FULL PAPER

Efficient Concomitant and Remanence Field ArtifactReduction in Ultra-Low-Field MRI Using aFrequency-Space Formulation

Yi-Cheng Hsu,1,2 Panu T. Vesanen,2 Jaakko O. Nieminen,2 Koos C.J. Zevenhoven,2

Juhani Dabek,2 Lauri Parkkonen,2 I-Liang Chern,1 Risto J. Ilmoniemi,2 and

Fa-Hsuan Lin2,3*

Purpose: For ultra-low-field MRI, the spatial-encoding mag-netic fields generated by gradient coils can have strong con-comitant fields leading to prominent image distortion.

Additionally, using superconducting magnet to pre-polarizemagnetization can improve the signal-to-noise ratio of ultra-low-field MRI. Yet the spatially inhomogeneous remanence

field due to the permanently trapped flux inside a supercon-ducting pre-polarizing coil modulates magnetization and

causes further image distortion.Method: We propose a two-stage frequency–space (f–x) for-mulation to accurately describe the dynamics of spatially-

encoded magnetization under the influence of concomitantand remanence fields, which allows for correcting image dis-

tortion due to concomitant and remanence fields.Results: Our method is computationally efficient as it uses acombination of the fast Fourier transform algorithm and a lin-

ear equation solver. With sufficiently dense discretization insolving the linear equation, the performance of this f–x method

was found to be stable among different choices of the regula-rization parameter and the regularization matrix.Conclusion: We present this method together with numerical

simulations and experimental data to demonstrate how con-comitant and remanence field artifacts in ultra-low-field MRIcan be corrected efficiently. Magn Reson Med 000:000–000,2013. VC 2013 Wiley Periodicals, Inc.

Key words: low field; distortion; concomitant field; remanencefield; time domain reconstruction

INTRODUCTION

In contrast to high-field MRI, ultra-low-field (ULF) MRIuses a magnetic field in the microtesla range for magnet-ization precession (1). To increase the signal-to-noise ra-tio (SNR), the pre-polarization technique (2) has beenused to magnetize first the sample in a stronger millite-sla-range field and then to allow magnetization preces-sion in the microtesla range for signal detection. Inaddition, superconducting quantum interference devices(SQUIDs) have been used to detect the weak precessingmagnetization (for review, see Ref. 1)). Acquiring MRI inthe microtesla range gives ULF MRI several advantages:acoustically silent acquisition, low projectile danger andthus safe operation, imaging compatibility with metalobjects, and an MRI system with open access. In addi-tion, it has been suggested that the T1 contrast in healthy(3) and malignant (4) tissues is higher in low magneticfields than in the tesla range. Finally, it has been demon-strated that magnetoencephalography can be combinedwith ULF MRI in a single hybrid system (4,5).

A technical challenge of ULF MRI is the concomitant-field effect (4,6–8), which is typically ignored in high-field MRI except in some special cases (such asphase-contrast MR (9), echo planar imaging (10,11), spi-ral scan (12), balanced steady-state free precession (13),and fast spin echo imaging (14)). According to Maxwell’sequations, the curl and divergence of a source-free staticmagnetic field are zero. Thus, a uniform magnetic fieldgradient in one direction must be accompanied with atleast one other magnetic field gradient in a differentdirection. In high-field MRI with a strong B0, this con-comitant field is usually neglected, because B0 is muchstronger than the concomitant field. However, in ULFMRI, B0 is often of the same order of magnitude as themaximum of the gradient fields used for spatial encod-ing. Thus, artifacts, including blurring and distortion,appear if image reconstruction only takes linear gradientsinto consideration.

Concomitant field artifacts can be reduced by usingcustomized MRI hardware or pulse sequences (15–18).However, it can be more convenient to use image recon-struction algorithms to correct for these artifacts. Forexample, when the concomitant field is large only alongthe frequency encoding direction, the distortion artifactscan be corrected by providing the image reconstructionthe correct local precession frequencies (19). When theconcomitant field is non-negligible in both frequency

1Department of Mathematics, National Taiwan University, Taipei, Taiwan.2Department of Biomedical Engineering and Computational Science, AaltoUniversity School of Science, Espoo, Finland.3Institute of Biomedical Engineering, National Taiwan University, Taipei,Taiwan.

Grant sponsor: National Science Council, Taiwan (NSC); Grant number:100–2917-I-564-027; Grant sponsor: National Science Council, Taiwan(NSC); Grant number: 101–2628-B-002–005-MY3; Grant sponsor: Ministryof Economic Affairs, Taiwan; Grant number: 100-EC-17-A-19-S1–175; Grantsponsor: National Health Research Institute, Taiwan; Grant number: NHRI-EX102–10247EI; Grant sponsor: Academy of Finland (the FiDiPro program)and the European Community’s Seventh Framework Programme (FP7/2007–2013); Grant number: 200859.

