MRI-Based Quantification of Magnetic Susceptibility in Gel...

Research ArticleMRI-Based Quantification of MagneticSusceptibility in Gel Phantoms Assessment ofMeasurement and Calculation Accuracy

Emma Olsson RonnieWirestam and Emelie Lind

Department of Medical Radiation Physics Lund University Skane University Hospital Lund 22185 Lund Sweden

Correspondence should be addressed to Ronnie Wirestam ronniewirestammedluse

Received 23 May 2018 Accepted 5 July 2018 Published 30 July 2018

Academic Editor Paul Sijens

Copyright copy 2018 Emma Olsson et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The local magnetic field inside and around an object in a magnetic resonance imaging unit depends on the magnetic susceptibilityof the object being magnetized in combination with its geometryorientation Magnetic susceptibility can thus be exploited asa source of tissue contrast and susceptibility imaging may also become a useful tool in contrast agent quantification and forassessment of venous oxygen saturation levels In this study the accuracy of an established procedure for quantitative susceptibilitymapping (QSM) was investigated Three gel phantoms were constructed with cylinders of varying susceptibility and geometryExperimental results were compared with simulated and analytically calculated data An expected linear relationship betweenestimated susceptibility and concentration of contrast agent was observed Less accurate QSM-based susceptibility values wereobserved for cylindrical objects at angles relative to the main magnetic field that were close to or larger than the magic angleResults generally improved for large objectshigh spatial resolution and large volume coverage For simulated phase maps accuratesusceptibility quantification by QSM was achieved also for more challenging geometries The investigated QSM algorithm wasgenerally robust to changes in measurement and calculation parameters but experimental phase data of sufficient quality may bedifficult to obtain in certain geometries

1 Introduction

An object in an external magnetic field will become magne-tized to a degree that is determined by the magnetic suscepti-bility of the object The local magnetic field inside and in thesurroundings of the object will thus depend on the magneticsusceptibility in combination with the geometryorientationof the object being magnetized In susceptibility-weightedmagnetic resonance imaging (SWI) the local magnetic fielddistribution is explored to enhance image contrast and toimprove the visibility of various structures on the basis oftheir magnetic susceptibility [1] Additionally quantitativesusceptibility mapping (QSM) has developed into a promis-ing method for calculating arbitrary magnetic susceptibilitydistributions from measured magnetic resonance imaging(MRI) phase data [2ndash4] A more complete understanding ofthe phase behaviour in vivomay also require biophysical con-siderations ofmicrostructural tissue anisotropy andmagneticsusceptibility anisotropy [5]

In vivo the magnetic susceptibility differs among tissuetypes and tissue regions and it can thus be exploited asa source of contrast in MRI For example in reference tothe cerebrospinal fluid (CSF) ie when the CSF magneticsusceptibility is set to 0 ppm the globus pallidus which ispart of the basal ganglia has amagnetic susceptibility of 0105ppm while white matter has a lower susceptibility of -0030ppm [6] In addition to the effects of normal ageing [7] anumber of conditions and processes can alter the magneticsusceptibility of tissue For example iron accumulation ininflamed myelin cells as in a multiple sclerosis (MS) plaqueincreases the susceptibility of the myelin [8] Iron accu-mulation is also seen in other neurodegenerative diseasesfor example Alzheimerrsquos and Parkinsonrsquos diseases [9] Themagnetic susceptibility is also dependent on the oxygensaturation level of blood and the susceptibility increaseswith increasing levels of deoxyhemoglobin Hence venousblood will show a higher susceptibility than arterial bloodand quantification of magnetic susceptibility can thus be

HindawiRadiology Research and PracticeVolume 2018 Article ID 6709525 13 pageshttpsdoiorg10115520186709525

2 Radiology Research and Practice

useful in estimations of oxygen extraction fraction (OEF)[10] and cerebral metabolic rate of oxygen (CMRO

2) [11]

The change in susceptibility with deoxygenation can alsobe manifested for example in extravasated blood from anintracranial haemorrhage [12] Quantitativemeasurements ofthe susceptibility could also be potentially useful to determinethe concentration of an external MRI contrast agent (CA)Relaxivity-based CA quantification which is the currentlymost common approach is associated with several method-ological complications in for example perfusion and per-meability measurements using dynamic contrast-enhancedMRI (DCE-MRI) and dynamic susceptibility contrast MRI(DSC-MRI) [13] Hence more accurate CA concentrationquantification in vivo would indeed be beneficial and a fewexamples of dynamic contrast-enhanced QSM studies havebeen presented [14ndash16]

In QSM applications quantification of magnetic suscep-tibility in absolute terms is becoming increasingly importantand extensive validation is thus warranted A number ofQSMreconstruction tools exist and the process of systematicallycharacterizing differences in accuracy between algorithmshas recently been initiated by other groups [17ndash19] Inexperimental evaluations phantoms have the advantage ofoffering well-defined contents and geometries and constitutean important though not complete part of the validationprocess and the present investigation serves as a supplementto previous investigations related to the accuracy of phaseand susceptibility quantification [eg [20ndash25]] In the presentstudy previously described in preliminary terms by Olsson(unpublished report) [26] the QSM approach was evaluatedin gel phantoms with inserted cylinders containing knownconcentrations of gadolinium CA to establish whether theQSM method can deliver accurate results with respect toquantitative magnetic susceptibility values in absolute termsMeasured phase values and the corresponding magneticsusceptibility estimates calculated by an established QSMalgorithm were compared to values based on theoreti-cal relationships Various phantom designs and simulatedsusceptibility distributions as well as different parametersand settings in the measurements and in the susceptibilitycalculation were investigated in order to establish optimalsettings and important sources of error in the attempts toproduce accurate magnetic susceptibility maps

2 Theory

21 The Dipole Field and the Magic Angle A magneticmoment with magnitude 119898 pointing in the 119911 directionproduces a magnetic flux density component 119861

119911

119861119911prop 119898

1198893(3 cos2Θ minus 1) (1)

where d is the distance from m and Θ is the angle relative tothe z-axis The angle at which the factor (3 cos2Θ minus 1) equalszero is called the magic angle (ie approximately plusmn547∘or 180∘plusmn547∘) At the magic angle positions the magneticflux density component 119861

119911will be zero independently of the

magnitude of the magnetic moment

22 Phase Shift and Magnetic Field Variations in the localmagnetic field with position r lead to differences in MRI res-onance frequency and to subsequent phase-shift variationsand the MRI phase evolution 120601(r) is given by

120601 (119903) = 120596 (119903) sdot 119879119864 = 120574 sdot 119861 (119903) sdot 119879119864 (2)

where TE is the echo time In order for the measuredphase images to be useful unwrapping and filtering ofbackground field variations are required The unwrappingcan be accomplished by a region growing algorithm whichidentifies phase gradients that correspond to a difference bya multiple of 2120587 and subsequent addition or subtraction of2120587 [27] Filtering is needed because the unwrapped imageusually contains a remaining background phase gradientover the entire image This phase does not arise from thesusceptibility distribution inside the object but from forexample imperfect shimming or susceptibility sources out-side the imaging volume Projection onto Dipole Fields (PDF)[28 29] is one method for background field removal thatcompares magnetic fields generated from magnetic dipolesinside and outside a region of interest Other examples offiltering methods are Laplacian Boundary Value (LBV) [30]and Regularization Enabled Sophisticated Harmonic ArtefactReduction for Phase data (RESHARP) [31]

23 Cylindrical Objects The internal (in) and external (ex)magnetic field alterations ΔB caused by an infinitely longcylinder are given by the following analytical expressions

Δ119861119894119899

=Δ1205946

(3cos2120579 minus 1) 1198610

(3)

Δ119861119890119909

=Δ1205942

1198862

1205882sin2120579 cos 2120593119861

0(4)

whereΔ120594 is the difference in susceptibility between the insideand the outside of the cylinder 119886 is the radius of the cylinder120579 is the angle between the direction of the B

0field and the

cylinder axis and 120588 and 120593 are the cylindrical coordinatesdescribing a point at distance 120588 and at an angle 120593 relative to apoint at the centre of the cylinder

