts (white arrows in Fig. 7)resulting from artifactual discontinuities in their estimated fieldmaps. The best QSM results were obtained when using 3-DSPURS for field map estimation. Beyond the reasons mentionedabove, 3-D SPURS further enforces continuity along the thirddimension. Streaking artifacts that were often seen in QSM re-sults were well suppressed due to the spatial regularization em-ployed in MEDI, and the Laplacian also suppresses streakingfrom remnant background field.The following factors need to be considered when choosing

practical TEs to lead to optimal results. 1) If water and fat needto be separated, the minimum number of echoes for SPURS is3, as -IDEAL is needed. The choice of TE should not makethe absolute value of (effective phase of fat chemical shift)too small. In a theoretical case, where the angle between waterand fat remains exactly the same for all echoes, andwater and fat cannot be separated. 2) When focusing on thefield map only, acquisition with a long last TE is preferred. Thisis evident by analyzing the model error in SPURS. The modelerror is roughly inversely proportional to thenumber of echoes multiplied by echo spacing . A long lastTE also allows adequate phase accrual to improve the SNR onthe field map. The upper limit for the last TE should be deter-mined by the of the tissue being imaged in order to avoid fit-ting pure noise. Considering the preference for a long last echoand a realistic of tissue, current QSM protocols employinga last echo ranging from 20–50 ms appear to offer a balance be-tween the two competing factors.In this study, we demonstrated that the field map estimated

using SPURS is useful for liver QSM reconstruction. The liveris a major iron storage organ in the human body. Excessive irondeposition in the liver may indicate pathologies, such as thoserelated to thalassemia. Although mapping has been used toestimate iron concentration, also includes contribution from, making it less specific to the magnetic field inhomogeneity

caused by iron. In addition, QSM provides an opportunity tostudy the oxygen level in the vessels, which may be useful toinvestigate the oxygen extraction process in the liver. The use


of susceptibility provides a more direct measure of fundamentalphysiological tissue parameters [43] that are related to diseasediagnosis, monitoring, and treatment. Conceptually, the SPURSmethod can be easily applied to imaging other organs with adi-pose tissue, such as joint imaging, breast imaging [44], or wholebody mouse imaging.Several limitations need to be acknowledged. We did not aim

to address the water–fat separation problem when the fat-onlyor water-only region is absolutely isolated from other regionsby air. This is because such a scenario is not very commonin human body imaging and most existing water-fat separationmethods, including VARPRO-GC and MRGS, are not designedfor this purpose. If an accurate identification of fat is desirablein this case, other methods such as short tau inversion recoverycould be employed. From the model error analysis of the singlespecies fitting, it is found that the largest error occurs in SPURSwhen the water and fat ratio in a voxel is 1:1. Fortunately, itis observed that most of the voxels in an imaging field of viewhave one dominant species. We restrict step size to be posi-tive in order to make the algorithm easier to implement. Therestriction of only introduces a constant overall offset ofin the final field map, which does not affect the water fat sep-aration and the QSM reconstruction results. Another limitationin this study is that a multi-peak fat model is not considered[45]. Nevertheless, multi-peak IDEAL may be adopted in thelast fine-tuning step [46].

VI. CONCLUSION

In this paper, we proposed a 3-D joint field map estimationand water/fat separation method. A field map is first estimatedvia solving a single species fitting problem. Simultaneous phaseunwrapping and chemical shift removal was then performedusing graph cuts with conditional jump moves. Finally, the fieldmap was fine-tuned by -IDEAL. Experiments on numericalphantom and human data showed that the SPURS method pro-vides accurate water/fat separation results in MR datasets withlong echo spacing and in areas where field map varied rapidly.The 3-D continuous field map without wrapping is well suitedfor QSM reconstruction.

APPENDIX ASINGLE SPECIES FITTING ERROR ANALYSIS

If

. Following (3), we use the followingsingle species model to fit the two species signal equation:

(A1)

For any complex-valued number , we have , sothe above objective function can be written as

(A2)

Invoking , this cost function can be eval-uated as

(A3)

Its minimum is found by setting

(A4)

When the phases in the sine functions are close towith as integers depending on echo index , a Taylor ex-pansion can be used

(A5)

Then (A4) becomes

(A6)

(A7)

For ease of explanation, we assume . Thisis not a necessary condition as any additional time constantin equally spaced TEs can be absorbed by the initial phase of

. With ( an integer), (A7) becomes

(A8)

Here can be interpreted as a wrap caused by the limiteddynamic range.If , the signal equation can be written

as ,shown in Fig. 2(b). Then

. Similar to (A1) and (A8), we have

(A9)

(A10)

So, the approximate model error of using single species fittingis

(A11)


APPENDIX BPROOF OF THE SUBMODUALR CONDITION

Theorem 1: For 1-jump move optimization, as long asis convex, the submodular condition holds for all binary energyterm .

Proof: From (11),

, where .For any convex function , we have

, where .In our problem, is a convex function. Let

, we have .So we have the following inequality from the definition ofconvex function:

(A12)

Similarly, , we have

(A13)

Summing the above two inequalities, we get

(A14)

Multiplying the positive value to both sides of theabove inequality, we get the inequality (12). So the submodularcondition holds for all binary energy terms in 1-jump move op-timization. We introduce the function to emphasize thatany convex function satisfying the submodular condition canbe used.

ACKNOWLEDGMENT

The authors would like to thank A. J. Luzzi for proof-reading. We acknowledge the use of the Fat-Water Toolbox(http://ismrm.org/workshops/FatWater12/data.htm).

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spurs using graph cuts

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