*Correspondence to: Fa-Hsuan Lin, Ph.D., Institute of Biomedical Engineer-ing, National Taiwan University, 1, Sec. 4, Roosevelt Road, Taipei, 106, Tai-wan. E-mail: fhlin@ntu.edu.tw

Received 17 August 2012; revised 28 February 2013; accepted 8 March2013

DOI 10.1002/mrm.24745Published online in Wiley Online Library(wileyonlinelibrary.com).

Magnetic Resonance in Medicine 00:000–000 (2013)

VC 2013 Wiley Periodicals, Inc. 1

and phase encoding directions, the associated blurringand distortion artifacts can be reduced after carefully cal-ibrating the local magnetization phase and the corre-sponding precession frequency (20). However, bothimage processing methods cannot reduce the concomi-tant field artifacts satisfactorily when the concomitantfield is strong (relative to B0) because the irregular phaseencoding causes the encoded phase variation rate spreadinto a wide bandwidth. One way to reconstruct imagesusing measurements under a strong concomitant field isto carefully describe the relationship between thereceived signal and magnetization dynamics over spaceusing linear equations (21). However, without furthersimplification, the encoding matrix of such an approachis often too large to be inverted practically.

To compensate for the low native SNR of ULF MRI, aseparate strong polarization field has been used to boostthe magnetization (1). Recently, it has been demonstratedthat a superconducting magnet can be used to generate apre-polarizing field (5). Benefits of a superconductingmagnet are, e.g., its compact size and zero resistanceleading to negligible heat generation (5). However, theuse of a superconducting pre-polarization magnet alsocauses artifacts due to a remanence field, which is themagnetic field generated by permanently trapped fluxinside the superconducting pre-polarizing coil. Thus,even after turning off the pre-polarizing magnet, the spa-tially inhomogeneous remanence field still exists andmodulates the magnetization dynamics. To our knowl-edge, correction of this kind of a distortion has so far notbeen addressed in the ULF MRI image reconstruction.

A potential advantage of ULF MRI is to produce quan-titative measurements. For example, susceptibility, con-ductivity-, and permittivity-related artifacts are greatlyreduced in the ULF range. However, to take advantage ofthese quantitative measurements in practice, imagedistortion arising from the concomitant and remanencefields must be corrected. Furthermore, ULF MRI holdsthe promise of measuring magnetoencephalography andMRI using the same device and thereby improvingcoregistration between magnetoencephalography andMRI (22). However, this also requires distortion-freeULF MRI.

Here, we propose a two-stage frequency-encoding real-space (f–x) formulation to accurately describe thedynamics of the spatially-encoded magnetization. Recon-structing images using data described by the f–x formula-tion can correct distortion due to concomitant andremanence fields, because these disturbances are incor-porated in the formulation explicitly. Compared with thefull time-domain reconstruction (TDR) (21), the complex-ity of this f–x formulation is lower. We present thismethod together with numerical simulations and experi-mental data to demonstrate how concomitant and rema-nence field artifacts in ULF MRI can be correctedefficiently.

THEORY

In our ULF-MRI setup, the total magnetic field b(r) expe-rienced by the magnetization is

b rð Þ ¼ bideal rð Þ þ bcon rð Þ þ brem rð Þ; [1]

with the ideal magnetic field

bideal rð Þ ¼ B0 þ gTr� �

ez; [2]

where B0 is the main field and g¼ [gx gy gz]T is the ideal

linear magnetic field gradient for spatial encoding. Theconcomitant field is approximately

bcon rð Þ ¼ –gzx=2þ gxzð Þex þ –gzy=2þ gyz� �

ey ; [3]

where ex, ey, and ez are orthogonal unit vectors in theCartesian coordinate system. brem(r) denotes thespatial distribution of the remanence field. The strengthof the concomitant-field effects depends on the ratiobetween |bideal(r)| and |bcon(r)|. Therefore, e¼ (gmax�FOV)/B0 is used to quantify the effect of the concomitantfield (20,21), where gmax is the maximal gradient strengthused during the pulse sequence and FOV is the sidelength of a square centered at the origin, wherebcon(r)¼ 0, which enclose the field of view (FOV). Typi-cally, ULF MRI has 0.1< e<1 (19,20).