24 Magnetic Susceptibility and Magnetic Field For morecomplicated geometries or shapes the local field changecaused by the introduction of an object in the externalmagnetic field can be described more generally [32 33] andis often formulated as a convolution (denoted ldquootimesrdquo) of thearbitrary susceptibility distribution with a dipole field kernelie the corresponding phase is given by

Δ120601 (119903) = 120574 sdot 119879119864 sdot 3cos2120579 minus 1

41205871199033otimes 120594 (119903) (5)

where 119903 and 120579 are spherical coordinates and 120594 denotes themagnetic susceptibility The main idea of QSM is to extractthe susceptibility distribution according to (5) using theinformation of the local magnetic field from the measuredphase images However problems arise because the dipolekernel is zero at the magic angle A convolution in real space

Radiology Research and Practice 3

represents a multiplication in k-space and extracting thesusceptibility distribution from (5) by deconvolution wouldtherefore imply a division by zero at some coordinates in k-space which would in principle affect every point of the 120594(r)solution in real space

Morphology EnabledDipole Inversion (MEDI) [29 34ndash36]is a QSM reconstruction method designed to solve the ill-posed inverse problem of resolving 120594(r) according to (5) Inthe MEDI approach the problem is formulated so that thedifference between an estimated field map and the measuredfield map should be of the order of the noise level 120576 This canbe written as

10038171003817100381710038171003817119882(120575 minus 119865119879minus1 (119863 sdot 119865119879 (120594)))100381710038171003817100381710038172 le 120576 (6)

where119882 is a weighting matrix 120575 is the measured field andDis the representation of the dipole field in k-space ldquoFTrdquo andldquoFTminus1rdquo denote the forward and inverse Fourier transformrespectively Additionally MEDI uses the fact that changes insusceptibility follow the morphological boundaries and thatthe susceptibility map therefore should have gradients in thesame locations as the magnitude image [35]

In brief the inverse problem is solved through an iter-ative process An initial guess is made for the susceptibilitydistribution Convolving this with the dipole kernel gives anestimated field map The estimated field map is comparedto the measured field map ie the phase image and thedifference the error is used to update the initial guess Theupdated susceptibility distribution is then used as input whenthis procedure is repeated Iterations are made until theresult fulfils the requirements A regularization parameter120582 determines how much magnitude versus phase imageinformation is prioritized The Lagrange multiplier methodis used to reformulate the problem in (6) as a minimizationof a cost function [35]

3 Materials and Methods

31 Phantom Design In order to evaluate the QSM methodwith respect to phase measurement as well as mathematicalreconstruction three different phantoms were constructedThin-walled plastic cylinders were filled with a paramagneticgadolinium (Gd) contrast agent solution (Dotarem GuerbetFrance)The employed plastic material and the low thicknessof the cylinder walls (of the order of 100 120583m) imply thatthe susceptibility effects created by the cylinders should benegligible The cylinders were sealed and glued onto theinside of a larger container The container with cylinderswas subsequently filled with agarose gel doped with a smallamount of nickel in the form of nickel(II)nitrate hexahydrateNi(NO

3)2sdot6H2OThe gel was designed according to a locally

developed preparation routine using in this study 1 agaroseand 024mMNi2+ [37]The susceptibility of the gel was calcu-lated using Wiedemannrsquos additivity law for the susceptibilityof mixtures ie 120594 = 119901

11205941+ 11990121205942+ sdot sdot sdot 119901

119899120594119899 where 119901

119899is the

concentration of substance 119899 [38]The purpose of the contrast agent was to obtain a

controlled increase of the susceptibility inside the cylindersto achieve a difference in susceptibility between the cylinders

Table 1 Reference magnetic susceptibility values of the differentcomponents of the phantoms

120594(water) -9022 ppm120594mol(Ni(NO3)2 sdot 6H2O) 54 ppmM120594mol(Gd) 326 ppmM in reference to water120594(gel) -9017 ppm120594(05 mM Gd) -8859 ppm

Table 2 Imaging parameters in the standard protocol used forQSMphase measurements

Sequence 3D Multi-TE Gradient EchoNumber of echoes 11Echo spacing ΔTE 678 msFlip angle 20∘

Band width 150 HzpixelField of view FOV 24 cmSlice thickness 2 mmNumber of averages 1Matrix size 256 x 240

and the background resembling different compartmentsin the human body (including cases of injected externalcontrast agent for example for the purpose of perfusionimaging) Table 1 shows the theoretical absolute values usedfor the susceptibility of water and nickel as well as themost commonly reported value of the molar susceptibilityfor gadolinium (used as a reference value in this study [39])The calculated susceptibility values for the gel and the 05mMgadolinium solution are also included

(i) In the first phantom design cylinders with 5 mmdiameter filled with 05 mM Gd solution werepositioned at five different angles relative to the mainmagnetic field (approximately 0 30 55 75 and 90∘)The actual angles were measured in the resultingimages

(ii) The second phantom design consisted of 5 mm diam-eter cylinders in parallel with varying concentrationsof Gd contrast agent ie [0 02 04 06 08 1 2 4 68 10] mM

(iii) In the third phantom cylinders in parallel containing05 mM Gd solution with diameters of [2 26 47 574 9 108] mm were used

32 Measurements Measurements were carried out at roomtemperature on a 3T MRI unit (Magnetom Trio SiemensHealthcare GmbH Erlangen Germany) using an imagingprotocol described in Table 2 The parameters were selectedaccording to the recommendations for QSM of human braingiven by the Cornell MRI Research Lab [40] A multi-TE gradient echo sequence was used because a single TEacquisition is regarded not to be sufficient for deriving themagnetic field from the phase due to an offset in themagneticfield depending on the conductivity of thematerialThephaseshifts reported below correspond to the difference in phase


between two subsequent TEs ie to a time period ΔTE (cfTable 2) and are based on multiecho data [34 39 41] Inthe evaluation process this protocol was subsequently alteredto accomplish measurements with isotropic voxels differentspatial resolutions and shorter TE The different parameterchanges are described more in detail below

Spatial Resolution Measurements were performed with vary-ing spatial resolution (altered matrix size and fixed volume ofinterest) A FOV of 205times205 mm2 and an excited slab of 32mm were used The matrix sizes used were 64times64 128times128256times256 and 512times512 corresponding to isotropic voxels withsides 32 16 08 and 04 mm respectively

Volume Coverage Measurements in which only the numberof slices was varied were carried out implying varied volumeof interest The voxel size was 08times08times08 mm3 and thenumber of slices was 40 60 104 and 144 The phantomwith different diameters was used placed with the cylindersperpendicular to the main magnetic field

33 Image Processing and QSM Calculation For postpro-cessing of the measured images a MEDI MATLAB codepackage for QSM from Cornell MRI Research Lab [40] wasemployed Magnitude and phase images one set for each TEwere obtained from the MRI experiments The phase imageswere unwrapped [27] and subsequently filtered with the PDFapproach [28 29]The phase images were also masked beforethey were supplied to the MEDI algorithm using a thresholdapproach based on information from the magnitude imageto define the object region to be included in the susceptibilitycalculation

34 Variations in Postprocessing and QSMCalculation Procedures

Filtering Method Different methods for phase backgroundremoval were compared ie ldquoProjection onto Dipole Fields(PDF)rdquo [28 29] ldquoLaplacian BoundaryValue (LBV)rdquo [30] andldquoRegularization Enabled Sophisticated Harmonic ArtefactReduction for Phase data (RESHARP)rdquo [31]

Variation of 120582 The susceptibility images were calculatedusing 120582 settings 1 10 100 1000 10 000 and 50 000

Zero Padding Zero padding to potentially reduce artefactswas performed in the spatial domain by padding the matrixsymmetrically with 200 zeros in all three dimensions