To better understand how concomitant field can leadto image distortion, we consider the magnetization atlocation (0, y, z) in a two-dimensional spin-echo pulsesequence (Fig. 1). We first discuss the precession of mag-netization in frequency encoding. The magnetizationwill experience the magnetic field

Bx ¼ 0By ¼ �gzy=2Bz ¼ B0 þ gzz

and precess at frequency

v ¼ g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB0 þ gzzð Þ2 þ gzy=2ð Þ2

q; [4]

higher than the ideal situation, where the concomitantfield is neglected (gzy ¼ 0). Such a deviation from theresonance frequency g(B0þ gzz) will cause mis-localiza-tion of the magnetization since its physical precessionfrequency is higher.

FIG. 1. The spin-echo pulse sequence used in the simulation andexperiments. The B0 field and the frequency-encoding gradient

were constantly on. The polarizing time Tp was 3 s and the echotime TE was 121 ms.

2 Hsu et al.

Then we consider the phase encoding process. Duringthe ith phase encoding step (1� i�Nph), the magnetiza-tion at location (0, y, z) experiences a magnetic field:

Bx ¼ 0By ¼ �gzy=2þ gy

max z � i �Nph=2� 1� �

= N=2ð ÞBz ¼ B0 þ gzz þ gy

max y � i �Nph=2� 1� �

= Nph=2� �

;[5]

which indicate that (1) most of the time, the precessionaxis is different from the precession axis of frequencyencoding, (2) the precession axis varies at different phaseencoding steps, and (3) the precession frequency betweenadjacent phase-encoding does not increase at constantrate. These effect will together cause image blurringif Fourier transform is directly used for imagereconstruction.

Given the spatial distribution of the total magneticfield, we attempt to describe accurately the dynamics ofmagnetization precession during the read-out of an N-step phase-encoded ULF-MRI experiment (1�n�N). Wefirst define a unit vector em(r, n) as the direction of themagnetization m(r,n) ¼ r(r)em(r, n) at the beginning ofthe read-out after the nth phase encoding step, wherer(r) denotes the net polarized spin density at location r.Furthermore, let c(r) denotes the vector of the B1

field sensitivity of a pick-up coil at location r. Thedetected AC signal s(t,n) at a time t after the nth phaseencoding is

s t;nð Þ ¼Z

V

r rð Þjem? r;nð Þjcos gjbreadout rð Þjt þ f r;nð Þð Þjc? rð Þjdr;

[6]

where |em?(r, n)| and |c?(r)| are the magnitudes ofem(r, n) and c(r) perpendicular to the total magnetic fieldduring readout breadout(r), which includes the ideal spa-tial encoding field, concomitant field, and the remanencefield. g is the gyromagnetic ratio and f(r,n) is the

accrued phase from the perpendicular component of themagnetization em?(r, n) by the end of the phase encodingstep n and the perpendicular component of the coil sen-sitivity c?(r). Note that because of concomitant fields,the magnetization precession axis in phase encoding isnot in parallel with the magnetization precession axis infrequency encoding. Consequently the evolution of thephase of magnetization becomes complicated in phaseencoding and frequency encoding. In other words,although f(r,n) is a phase of the magnetization, it is dif-ferent from the definition of the phase in high-field MRIduring phase encoding. Figure 2 shows these vectors.

Given |breadout(r)|, we can generate a map of iso-frequencies, each one of which delineates a curvilinearsurface (lines in two-dimensional imaging) of magnetiza-tions precessing at identical frequencies (Fig. 3, upperleft panel). The maximum, minimum, and the spacing ofthese iso-frequency curves depends on the upper/lowerbounds of the |breadout(r)| within the FOV and the dura-tion of the data acquisition.

For simplicity, we consider MRI measurements con-sisting of steps of phase-encoded readouts. Thus, our for-mulation can be applied to either two-dimensional orthree-dimensional imaging. Provided with (i) a map oftotal magnetic field strength during readout, and (ii) avector map of total magnetic field during each phaseencoding step, the f–x formulation consists of two stages:first, at each phase-encoding step, we apply fast Fouriertransform to s(t,n) to estimate amplitude and phases ofeach discretized frequency (the “f” step). At each discre-tized frequency, this complex-valued signal is the

FIG. 2. Directions of the magnetization at the beginning of the

read-out, the total magnetic field during the read-out, and the fieldsensitivity of a pick-up coil.

FIG. 3. Flowchart of the f–x reconstruction method. In upper leftpanel, the map of the iso-frequency curves during the signal read-out. In lower left panel, the phase-encoding weightings

jem? r;nð Þjexp jf r;nð Þð Þjc? rð ÞjJ rð Þ along the chosen iso-frequencycurve (1800 Hz). These weightings across the phase-encodingscan be combined to form a matrix A. In upper right panel, the FFT

of the signal during read-out after the various phase encodings.The complex signals at the chosen frequency (1800 Hz) across

the phase-encodings can be used to form a signal vector s. Thematrix A and the signal s are related by s¼Aq, where q is thevector of spin-densities along the chosen iso-frequency curve.