35 Simulation A simulated set of phase images was con-structed by creating a template based on the magnitudeimages that distinguishes between cylinders and gel Artificialsusceptibility images were then constructed by assigning thetheoretical values (cf Table 1) of the susceptibility for agarosegel and gadolinium solution to the respective pixels From theartificial susceptibility images a set of simulated phase imageswas calculated using (5) This set of simulated phase imageswas then used as input to the MEDI algorithm and simulated

eoreticalsusceptibility

Dipolefield

Constants Simulated phaseimage

Magnitudeimage

Template

otimesmiddot middot middot TE =

Figure 1 Illustration of the construction of a simulated phase image

QSMmaps were obtainedThe construction of the simulatedphase image is illustrated in Figure 1

36 Image Analysis Experimental as well as simulated phaseand QSM images were evaluated by measuring the value ofinterest (mean and standard deviation) in ROIs placed inthe cylinders In the output data from the MEDI softwarethe background gel region was assigned values which werevery close to zero (cf Figure 2) and the background gelthus served as a zero reference (with known susceptibilityaccording to Table 1) Experimental values were comparedwith the corresponding theoretical andor simulated valuesThe calculations of theoretical phase displayed for the phan-tom with 05 mM Gd cylinders of varying angles and forthe phantom with varying Gd contrast agent concentrationswere based on (3) under the assumption that the respectivemagnetic susceptibility differences Δ120594 were known basedon the reference values in Table 1 Conversely (3) canbe used to calculate an unknown magnetic susceptibilitydifference based on measured phase and such calculated Δ120594estimates based onmeasured phase and the infinite-cylinderapproximation of (3) are also for completeness included inthe results

4 Results

41 Cylinder Angle Figure 2 shows (a) a phase image and(b) a corresponding susceptibility image of the phantomwith cylinders of varying angles relative to the B

0field So-

called blooming effects in the phase image related to theproperties of a dipole field are seen around cylinders notoriented parallel to the main magnetic field Some unwantedresidues of this blooming effect can be seen in the suscep-tibility image In Figure 2(c) analytically calculated phasevalues based on the infinite-cylinder approximation (see (3))and the assumption of known reference values of magneticsusceptibility (Table 1) are compared with the measuredphase data for cylinders at different angles Figure 2(d) showsthe expected magnetic susceptibility values (based on thereference values in Table 1) as well as the correspondingresults based on experimental phase data ie employing theinfinite-cylinder approximation (see (3)) as well as the QSMalgorithm The phase inside the cylinders corresponded wellwith the theoretical values and accordingly the infinite-cylinder approximation yielded quite reasonable susceptibil-ity estimates based on experimental phase data However theQSM calculation returned susceptibility values that deviated


oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

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minus02

[ppm]

(b)

0 20 40 60 80 100

Experimenteory

minus15

minus10

minus5

0

5

10

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20

Δ

[∘]

[∘]

(c)

000

005

010Δ

[ppm

] 015

020

025

0 20 40 60 80 100

eoryExperiment QSMExperiment Eq 3

[∘]

(d)

Figure 2 (a) Phase image and (b) the corresponding calculated susceptibility image of the phantom with varying angles of the cylinders(slice 17 of 48) The graphs in (c) and (d) show theoretical as well as experimental phase and susceptibility in the cylinders as a function ofthe angle relative to the main magnetic field (c) Analytically calculated phase values (based on the infinite-cylinder approximation and thereference values in Table 1) and the corresponding measured phase values (d) Expected theoretical magnetic susceptibility values (based onthe reference values in Table 1) as well as the estimates based on experimental phase data ie employing the infinite-cylinder approximationas well as the QSM algorithm Error bars correspond to the standard deviation within the region of interest for the experimental QSM data

considerably from the expected values for angles larger thanthe magic angle

42 Concentration of Gadolinium Solution An expectedlinear dependence for both phase and susceptibility on theconcentration of gadolinium was observed (Figure 3) TheQSM algorithm yielded a measured slope of 390 ppmMwhichwas slightly higher than the corresponding experimen-tal slope based on the infinite-cylinder approximation (382ppmM) (Figure 3(b)) Insufficient phase unwrapping wasobserved at high concentrations and for the phase resultsthis was compensated for manually by simply adding 2120587 to

the extracted numerical values It was however not possibleto evaluate the QSM output for concentrations above 4 mM

43 Cylinder Diameter QSM-based susceptibility estimatesfor the seven cylinders of various diameters at parallel andperpendicular orientations are presented in Figure 4

44 Variation of Measurement Parameters

Spatial Resolution The phantom with different diameterswas measured with the cylinders oriented perpendicularto the main magnetic field at different spatial resolutions


Experimenteory

Linear (Experiment)Linear (eory)

2 4 6 8 10 120

Concentration of Gd [mM]minus50

0

50

100

150

200

250

300

350

400

450ΔΦ

[∘]

(a)

1 2 3 4 50Concentration of Gd [mM]

minus02

0

02

04

06

08

1

12

Δ

[ppm

]

14

16

18

2


Linear (eory)Linear (Experiment QSM)Linear (Experiment Eq 3)

(b)

Figure 3 Phase andmagnetic susceptibility as a function of the concentration of gadolinium contrast agent (a) Analytically calculated phase(based on the infinite-cylinder approximation and the reference values in Table 1) and measured phase (b) Expected theoretical magneticsusceptibility values (based on the reference values in Table 1) as well as the estimates based on experimental phase data ie employing theinfinite-cylinder approximation as well as theQSMalgorithm For the three highest concentrations in (a) the phase wasmanually unwrappedError bars correspond to the standard deviation within the region of interest for the experimental QSM data

The susceptibilities in the 5 mm and 108 mm cylinders arepresented as a function of pixel size in Figure 5

VolumeCoverage Figure 6 shows the result ofmeasuringwithdifferent volume coverageThe same slice thickness (08 mm)was used for each acquisition


Filtering Methods Susceptibility estimates obtained usingphase data filtered with three different methods for back-ground field removal (PDF LBV and RESHARP) as well aswithout any filtering are presented in Figure 7

Variation of 120582 Although clear differences in QSM imagequality were observed for different 120582 settings the numericalsusceptibility values inside cylinders did not vary substan-tially (approximately plusmn001 ppm from the measured meanvalue) when 120582 varied between 1 and 50000

Zero Padding The zero padding did not have any observableeffect on the estimated absolute susceptibility values for thephantomwith cylinders in various angles relative to the mainmagnetic field

46 Simulations Simulated phase images and the corre-sponding artificial susceptibility maps calculated from sim-ulated phase data using the MEDI algorithm are shown inFigures 8(a)ndash8(d) Simulated phase images appeared visually

similar to corresponding measured phase images but inthe simulated QSM images (ie calculated from simulatedphase maps) no susceptibility dependence on the angle ofthe cylinder axis relative to the B

0field was observed (see

Figure 8(e)) In the phantomwith varying cylinder diametersthe simulated phase images resulted in susceptibility valuesthat were in much better agreement with theory than the sus-ceptibility values based on measured phase (see Figure 8(f))These findings can be compared to results from measureddata in Figures 2(d) and 4

47 Phase Profiles A profile was positioned through the 5mm cylinder in measured and simulated phase images ofthe phantom with cylinders of varying diameters Profileswere plotted both in-plane and along the slice direction asillustrated in Figures 9(a) and 9(b) ie the measurementswere identical except for the use of different slice direc-tions The results are presented in Figures 9(c)ndash9(f) Specialattention was paid to the amplitude of the peaks at thecylinder edges no marked difference in peak amplitude wasseen between measured and simulated phase for the in-planeprofile However with the phase profile in the slice directionthe simulated peak phase value was considerably higher thanthe measured phase

5 Discussion

Cylindrical objects were used in this investigation due to theirgeometrical resemblancewith blood vessels Also the distinctangular dependence of the phase in a cylinder (cf Figure 2(a))


Experiment parallelExperiment perpendicular

eory

2 4 6 8 10 120Diameter [mm]

0

005

01

015

Δ

[ppm

]

02

025

03

Figure 4 Measured susceptibility estimated from the QSM algo-rithm as a function of cylinder diameter Results for cylindersparallel and perpendicular to the main magnetic field are comparedwith the theoretical value Error bars correspond to the standarddeviation within the region of interest for the experimental data