This procedure is repeated to cycle through all the iso-frequencycurves.

Concomitant and Remanence Field Artifacts in ULF MRI 3

integral of the unknown magnetization distribution (pre-cessing at the same frequency) along an iso-frequencycurve weighted by a spatial function determined by thetotal magnetic field during the phase encoding step (Fig.3, upper right panel). Second, at each discretized fre-quency, complex-valued signals across the phase-encod-ing steps are collected. Each signal is the spatial integralof the product between the unknown magnetizations anda spatial weighting modulated by the total magnetic fieldduring one phase encoding step (Fig. 3, lower left panel).Subsequently, we solve a linear equation consisting ofthese signals and their associated spatial weightings (Fig.3, lower right part) to reveal the spatial distribution ofmagnetization along each discretized frequency sepa-rately (the “x” step).

Specifically, the Fourier transform of s(t,n) is

s v;nð Þ¼Z

g jbreadout rð Þj¼v

r rð Þjem? r;nð Þjexp jf r;nð Þð Þjc? rð ÞjJ rð Þdl;

[7]

Here, we describe the signal as the spatial integral ofthe magnetization with a particular precession frequencyg|breadout(r)|¼v spatially weighted by jem? r;nð Þjexpjf r;nð Þð Þjc? rð ÞjJ rð Þ, where J(r)¼1/|!(g|breadout(r)|)| is

the Jacobian corresponding to the changes between thespatial coordinate r and the frequency coordinate v.

At each frequency v, there are N phase-encoded meas-urements. Thus, Eq. [6] can be converted into a discretelinear equation Aq¼ s. Each row of A is the associatedspatial weighting jem? r;nð Þjexp jf r;nð Þð Þjc? rð ÞjJ rð Þ at aphase encoding step n. s is a vector of the measuredcomplex-valued signal for frequency v.

Conventional MRI, neglecting concomitant and rema-nence fields, typically uses N measurements to estimatethe spatial distribution of the magnetization at Nequally-spaced locations when the maximum, minimum,the increment, and the duration of the phase-encodinggradient satisfies the Nyquist theorem. Considering con-comitant and remanence fields, in practice, it is still pos-sible to estimate q at N equispaced locations (withrespect to an orthogonal rectilinear Cartesian coordinatesystem) from N measurements. Yet, as demonstrated bythe following simulations, we found that each iso-fre-quency curve in the image domain has to be discretizedinto Np, Np� 2N, equally-spaced locations in order toaccurately capture the distribution of magnetizationphase due to the inhomogeneous phase encoding causedby the concomitant and remanence fields. However, solv-ing for 2N unknowns using N measurements is mathe-matically ill-posed. Thus, we propose to solve theunderdetermined equation Aq ¼ s by imposing a regula-rization constraint to minimize the ‘2-norm of Dq.D could be an identity matrix or a discrete difference op-erator taking the difference between neighboring pixels.The regularization using a discrete difference operatorfavors spatially smooth signals. Such a preference stemsfrom the fact that typically N phase-encoding steps pro-vide spatial weightings of the N lowest frequency modes.Therefore, by choosing a suitable regularization

parameter l, q can be estimated by minimizing the fol-lowing cost function:

jjAq� sjj2 þ ls AHA� �

jjDqjj2 [8]

Here r(AHA) denotes the largest eigenvalue of AHA.For the simulations and experiments in this paper with awell-behaving A, we chose l< 10. Eq. [8] is solved repet-itively for different v to sweep over the precession fre-quencies determined by the FOV, the strength of theread-out gradient, and the duration of data acquisition.Finally, the reconstructed image q is interpolated over atwo-dimensional or three-dimensional rectilinear imagegrid.

METHODS

Simulations

For the simulations, we calculated the magnetic fieldsgenerated by two gradient coils with the following fieldpatterns: (gyz)eyþ (gyy)ez and (–gzx/2)exþ (–gzy/2)eyþ(gzz)ez. The coordinate origin was set at the center of theFOV. To solely focus on the concomitant field artifact, weset c(r)¼ (1, 0, 0) in the simulation. The concomitantfields were simulated by setting e¼0, 1, and 2, correspondto main field B0¼1, 64 and 32 mT with gradient strength256 mT/m. A digital human brain phantom was con-structed from a high-resolution three-dimensional T1-weighted structural MRI data set acquired at 3T (Tim Trio,Siemens Medical Solutions, Erlangen, Germany). Thepulse sequence used to acquire this data was a standardMPRAGE (TR/TE/flip¼2530 ms/3.49 ms/7

�, partition

thickness¼1.0 mm, matrix¼ 256� 256, 256 partitions,FOV¼ 256� 256 mm2). To accurately simulate the spatialintegral describing the ULF-MRI signal (Eq. [2]), we simu-lated s(t,n) using a two-dimensional spin-echo sequencewith 128 phase encoding steps and 256 sampled timepoints by numerical integration over a two-dimensionalspatial grid of 4096� 4096 voxels. To approximate realis-tic measurements, Gaussian noise was added to the simu-lated signal such that the SNR, which was defined as theratio between the standard deviation of the signal and thestandard deviation of the noise, was 5.

To demonstrate the effectiveness of f–x reconstruction,we compare our method with previously proposed con-comitant field correction algorithm. The first is the post-acquisition, pre-reconstruction correction algorithm pro-posed by Myers et al (20). We divided image domaininto 32� 32 regions and combine the corrected image ineach region to form the final image. The second is theTDR method (21), we reconstruct the signal with128� 128 grid points and use properly chosen regulari-zation parameter.

In addition, we used simulations to study the requirednumber of discretization points Np in order to accuratelycapture the distribution of magnetization phases due toconcomitant and remanence fields described by Eq. [3].Specifically, we reconstructed images with Np¼N and2N separately. We also studied the effect of the regulari-zation parameter l with values 10�5, 10�3, 10�1, 10. Wefurther studied the effect of the matrix D in Eq. [8] using

4 Hsu et al.

either an identity matrix or a difference operator takingthe difference between image voxels to understand theeffect on the reconstructed image.

Finally, the reconstructed images were linearly inter-polated over 512�512 image voxels based on Delaunaytriangulation using Matlab (Mathworks, Natick, MA).

Experiments

Experimental data were acquired with our customizedsystem (5), which has a coil for magnetization precessionB0¼ 54.7 mT, three gradient coils, an excitation coil, anda polarizing coil (Bp¼ 24 mT). The details of this proto-type system are described in (5). The data were acquiredfrom a spin-echo sequence in Figure 1. The readoutlength was 107 ms with a constant gradient strengthgz¼85 mT/m driven by a current of 2390 mA. Phaseencoding was achieved by 58 steps with the maximalmagnitude gx¼ 121 mT/m and duration 38 ms with a6-ms ramp time. The nominal resolution was 2.6� 2.8mm2 and the corresponding strengths of the con-comitant field effect in our experiment were e¼0.57.The imaging time was 93 minutes including a 30-foldaveraging.

Remanence Field and Concomitant Field Measurements

We first measured the x-, y-, and z-components of theremanence fields over the 100� 82 mm2 FOV by using afluxgate magnetometer (MAG03-MC, Bartington Instru-ments Ltd., Oxford, England) positioned in a grid com-prised of LEGO bricks (LEGO Systems, Billund,Denmark) resulting in a spatial resolution of 8� 8 mm2.

The sum of the gradient (z-directional component) andconcomitant fields (x- and y-directional components) ofthe frequency and phase encoding coils were then meas-ured separately using the same fluxgate magnetometer atsteps of 16 mm. Since the remanence field was measuredseparately, the pure concomitant field pattern wasderived empirically from the difference between thesetwo measurements. To suppress noise, both frequencyencoding and concomitant fields were linearly fitted andscaled according to the strengths used in the experimen-tal sequence.

The f–x Formulation

To compare the computational efficiency, we simulatedand reconstructed 15 square images of n�n pixels,where n was incremented from 8 to 120 in steps of 8using our method and a direct time-domain reconstruc-tion (TDR) algorithm (21). All images were reconstructedon the same computer (Quad-core CPU at 2.83 GHz,Intel, Santa Clara, CA; 8 Gbytes memory) using Matlab.

Given the empirically measured magnetic field data,we first separately interpolated the remanence field, con-comitant field, phase- and frequency-encoding fields intothe 0.1� 0.1 mm2 resolution using a cubic spline. Thefirst “f” stage requires a map of iso-frequency curves ofg|breadout(r)|. Twenty-nine such curves ranging between2211.9 Hz and 2472.1 Hz in steps of 9.29 Hz were con-structed by identifying the nearest frequency over thefield maps of 0.1�0.1 mm2 resolution.