108 mm5 mm

0

002

004

006

008

Δ

[ppm

]

01

012

014

1 2 3 40Voxel side length [mm]

Figure 5 Measured susceptibility estimates from the QSM algo-rithm in the centres of the 108 mm and 5 mm diameter perpendic-ularly oriented cylinders for different isotropic spatial resolutions

made it reasonable to assume that cylinders could be achallenge for theQSMalgorithmAnother advantagewith theuse of cylinders is the availability of theoretical relationshipsbetween the local magnetic field change and the magneticsusceptibility as seen in (3) and (4) In the interpretation ofthe current results it should however be remembered thatcylinders show rather limited resemblance with most in vivostructures of relevance to clinical investigations The initialpresumption that cylindrical objects could be problematic

for the QSM algorithm seemed at first to be valid basedon the observation of the measured and theoretical data ofthe phase and susceptibility inside the cylinders at differentangles relative to the main magnetic field (Figure 2(d)) Thesimulated data however indicated that the QSM algorithmdid in fact generate magnetic susceptibility values that werevery close to theory even for angles larger than the magicangle (cf Figure 8(e)) Furthermore the simulated data didnot show any dependence of estimated susceptibility on thediameter of the phantom (cf Figure 8(f))

From themeasurements on cylinders filled with solutionsof varying gadolinium concentrations it was concludedthat the estimated susceptibility varied linearly with theconcentration of contrast agent with an estimated slopeof approximately 390 ppmM using the QSM algorithmand 382 ppmM using the infinite-cylinder approximationie the QSM algorithm generated a slightly higher slopethan the infinite-cylinder approximation based on the sameexperimental phase data Our current estimates are in goodagreement with previous findings for gadoterate meglumine(Dotarem) by Fruytier et al [42] but higher than thereference (Magnevist) Gd molar susceptibility value of 326ppmM [39]

For the 05 mM gadolinium solution also used in theother phantom designs the susceptibility value was slightlyhigher than the reference value in accordancewith the resultsshown in Figure 3(b) as discussed above Comparing theresults of Figures 2(d) and 4 with Figure 3(b) indicatesthat the deviation of the slope from the reference value (inFigure 3(b)) corresponds well to the slight apparent over-estimation of the susceptibility for cylinders approximatelyparallel to the main magnetic field Since the phase shift (inFigure 3(a)) and the susceptibility estimates based on theinfinite-cylinder approximation (in Figure 3(b)) were alsohigher than expected from the molar susceptibility referencevalue it seems reasonable to conclude that the overestimationwas not at least not entirely related to the QSM-basedsusceptibility calculationThe result from the use of simulatedphase data confirmed that the problem of estimating the truesusceptibility values for different angulations relative to themainmagnetic field was not inherent to theMEDI algorithm

The choice of the regularization parameter 120582 has previ-ously been shown to influence the quantitative accuracy of theQSM method [35] However in the present study the valueof 120582 seemed to influence primarily the image quality but theabsolute susceptibility estimates did not vary substantiallyThe fact that the results were not much affected by the choiceof 120582 also suggests that the observed error arises from someother step in the procedure Hence three different filteringmethods were applied to see if the problems were causedin the background phase removal procedure Although theresults for different filtering methods were not identical nofiltering method returned a systematically more accurateresult than the others An attempt was also made to calculatethe susceptibility without filtering This resulted in images ofpoor quality but the angular dependence of the susceptibilityestimates remained

As shown the phase was measured correctly inside thecylinders but measured and simulated phase data did not


14410460

40

0 50 100 150minus50

Sliceminus02

minus015minus01

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00501

01502

02503

035

ΔΦ

[rad

]

(a)

0

002

004

006

008

01

012

014

016

018

144104

40eory

60

Δ

[ppm

]

0 50 100 150minus50

Slice

(b)

Figure 6 The diagrams show (a) the phase and (b) the QSM-based susceptibility measured in the largest cylinder of the phantom withdifferent diameters for a varying number of slices 40 slices correspond to an object coverage of 32 mm 60 slices correspond to 48 mm 104slices correspond to 832 mm and 144 slices correspond to 1152 mm

40

PDF RESHARPWithout filteringLBV

0

002

004

006

008

01

012

014

016

018

02

60 80 1000 20

Δ

[ppm

]

[∘]

Figure 7 QSM-based magnetic susceptibility in cylinders orientedat different angles relative to the main magnetic field calculatedusing three different filtering methods (PDF LBV and RESHARP)as well as without filtering

generate the same output from the QSM algorithm Hencea reasonable conclusion was that measured phase must differfrom simulated phase outside the cylinder The comparisonof profiles originating from measured and simulated phaseshowed that the difference in phase between measurementand simulation occurred just outside the cylinder and theunderestimation was more pronounced when phase was

recorded along the slice direction Partial volume effects mayhave been of importance in this context [43] considering thehigh spatial phase gradient in this region and it is not entirelystraightforward to predict the exact manifestation of theloss of phase information occurring in the complex sum ofvoxel components in a given imaging situation Dependingon the particular MRI unit and imaging protocol potentialeffects of applied deapodization filters might also be relevantto consider in this context Since the magnetization of thecylinder affects the phase not only inside the cylinder but alsoin the area surrounding the cylinder the background phasein the vicinity of the cylinder will influence the result of theQSM susceptibility calculation Hence if the phase was notcorrectly measured in the background region close to thecylinder this constitutes a plausible explanation to why QSMcalculations failed to return accurate values for the magneticsusceptibility This explanation is also in accordance with thefact that the infinite-cylinder approximation (see (3)) yieldedmore reasonable magnetic susceptibility estimates than theQSM algorithm

In this context it is also relevant to note that the phantomwith parallel cylinders of varying diameters was scanned attwo different orientations relative to the main magnetic field(0∘ and 90∘) while keeping the slice orientation orthogonal tothe cylinders in each acquisition ie with the measurementof the phase being consistent in terms of slice orientationrelative cylinder axis Comparing the QSM susceptibilityresults in Figure 4 (for 5 mm diameter) with those inFigure 2(d) from the phantom with cylinders of varyingangles (for 0∘ and 90∘) shows that the parallel as well as theperpendicular cylinder orientation yielded very similar QSMsusceptibility values between the two separate phantoms


oplus B0

(a)

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[ppm]

(b)

B0

(c)

B0

02

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[ppm]

(d)

eorySimulation

000

005

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015

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025

Δ

[ppm

]

20 40 60 80 1000 [∘]

(e)

0

005

01

015

Δ

[ppm

]

02

025

5 10 150Diameter [mm]

eorySimulation (perpendicular)

(f)

Figure 8 (a) Simulated phase image and (b) the corresponding susceptibility image calculated with MEDI for the phantom with cylindersat different angles (slice 17 of 48) (c) Simulated phase image and (d) the corresponding susceptibility image for the phantom with varyingdiameters (slice 40 of 80) Images were simulated without added noise for a matrix size of 512times512 (d) Susceptibility registered in cylinderswith different angles in simulated images (e) Calculated susceptibility using simulated phase data for varying cylinder diameter comparedwith the theoretical susceptibility Simulated cylinders corresponded to 05 mMGd and the assigned values of magnetic susceptibility in thesimulated phantom are given in Table 1


B0

(a)

oplus B0

(b)

0 50 100 150

Distance [mm]minus025

minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

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[rad

]

Distance [mm]

(d)

minus015

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005

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025

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ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

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[rad

]

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minus005

0

005

01

015

02

025

(f)

Figure 9 Top row Positioning of a profile through the 5 mm cylinder perpendicular to the main magnetic field (a) in-plane and (b) alongthe slice direction Note the measurements were identical except for the use of different slice directions Middle row values along the profile(when placed perpendicular to the slice direction) through the 5 mm cylinder in a slice in the middle of the volume showing (c) measuredand (d) simulated phase Bottom row slice direction phase profile through the 5 mm cylinder comparison between (e) measured phase and(f) simulated phase

(ie 016-017 ppm in the parallel case and 010 ppm in theperpendicular case)