The second “x” stage requires jem? r;nð Þjexp jf r;nð Þð Þjc? rð ÞjJ rð Þ on 334 discretization points along each iso-frequency curve. The spatial distributions of jem? r;nð Þjat each step of the phase encoding were derived frombreadout(r) and the magnetization at the beginning of thereadout after the nth phase encoding based on the wave-forms of the phase-encoding step and the estimated con-comitant and remanence fields. In theory, we need amap of the coil sensitivity c(r) to calculate f(r,n). How-ever, without knowing the actual c(r), we only estimatedthe incremental phase Df(r,n) in each phase-encodingstep: f(r,n)¼f0(r)þDf(r,n). In practice, Df(r,n) wasestimated from the angle between em? r;nð Þ andem? r;n ¼ nPEð Þ, where nPE represents the phase-encodingstep without any current in the phase-encoding coil.

Taken together, without further information on coilsensitivity c(r), in this study, we only reconstructed animage of r rð Þexp jf0 rð Þð Þjc? rð Þj. Finally, we linearly inter-polated the reconstructed magnetization distributedalong Np discretization points on each iso-frequencycurve into 274�334 image voxels using Delaunay trian-gulation by Matlab.

RESULTS

Computational Time and Complexity

We first studied the computational complexity of ourproposed method compared to conventional 2D fast Fou-rier transform algorithm and direct time-domain method(21). For an image of m-by-n pixels (thus spatiallyencoded by n phase-encoding steps and m frequency-encoded samples), the fast Fourier transform algorithmwith the complexity of O(m logm) has to be repeated ntimes (across n phase encoding steps) in the first “f”stage. This accounts for the complexity of O(n m logm).Subsequently, we need to solve m sets (across differentiso-frequency curves) of n linear equations, each set ofwhich amounts to the complexity of O(n3). The second“x” stage thus has the complexity of O(m n3). Taken to-gether, since the latter “x” stage dominates the computa-tional load, the total complexity of our method is O(mn3), which is much improved compared to the TDRmethod with computational complexity of O(m3 n3). Fig-ure 4 shows the empirical computational time for differ-ent image matrix sizes. We extrapolated the computationtime required for larger image matrices by linearly fittingthe computation time for data with n� 48. The ratiobetween slopes using TDR and the f–x method was 1.6,which approximates to the O(n2) difference (ratiobetween slopes¼1.5) between the two methods. Thisclearly shows the computational efficiency of the f–xhybrid method, especially at a large image matrix size.

Figure 5A–D shows the reconstructed simulationimages of the human brain phantom including the rema-nence field and the concomitant field with e¼ 0, 1, and2 using Fourier transform, post-acquisition pre-recon-struction method (20), TDR method, and the f–x hybridmethod with Np¼ 256, D as a difference operator and l¼10�2 (see Eq. [8]). At e ¼ 1, the Fourier-reconstructedimage shows compressed frontal lobes and inflated occi-pital lobes along the left-right direction due to the con-comitant fields (Fig. 5E). Such artifacts become even

Concomitant and Remanence Field Artifacts in ULF MRI 5

stronger at e¼ 2. The post-acquisition pre-reconstructionmethod reconstructed images without much distortionand blurring despite a little intensity drop at the lower

part of the image at e¼ 1. This artifact became more seri-ously at e¼ 2 (Fig. 5B). The TDR method reconstructedthe image without distortion or blurring at e¼ 1. At e¼ 2,even without noise, we observe that lower part of theimage became noisy. This is due to the fact that the rec-tangular grid used in the TDR method could not approxi-mate the signal equations well when the encoded spatialfrequencies are higher than expected. The f–x hybridmethod can reconstruct the image without much distor-tion. However at e¼ 2, prominent residuals were evidentin the lower left and right corners. This is likely due toaliasing. Note that the noise distribution is spatially in-homogeneous, most likely because of the spatially vary-ing conditioning of the encoding matrix A in Eq. [8]across the different iso-frequency curves. Images with astronger concomitant field (e¼ 2) show a higher noiselevel than in the case e¼1. Including the simulated rem-anence field (Fig. 5E) further distorted the image by com-pressing the right frontal lobe in the anterior-posteriordirection. Still, the f-x hybrid method reconstructed theimage without visible distortion, except that the aliasingartifacts appear at the lower left and right corners.

Figure 6 shows the simulated reconstructions using dif-ferent numbers of discretization points Np in evaluating

FIG. 4. The fitted (dashed lines) and the empirical (solid lines)

computational time for an image of n�n pixels using TDR (blueand cyan colors) and the f–x method (red and magenta colors).