Slice spacing slice thickness and volume coverage arealso important issues in QSM [44] In our study a largervolume coverage of the object resulted in susceptibility values

closer to the theoretical value and in susceptibility imageswith a more uniform slice direction profile in accordancewith previous recommendations for an extended spatial cov-erage in QSM of deep grey matter [45] Preliminary attemptsto remove slices at the edges of the phase image stack after


the PDF but before the MEDI QSM reconstruction alsoresulted in images with less artefacts and a more uniformslice direction profile (data not shown) and this suggeststhat phase values along the slice direction or at the edges ofthe imaged volume are influenced by factors which are stillunknown (eg residual slice aliasing effects etc)

Furthermore the spatial resolution seemed to be of someimportance for the accuracy of the estimated susceptibilitysince the estimated susceptibility values appeared to varybetween different voxel sizes For the smallest cylinders(diameters of 2 mm and 26 mm) some degree of partialvolume effects can be expected since the voxel size used was1times1times2 mm3 but even for the larger cylinders the estimatesdiffered from the expected values Figure 5 indicates thatthere is no dependence on the voxel size for the largestcylinder (108 mm in diameter) but for the 5 mm cylindera higher resolution resulted in a value closer to theory Thisimplies not surprisingly that sufficient spatial resolutionis needed to obtain optimal results At a certain pointthe combination of object size and spatial resolution givessufficiently goodmeasurement conditions and the systematicerror is minimal For a smaller object a higher resolution isobviously required to reach that point However to establisha clear relationship between image resolution and estimatedsusceptibility values more data points would be needed

Finally it should be noted that even if experimentalphase data were to be accurate the QSM results would stillin practice be relative The assignment of zero phase inthe phase maps would imply zero susceptibility which isnormally not the true value for the compartment in questionand furthermore the QSM algorithm will typically shift thevalues of the output data so that the mean value of thevolume is close to zero In the phantoms the true backgroundsusceptibility value is known and thus the expected phasewithin the cylindrical object can be calculated In the humanbody we do not normally have any such information Hencesome reference region with known susceptibility is requiredin order to obtain the correct absolute level of susceptibilitywithin the dataset CSF has been proposed for such referencepurposes but such an approach is far from straightforward[46]

6 Conclusions

TheMEDI algorithm was demonstrated to be quite stable forQSM calculations The choice of parameters and settings forexample the regularization parameter 120582 the zero paddingand the choice of filtering method seemed not to have a largeimpact on the quantitative results Most importantly whenapplied to simulated phase maps MEDI returned accuratesusceptibility quantification also for challenging geometriesFor experimental phase data theQSMalgorithmdid result ina linear relationship between susceptibility and concentrationof contrast agent but correct susceptibility was not obtainedfor cylindrical objects at an angle close to or larger than themagic angle The error seemed to originate from the phasemeasurement rather than from imperfections in the QSMsusceptibility calculation In our MRI installation deviationfrom theory was observed primarily along the slice direction

in the phantom background (ie gel) region of the measuredphase images The QSM-based susceptibility results seemedto be somewhat more accurate for large objects andor goodspatial resolution large volume coverage of the object andwith the slice direction applied along the long axis of theobject of interest

Data Availability

Requests for data will be considered by the correspondingauthor

Disclosure

This study has previously been described in preliminaryterms by Olsson (unpublished report) [26] The currentresearch article contains extended data analysis additionalresults and a correspondingly expanded discussion com-pared with [26]

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This study was supported by the Swedish Research Council[2017-00995] the Swedish Cancer Society [2015567] and theCrafoord Foundation in Lund [20150753]

References

[1] S Liu S Buch Y Chen et al ldquoSusceptibility-weighted imagingcurrent status and future directionsrdquo NMR in Biomedicine vol30 no 4 Article ID e3552 2017

[2] L Li and J S Leigh ldquoQuantifying arbitrary magnetic suscepti-bility distributions with MRrdquo Magnetic Resonance in Medicinevol 51 no 5 pp 1077ndash1082 2004

[3] K Shmueli J A de Zwart P van Gelderen T-Q Li S J Doddand JHDuyn ldquoMagnetic susceptibilitymapping of brain tissuein vivo usingMRI phase datardquoMagnetic Resonance inMedicinevol 62 no 6 pp 1510ndash1522 2009

[4] SWharton A Schafer and R Bowtell ldquoSusceptibility mappingin the human brain using threshold-based k-space divisionrdquoMagnetic Resonance in Medicine vol 63 no 5 pp 1292ndash13042010

[5] D A Yablonskiy and A L Sukstanskii ldquoEffects of biologicaltissue structural anisotropy and anisotropy of magnetic sus-ceptibility on the gradient echo MRI signal phase theoreticalbackgroundrdquo NMR in Biomedicine vol 30 no 4 Article IDe3655 2017

[6] I A L Lim A V Faria X Li et al ldquoHuman brain atlas for auto-mated region of interest selection in quantitative susceptibilitymapping Application to determine iron content in deep graymatter structuresrdquo NeuroImage vol 82 pp 449ndash469 2013

[7] Y Zhang H Wei M J Cronin N He F Yan and C LiuldquoLongitudinal atlas for normative human brain developmentand aging over the lifespan using quantitative susceptibilitymappingrdquo NeuroImage vol 171 pp 176ndash189 2018


[8] C Stuber D Pitt andYWang ldquoIron inmultiple sclerosis and itsnoninvasive imaging with quantitative susceptibility mappingrdquoInternational Journal of Molecular Sciences vol 17 no 1 article100 2016

[9] J Stankiewicz S S Panter M Neema A Arora C E Battand R Bakshi ldquoIron in chronic brain disorders imaging andneurotherapeutic implicationsrdquoNeurotherapeutics vol 4 no 3pp 371ndash386 2007

[10] K Kudo T Liu T Murakami et al ldquoOxygen extraction frac-tion measurement using quantitative susceptibility mappingComparison with positron emission tomographyrdquo Journal ofCerebral Blood FlowampMetabolism vol 36 no 8 pp 1424ndash14332016

[11] J Zhang T Liu A Gupta P Spincemaille T D Nguyen and YWang ldquoQuantitative mapping of cerebral metabolic rate ofoxygen (CMRO

2) using quantitative susceptibility mapping

(QSM)rdquoMagnetic Resonance inMedicine vol 74 no 4 pp 945ndash952 2015

[12] J H Duyn and J Schenck ldquoContributions to magnetic suscep-tibility of brain tissuerdquo NMR in Biomedicine vol 30 no 4 pe3546 2017

[13] R Wirestam ldquoUsing contrast agents to obtain maps of regionalperfusion and capillary wall permeabilityrdquo Imaging inMedicinevol 4 no 4 pp 423ndash442 2012

[14] D Bonekamp P B Barker R Leigh P C M Van Zijl and XLi ldquoSusceptibility-based analysis of dynamic gadolinium bolusperfusion MRIrdquoMagnetic Resonance in Medicine vol 73 no 2pp 544ndash554 2015

[15] B Xu P Spincemaille T Liu et al ldquoQuantification of cerebralperfusion using dynamic quantitative susceptibility mappingrdquoMagnetic Resonance in Medicine vol 73 no 4 pp 1540ndash15482015

[16] X L Xie A T Layton N Wang et al ldquoDynamic contrast-enhanced quantitative susceptibility mapping with ultrashortecho timeMRI for evaluating renal functionrdquoAmerican Journalof Physiology-Renal Physiology vol 310 no 2 pp F174ndashF1822016

[17] C Langkammer F Schweser K Shmueli et al ldquoQuantitativesusceptibility mapping Report from the 2016 reconstructionchallengerdquo Magnetic Resonance in Medicine vol 79 no 3 pp1661ndash1673 2018

[18] L Bao X Li C Cai Z Chen and P C M Van ZijlldquoQuantitative susceptibility mapping using structural featurebased collaborative reconstruction (SFCR) in the human brainrdquoIEEE Transactions on Medical Imaging vol 35 no 9 pp 2040ndash2050 2016

[19] Y Wang and T Liu ldquoQuantitative susceptibility mapping(QSM) decoding MRI data for a tissue magnetic biomarkerrdquoMagnetic Resonance in Medicine vol 73 no 1 pp 82ndash101 2015

[20] D Zhou J Cho J Zhang P Spincemaille and Y Wang ldquoSus-ceptibility underestimation in a high-susceptibility phantomDependence on imaging resolution magnitude contrast andother parametersrdquoMagnetic Resonance in Medicine vol 78 no3 pp 1080ndash1086 2017

[21] M C Langham J F Magland C L Epstein T F Floyd andF W Wehrli ldquoAccuracy and precision of MR blood oximetrybased on the long paramagnetic cylinder approximation of largevesselsrdquoMagnetic Resonance inMedicine vol 62 no 2 pp 333ndash340 2009

[22] M J Cronin N Wang K S Decker H Wei W-Z Zhuand C Liu ldquoExploring the origins of echo-time-dependent

quantitative susceptibility mapping (QSM) measurements inhealthy tissue and cerebral microbleedsrdquo NeuroImage vol 149pp 98ndash113 2017

[23] C-Y Hsieh Y-C N Cheng J Neelavalli E M Haacke andR J Stafford ldquoAn improved method for susceptibility andradius quantification of cylindrical objects fromMRIrdquoMagneticResonance Imaging vol 33 no 4 pp 420ndash436 2015

[24] H Xie Y-C N Cheng P Kokeny et al ldquoA quantitative study ofsusceptibility and additional frequency shift of three commonmaterials in MRIrdquoMagnetic Resonance in Medicine vol 76 no4 pp 1263ndash1269 2016

[25] J Li S Chang T Liu et al ldquoReducing the object orien-tation dependence of susceptibility effects in gradient echoMRI through quantitative susceptibility mappingrdquo MagneticResonance in Medicine vol 68 no 5 pp 1563ndash1569 2012

[26] E OlssonMRI-Based Quantification of Magnetic SusceptibilityAssessment of Measurement And Calculation Accuracy (Unpub-lished MSc Dissertation) Lund University Lund 2016

[27] R Cusack and N Papadakis ldquoNew robust 3-D phase unwrap-ping algorithms application to magnetic field mapping andundistorting echoplanar imagesrdquoNeuroImage vol 16 no 3 pp754ndash764 2002

[28] T Liu I Khalidov L de Rochefort et al ldquoA novel backgroundfield removal method for MRI using projection onto dipolefields (PDF)rdquo NMR in Biomedicine vol 24 no 9 pp 1129ndash11362011

[29] L de Rochefort T Liu B Kressler et al ldquoQuantitative suscep-tibility map reconstruction fromMR phase data using bayesianregularization validation and application to brain imagingrdquoMagnetic Resonance inMedicine vol 63 no 1 pp 194ndash206 2010

[30] D Zhou T Liu P Spincemaille andYWang ldquoBackground fieldremoval by solving the Laplacian boundary value problemrdquoNMR in Biomedicine vol 27 no 3 pp 312ndash319 2014

[31] H Sun and A H Wilman ldquoBackground field removal usingspherical mean value filtering and Tikhonov regularizationrdquoMagnetic Resonance in Medicine vol 71 no 3 pp 1151ndash11572014

[32] R Salomir B D De Senneville and C T W Moonen ldquoAfast calculation method for magnetic field inhomogeneity dueto an arbitrary distribution of bulk susceptibilityrdquo Concepts inMagnetic Resonance B vol 19 no 1 pp 26ndash34 2003

[33] J P Marques and R Bowtell ldquoApplication of a Fourier-basedmethod for rapid calculation of field inhomogeneity due to spa-tial variation of magnetic susceptibilityrdquo Concepts in MagneticResonance vol 25 no 1 pp 65ndash78 2005

[34] T Liu C Wisnieff M Lou W Chen P Spincemaille and YWang ldquoNonlinear formulation of the magnetic field to sourcerelationship for robust quantitative susceptibility mappingrdquoMagnetic Resonance in Medicine vol 69 no 2 pp 467ndash4762013

[35] J Liu T Liu L de Rochefort et al ldquoMorphology enableddipole inversion for quantitative susceptibility mapping usingstructural consistency between the magnitude image and thesusceptibility maprdquo NeuroImage vol 59 no 3 pp 2560ndash25682012

[36] T Liu J Liu L De Rochefort et al ldquoMorphology enabled dipoleinversion (MEDI) from a single-angle acquisition Comparisonwith COSMOS in human brain imagingrdquo Magnetic Resonancein Medicine vol 66 no 3 pp 777ndash783 2011

[37] J O Christoffersson L E Olsson and S Sjoberg ldquoNickel-doped agarose gel phantoms in MR imagingrdquo Acta Radiologicavol 32 no 5 pp 426ndash431 1991


[38] J A PopleWG Schneider andH J BernsteinHigh-ResolutionNuclearMagnetic ResonanceMcGraw-Hill BookCompany IncNew York NY USA 1959

[39] L De Rochefort R Brown M R Prince and Y Wang ldquoQuan-titative MR susceptibility mapping using piece-wise constantregularized inversion of themagnetic fieldrdquoMagnetic Resonancein Medicine vol 60 no 4 pp 1003ndash1009 2008

[40] Cornell MRI Research Lab 2018 httpweillcornelledumripagesqsmhtml

[41] B Kressler L de Rochefort T Liu P Spincemaille Q Jiangand YWang ldquoNonlinear regularization for per voxel estimationof magnetic susceptibility distributions from MRI field mapsrdquoIEEE Transactions on Medical Imaging vol 29 no 2 pp 273ndash281 2010

[42] A-C Fruytier J Magat F Colliez B Jordan G Cron and BGallez ldquoDynamic contrast-enhancedMRI in mice at high fieldEstimation of the arterial input function can be achieved byphase imagingrdquoMagnetic Resonance in Medicine vol 71 no 2pp 544ndash550 2014

[43] P G Ward A P Fan P Raniga et al ldquoImproved quantificationof cerebral vein oxygenation using partial volume correctionrdquoFrontiers in Neuroscience vol 11 p 89 2017

[44] A Karsa E Biondetti S Punwani and K Shmueli ldquoTheeffect of large slice thickness and spacing and low coverage onthe accuracy of susceptibility mappingrdquo in Proceedings of the24th Annual Meeting of the International Society for MagneticResonance in Medicine (ISMRM) p 1555 Singapore 2016

[45] A M Elkady H Sun and A H Wilman ldquoImportanceof extended spatial coverage for quantitative susceptibilitymapping of iron-rich deep gray matterrdquo Magnetic ResonanceImaging vol 34 no 4 pp 574ndash578 2016

[46] E Lind L Knutsson R Kampe F Stahlberg and R WirestamldquoAssessment of MRI contrast agent concentration by quantita-tive susceptibility mapping (QSM) application to estimation ofcerebral blood volume during steady staterdquoMagnetic ResonanceMaterials in Physics Biology and Medicine vol 30 no 6 pp555ndash566 2017

Stem Cells International

Hindawiwwwhindawicom Volume 2018


MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine


Behavioural Neurology

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom


useful in estimations of oxygen extraction fraction (OEF)[10] and cerebral metabolic rate of oxygen (CMRO

2) [11]

The change in susceptibility with deoxygenation can alsobe manifested for example in extravasated blood from anintracranial haemorrhage [12] Quantitativemeasurements ofthe susceptibility could also be potentially useful to determinethe concentration of an external MRI contrast agent (CA)Relaxivity-based CA quantification which is the currentlymost common approach is associated with several method-ological complications in for example perfusion and per-meability measurements using dynamic contrast-enhancedMRI (DCE-MRI) and dynamic susceptibility contrast MRI(DSC-MRI) [13] Hence more accurate CA concentrationquantification in vivo would indeed be beneficial and a fewexamples of dynamic contrast-enhanced QSM studies havebeen presented [14ndash16]