FIG. 5. Brain-phantom images reconstructed using Fourier transform (a), method proposed by Myers et al. (b), TDR method (c), and thef–x method (d). The spatial distribution of the magnetic field during readout |breadout(r)| (e) with concomitant fields (e¼1 and e¼2), the

remanence field, and contaminating noise.

6 Hsu et al.

Eq. [7] with a concomitant field strength of e¼ 1. We cansee that when Np¼N without any regularization, thereconstructed image shows serious artifacts because (1)the discretization in Eq. [7] fails to represent the integra-tion accurately, and (2) the encoding matrix A in Eq. [8] isextremely ill-conditioned. Adding a regularization termcan mitigate the artifact arising from the latter reason.Still, at locations with a strong concomitant field (rightand left lower corners), we cannot describe the integrationaccurately using Np¼N. When Np is increased to 2N, theimage can be reconstructed with minimally visible resid-uals. Thus, all the following reconstructions in this workuse Np¼2N. Even higher N might be necessary in the caseof stronger distorting fields.

Figure 7 shows simulated reconstructions using differ-ent regularization parameters with SNR¼ 5. With l

between 10�1 and 10�3 the quality of the reconstructionswere similar, while less regularization (l¼ 10�3) resultedin an amplified background noise. This is reasonable,since a strong regularization parameter yields spatiallysmoother and thus less noisy images. Using l ¼ 10overly emphasized the prior information and thus thereconstructed image was blurred. When the regulariza-tion parameter was too small (l¼ 10�5), the conditioningof the encoding matrix A in Eq. [8] became unstable andthe reconstructed image shows significantly amplifiednoise. Generally, l ranging between 10�1 and 10�3 gavea stable reconstruction.

Figure 8 shows simulated reconstruction with l¼10�2

using either an identity matrix or a difference operator

taking the difference between neighboring pixels as thematrix D in Eq. [8]. Little difference between the recon-structions was found, except that the background noise atthe lower left and right corners was slightly amplifiedwhen the difference operator was used for D. This resultsuggests that the two choices for D only marginally modu-late the reconstruction with the chosen regularizationparameter.

The spatial distribution of the remanence field brem

and the strength of the total field with and without theremanence field brem are shown in Figure 9. The rema-nence field was found to be rather inhomogeneous forall magnetic field components. The magnitude of theremanence field was within 63 mT. The strength of thetotal field was maximally about 60 mT, much larger thanthe strength of the remanence field. However, the gradi-ent of brem is locally strong, which causes distortion ofthe iso-frequency lines.

Figure 10 shows the reconstructed images from fivechannels of the ULF-MRI system using the f–x method.The total computation time was less than 2 s. These fivechannels were closest to the grid phantom and they hadthe highest SNR. We reconstructed the image in two ways:using either only the z-component of brem and using bothbrem and bcon. This was done to compare our method tothe distortion correction methods in high-field MRI with afield map. Figure 8 also shows the sum-of-squares (SoS)image. Compared with the photo of the phantom, wefound that the image reconstructed by the Fourier trans-form readily shows prominent distortions structure.

FIG. 6. The reconstructed images

using the f–x method with a differ-ent number of discretization pointsNp with and without regularization

with concomitant fields e¼1.

FIG. 7. The images reconstructed

by the f–x method using differentregularization parameters (upperrow) with concomitant fields e¼1

and the magnified image around theleft temporal lobe (lower row).

Concomitant and Remanence Field Artifacts in ULF MRI 7

Specifically, the grid was dilated at the left side and com-pressed at the right side of the image. With only the effectof z-component of brem corrected, the distortion wasreduced but still noticeable. Provided with the completebrem and bcon, the f–x hybrid method corrected this distor-tion significantly.

DISCUSSION

This study presents a computationally efficient methodof reconstructing ultra-low-field MR images withreduced distortion caused by the concomitant and

remanence fields. With validations of the methodslargely done through simulations, we demonstrated thatthis method is capable of correcting distortions evenwith strong concomitant field artifacts (e¼ 2; Fig. 5),because our method describes the magnetization dynam-ics considering all the effects due to the remanence andconcomitant fields of three spatial components. Thismethod is computationally efficient as it uses a combina-tion of the fast Fourier transform algorithm and a linearequation solver (Fig. 4). With sufficiently dense discreti-zation in solving the linear equation (Fig. 6), the per-formance of this f–x method was found to be stable withdifferent choices of the regularization parameter (Fig. 7)and the regularization matrix (Fig. 8). Empirical datademonstrated that visible distortion was corrected effi-ciently (Fig. 10).