In QSM applications quantification of magnetic suscep-tibility in absolute terms is becoming increasingly importantand extensive validation is thus warranted A number ofQSMreconstruction tools exist and the process of systematicallycharacterizing differences in accuracy between algorithmshas recently been initiated by other groups [17ndash19] Inexperimental evaluations phantoms have the advantage ofoffering well-defined contents and geometries and constitutean important though not complete part of the validationprocess and the present investigation serves as a supplementto previous investigations related to the accuracy of phaseand susceptibility quantification [eg [20ndash25]] In the presentstudy previously described in preliminary terms by Olsson(unpublished report) [26] the QSM approach was evaluatedin gel phantoms with inserted cylinders containing knownconcentrations of gadolinium CA to establish whether theQSM method can deliver accurate results with respect toquantitative magnetic susceptibility values in absolute termsMeasured phase values and the corresponding magneticsusceptibility estimates calculated by an established QSMalgorithm were compared to values based on theoreti-cal relationships Various phantom designs and simulatedsusceptibility distributions as well as different parametersand settings in the measurements and in the susceptibilitycalculation were investigated in order to establish optimalsettings and important sources of error in the attempts toproduce accurate magnetic susceptibility maps

2 Theory

21 The Dipole Field and the Magic Angle A magneticmoment with magnitude 119898 pointing in the 119911 directionproduces a magnetic flux density component 119861

119911

119861119911prop 119898

1198893(3 cos2Θ minus 1) (1)

where d is the distance from m and Θ is the angle relative tothe z-axis The angle at which the factor (3 cos2Θ minus 1) equalszero is called the magic angle (ie approximately plusmn547∘or 180∘plusmn547∘) At the magic angle positions the magneticflux density component 119861

119911will be zero independently of the

magnitude of the magnetic moment

22 Phase Shift and Magnetic Field Variations in the localmagnetic field with position r lead to differences in MRI res-onance frequency and to subsequent phase-shift variationsand the MRI phase evolution 120601(r) is given by

120601 (119903) = 120596 (119903) sdot 119879119864 = 120574 sdot 119861 (119903) sdot 119879119864 (2)

where TE is the echo time In order for the measuredphase images to be useful unwrapping and filtering ofbackground field variations are required The unwrappingcan be accomplished by a region growing algorithm whichidentifies phase gradients that correspond to a difference bya multiple of 2120587 and subsequent addition or subtraction of2120587 [27] Filtering is needed because the unwrapped imageusually contains a remaining background phase gradientover the entire image This phase does not arise from thesusceptibility distribution inside the object but from forexample imperfect shimming or susceptibility sources out-side the imaging volume Projection onto Dipole Fields (PDF)[28 29] is one method for background field removal thatcompares magnetic fields generated from magnetic dipolesinside and outside a region of interest Other examples offiltering methods are Laplacian Boundary Value (LBV) [30]and Regularization Enabled Sophisticated Harmonic ArtefactReduction for Phase data (RESHARP) [31]

23 Cylindrical Objects The internal (in) and external (ex)magnetic field alterations ΔB caused by an infinitely longcylinder are given by the following analytical expressions

Δ119861119894119899

=Δ1205946

(3cos2120579 minus 1) 1198610

(3)

Δ119861119890119909

=Δ1205942

1198862

1205882sin2120579 cos 2120593119861

0(4)

whereΔ120594 is the difference in susceptibility between the insideand the outside of the cylinder 119886 is the radius of the cylinder120579 is the angle between the direction of the B

0field and the

cylinder axis and 120588 and 120593 are the cylindrical coordinatesdescribing a point at distance 120588 and at an angle 120593 relative to apoint at the centre of the cylinder

24 Magnetic Susceptibility and Magnetic Field For morecomplicated geometries or shapes the local field changecaused by the introduction of an object in the externalmagnetic field can be described more generally [32 33] andis often formulated as a convolution (denoted ldquootimesrdquo) of thearbitrary susceptibility distribution with a dipole field kernelie the corresponding phase is given by

Δ120601 (119903) = 120574 sdot 119879119864 sdot 3cos2120579 minus 1

41205871199033otimes 120594 (119903) (5)

where 119903 and 120579 are spherical coordinates and 120594 denotes themagnetic susceptibility The main idea of QSM is to extractthe susceptibility distribution according to (5) using theinformation of the local magnetic field from the measuredphase images However problems arise because the dipolekernel is zero at the magic angle A convolution in real space











11205941+ 11990121205942+ sdot sdot sdot 119901

119899120594119899 where 119901

119899is the
























Dipolefield


Magnitudeimage

Template





4 Results


0field So-



oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

0 20 40 60 80 100

Experimenteory

minus15

minus10

minus5

0

5

10

15

20

Δ

[∘]

[∘]

(c)

000

005

010Δ

[ppm

] 015

020

025

0 20 40 60 80 100


[∘]

(d)









Experimenteory


2 4 6 8 10 120


0

50

100

150

200

250

300

350

400

450ΔΦ

[∘]

(a)


minus02

0

02

04

06

08

1

12

Δ

[ppm

]

14

16

18

2



(b)













5 Discussion




eory

2 4 6 8 10 120Diameter [mm]

0

005

01

015

Δ

[ppm

]

02

025

03


108 mm5 mm

0

002

004

006

008

Δ

[ppm

]

01

012

014










14410460

40

0 50 100 150minus50

Sliceminus02

minus015minus01

minus0050

00501

01502

02503

035

ΔΦ

[rad

]

(a)

0

002

004

006

008

01

012

014

016

018

144104

40eory

60

Δ

[ppm

]

0 50 100 150minus50

Slice

(b)


40


0

002

004

006

008

01

012

014

016

018

02

60 80 1000 20

Δ

[ppm

]

[∘]






oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

B0

(c)

B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(d)

eorySimulation

000

005

010

015

020

025

Δ

[ppm

]

20 40 60 80 1000 [∘]

(e)

0

005

01

015

Δ

[ppm

]

02

025



(f)



B0

(a)

oplus B0

(b)

0 50 100 150


minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

minus025

minus015

minus005

005

015

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(d)

minus015

minus01

minus005

0

005

01

015

02

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

ΔΦ

[rad

]

minus015

minus01

minus005

0

005

01

015

02

025

(f)









6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





























11205941+ 11990121205942+ sdot sdot sdot 119901

119899120594119899 where 119901

119899is the
























Dipolefield


Magnitudeimage

Template





4 Results


0field So-



oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

0 20 40 60 80 100

Experimenteory

minus15

minus10

minus5

0

5

10

15

20

Δ

[∘]

[∘]

(c)

000

005

010Δ

[ppm

] 015

020

025

0 20 40 60 80 100


[∘]

(d)









Experimenteory


2 4 6 8 10 120


0

50

100

150

200

250

300

350

400

450ΔΦ

[∘]

(a)


minus02

0

02

04

06

08

1

12

Δ

[ppm

]

14

16

18

2



(b)













5 Discussion




eory

2 4 6 8 10 120Diameter [mm]

0

005

01

015

Δ

[ppm

]

02

025

03


108 mm5 mm

0

002

004

006

008

Δ

[ppm

]

01

012

014










14410460

40

0 50 100 150minus50

Sliceminus02

minus015minus01

minus0050

00501

01502

02503

035

ΔΦ

[rad

]

(a)

0

002

004

006

008

01

012

014

016

018

144104

40eory

60

Δ

[ppm

]

0 50 100 150minus50

Slice

(b)


40


0

002

004

006

008

01

012

014

016

018

02

60 80 1000 20

Δ

[ppm

]

[∘]






oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

B0

(c)

B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(d)

eorySimulation

000

005

010

015

020

025

Δ

[ppm

]

20 40 60 80 1000 [∘]

(e)

0

005

01

015

Δ

[ppm

]

02

025



(f)



B0

(a)

oplus B0

(b)

0 50 100 150


minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

minus025

minus015

minus005

005

015

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(d)

minus015

minus01

minus005

0

005

01

015

02

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

ΔΦ

[rad

]

minus015

minus01

minus005

0

005

01

015

02

025

(f)









6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






























Dipolefield


Magnitudeimage

Template





4 Results


0field So-



oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

0 20 40 60 80 100

Experimenteory

minus15

minus10

minus5

0

5

10

15

20

Δ

[∘]

[∘]

(c)

000

005

010Δ

[ppm

] 015

020

025

0 20 40 60 80 100


[∘]

(d)









Experimenteory


2 4 6 8 10 120


0

50

100

150

200

250

300

350

400

450ΔΦ

[∘]

(a)


minus02

0

02

04

06

08

1

12

Δ

[ppm

]

14

16

18

2



(b)













5 Discussion




eory

2 4 6 8 10 120Diameter [mm]

0

005

01

015

Δ

[ppm

]

02

025

03


108 mm5 mm

0

002

004

006

008

Δ

[ppm

]

01

012

014










14410460

40

0 50 100 150minus50

Sliceminus02

minus015minus01

minus0050

00501

01502

02503

035

ΔΦ

[rad

]

(a)

0

002

004

006

008

01

012

014

016

018

144104

40eory

60

Δ

[ppm

]

0 50 100 150minus50

Slice

(b)


40


0

002

004

006

008

01

012

014

016

018

02

60 80 1000 20

Δ

[ppm

]

[∘]






oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

B0

(c)

B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(d)

eorySimulation

000

005

010

015

020

025

Δ

[ppm

]

20 40 60 80 1000 [∘]

(e)

0

005

01

015

Δ

[ppm

]

02

025



(f)



B0

(a)

oplus B0

(b)

0 50 100 150


minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

minus025

minus015

minus005

005

015

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(d)

minus015

minus01

minus005

0

005

01

015

02

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

ΔΦ

[rad

]

minus015

minus01

minus005

0

005

01

015

02

025

(f)









6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

0 20 40 60 80 100

Experimenteory

minus15

minus10

minus5

0

5

10

15

20

Δ

[∘]

[∘]

(c)

000

005

010Δ

[ppm

] 015

020

025

0 20 40 60 80 100


[∘]

(d)









Experimenteory


2 4 6 8 10 120


0

50

100

150

200

250

300

350

400

450ΔΦ

[∘]

(a)


minus02

0

02

04

06

08

1

12

Δ

[ppm

]

14

16

18

2



(b)













5 Discussion




eory

2 4 6 8 10 120Diameter [mm]

0

005

01

015

Δ

[ppm

]

02

025

03


108 mm5 mm

0

002

004

006

008

Δ

[ppm

]

01

012

014










14410460

40

0 50 100 150minus50

Sliceminus02

minus015minus01

minus0050

00501

01502

02503

035

ΔΦ

[rad

]

(a)

0

002

004

006

008

01

012

014

016

018

144104

40eory

60

Δ

[ppm

]

0 50 100 150minus50

Slice

(b)


40


0

002

004

006

008

01

012

014

016

018

02

60 80 1000 20

Δ

[ppm

]

[∘]






oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

B0

(c)

B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(d)

eorySimulation

000

005

010

015

020

025

Δ

[ppm

]

20 40 60 80 1000 [∘]

(e)

0

005

01

015

Δ

[ppm

]

02

025



(f)



B0

(a)

oplus B0

(b)

0 50 100 150


minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

minus025

minus015

minus005

005

015

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(d)

minus015

minus01

minus005

0

005

01

015

02

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

ΔΦ

[rad

]

minus015

minus01

minus005

0

005

01

015

02

025

(f)









6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















Experimenteory


2 4 6 8 10 120


0

50

100

150

200

250

300

350

400

450ΔΦ

[∘]

(a)


minus02

0

02

04

06

08

1

12

Δ

[ppm

]

14

16

18

2



(b)













5 Discussion




eory

2 4 6 8 10 120Diameter [mm]

0

005

01

015

Δ

[ppm

]

02

025

03


108 mm5 mm

0

002

004

006

008

Δ

[ppm

]

01

012

014










14410460

40

0 50 100 150minus50

Sliceminus02

minus015minus01

minus0050

00501

01502

02503

035

ΔΦ

[rad

]

(a)

0

002

004

006

008

01

012

014

016

018

144104

40eory

60

Δ

[ppm

]

0 50 100 150minus50

Slice

(b)


40


0

002

004

006

008

01

012

014

016

018

02

60 80 1000 20

Δ

[ppm

]

[∘]






oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

B0

(c)

B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(d)

eorySimulation

000

005

010

015

020

025

Δ

[ppm

]

20 40 60 80 1000 [∘]

(e)

0

005

01

015

Δ

[ppm

]

02

025



(f)



B0

(a)

oplus B0

(b)

0 50 100 150


minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

minus025

minus015

minus005

005

015

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(d)

minus015

minus01

minus005

0

005

01

015

02

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

ΔΦ

[rad

]

minus015

minus01

minus005

0

005

01

015

02

025

(f)









6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















eory

2 4 6 8 10 120Diameter [mm]

0

005

01

015

Δ

[ppm

]

02

025

03


108 mm5 mm

0

002

004

006

008

Δ

[ppm

]

01

012

014










14410460

40

0 50 100 150minus50

Sliceminus02

minus015minus01

minus0050

00501

01502

02503

035

ΔΦ

[rad

]

(a)

0

002

004

006

008

01

012

014

016

018

144104

40eory

60

Δ

[ppm

]

0 50 100 150minus50

Slice

(b)


40


0

002

004

006

008

01

012

014

016

018

02

60 80 1000 20

Δ

[ppm

]

[∘]






oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

B0

(c)

B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(d)

eorySimulation

000

005

010

015

020

025

Δ

[ppm

]

20 40 60 80 1000 [∘]

(e)

0

005

01

015

Δ

[ppm

]

02

025



(f)



B0

(a)

oplus B0

(b)

0 50 100 150


minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

minus025

minus015

minus005

005

015

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(d)

minus015

minus01

minus005

0

005

01

015

02

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

ΔΦ

[rad

]

minus015

minus01

minus005

0

005

01

015

02

025

(f)









6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















14410460

40

0 50 100 150minus50

Sliceminus02

minus015minus01

minus0050

00501

01502

02503

035

ΔΦ

[rad

]

(a)

0

002

004

006

008

01

012

014

016

018

144104

40eory

60

Δ

[ppm

]

0 50 100 150minus50

Slice

(b)


40


0

002

004

006

008

01

012

014

016

018

02

60 80 1000 20

Δ

[ppm

]

[∘]






oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

B0

(c)

B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(d)

eorySimulation

000

005

010

015

020

025

Δ

[ppm

]

20 40 60 80 1000 [∘]

(e)

0

005

01

015

Δ

[ppm

]

02

025



(f)



B0

(a)

oplus B0

(b)

0 50 100 150


minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

minus025

minus015

minus005

005

015

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(d)

minus015

minus01

minus005

0

005

01

015

02

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

ΔΦ

[rad

]

minus015

minus01

minus005

0

005

01

015

02

025

(f)









6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















oplus B0

(a)

oplus B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(b)

B0

(c)

B0

02

015

01

005

0

minus005

minus01

minus015

minus02

[ppm]

(d)

eorySimulation

000

005

010

015

020

025

Δ

[ppm

]

20 40 60 80 1000 [∘]

(e)

0

005

01

015

Δ

[ppm

]

02

025



(f)



B0

(a)

oplus B0

(b)

0 50 100 150


minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

minus025

minus015

minus005

005

015

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(d)

minus015

minus01

minus005

0

005

01

015

02

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

ΔΦ

[rad

]

minus015

minus01

minus005

0

005

01

015

02

025

(f)









6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















B0

(a)

oplus B0

(b)

0 50 100 150


minus015

minus005

ΔΦ

[rad

]

005

015

025

(c)

minus025

minus015

minus005

005

015

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(d)

minus015

minus01

minus005

0

005

01

015

02

025

0 50 100 150

ΔΦ

[rad

]

Distance [mm]

(e)

0 50 100 150

Distance [mm]

ΔΦ

[rad

]

minus015

minus01

minus005

0

005

01

015

02

025

(f)









6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























6 Conclusions



Data Availability


Disclosure




Acknowledgments


References
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















MRI-Based Quantification of Magnetic Susceptibility in Gel...

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Transcript of MRI-Based Quantification of Magnetic Susceptibility in Gel...