When the concomitant field effect is moderate (e� 1)(20), the artifact can be corrected by image post-process-ing using the Fourier reconstruction method after respec-tively correcting the local magnetization phases andfrequencies disturbed by the concomitant fields in phase-and frequency-encoding directions. Stronger concomitantfield artifacts can be corrected by the TDR method, whichdirectly relates the detected MRI signal to magnetizationdynamics under the influence of a stronger concomitantfield (21). However, the signal equation in the TDR canbe too large to be solved (see our computational complex-ity analysis). Another limitation of the TDR method isthat due its limited computational efficiency, the spatialintegration (Eq. [6]) is typically evaluated over np�nf

equidistant discretized spatial locations, where np and nf

are the number of phase and frequency encoding stepsrespectively. However, such an equidistant spatial gridmay not be able to capture the magnetization dynamicsaccurately considering the disturbance from concomitantand remanence fields.

FIG. 8. The images reconstructed by the f–x method using eithera linear operator taking the difference between neighboring image

voxels or an identity matrix in the regularization matrix with con-comitant fields e¼1. The lower row shows magnified imagesaround the left temporal lobe.

FIG. 9. a: The spatial distribution of the x-, y-, and z-component of the remanence field. b: The spatial distribution of the strength of the

concomitant field and the strength of the sum of the concomitant and remanence fields.

8 Hsu et al.

Our method can reconstruct MRI signals affected by astrong concomitant field (e>1; Fig. 5). Similar to TDR,our method also describes magnetization dynamics insignal generation. However, we use the Fourier transformin the frequency-encoding direction and thus it is compu-tationally more efficient. Furthermore, based on the spa-tial distribution of iso-frequency curves derived from thetotal magnetic field (including concomitant and rema-nence fields), our method can directly reveal the spatialdistribution of magnetization precessing at different fre-quencies. This avoids the necessity of evaluating the sig-nal integration (Eq. [6]) in the frequency-encodingdirection over equidistant discretized spatial locations asrequired in TDR. Along the phase-encoding direction, wemitigate the risk of inaccurate signal integration overequidistant spatial locations by using a two times denserequidistant spatial locations than what is required forTDR.

High-precision measurement of the total magnetic fieldis crucial in a successful image reconstruction usingTDR or the f–x method. This is because TDR requiresknowledge of the direction of the magnetization over theFOV. The f–x method also requires the knowledge of|em?|. Such information is only available when thelocal oscillator and analog-to-digital converter have hightemporal accuracy and the gradient coils generate high-SNR magnetic fields. However, here e was approximately0.6. When also considering the remanence field, our ex-perimental data had |em?|> 0.99. In this case, we used|em?|¼ 1 in the reconstruction.

Most distortion correction methods, including thoseapplied to high-field MRI, require some calibration infor-mation, such as a field map. Our method also needssuch calibration information: the spatial distribution ofconcomitant and remanence fields. Fortunately, such in-formation needs only to be measured once and it can beapplied to all subsequent measurements, when the fol-lowing conditions are met: (1) the concomitant fields,including all directional components, are accurately

measured with precise spatial localization, (2) the con-comitant fields are linearly scaled with the current driv-ing the gradient, (3) the total concomitant fieldsgenerated by all gradient coils are the summation of theconcomitant fields generated by each gradient coil, (4)the gradient system is stable over time, and (5) the core-gistration between measured concomitant fields and theimaging area is accurate such that imaging FOV can beaccurately derived within the FOV used for concomitantfield measurements. Ensuring the validity of these fiveassumptions are clearly the foundation of using the f–xmethod to reduce ULF MRI distortions.

The concomitant fields can be derived from imaginggradient, if the geometry of the gradient coils and thedriving current are known accurately. However, in prac-tice, it is more convenient to measure the concomitantfields directly.

It is well known that both concomitant and remanencefields can have a through-plane effect. Because of thestructure of our phantom, we only show images over oneplane with clearly reduced in-plane distortion. Thereforepeople may speculate that the f–x method is only usefulin multi-slice imaging. However, it is important to notethat, due to the limitation of gradient strength, volumetricULF MRI is typically achieved by partition encodingrather than by multiple slice-selection. Our method maybe able to correct through-plane distortion when concom-itant and remanence fields distribution over a volume isavailable. However, this hypothesis requires further ex-perimental evidence to support it. We will work on thisas the future development of the f–x method.

Our method can reduce the artifacts due to the con-comitant and remanence fields; however, the recon-structed image is not perfect (Figs. 5 and 10). This maybe explained by two potential reasons: first, the local re-solution along the frequency-encoding direction is deter-mined by the magnitude of the gradient of the magneticfield during the frequency encoding. Considering thetotal field generated by the frequency-encoding magnetic

FIG. 10. a: The reconstructed

images of five channels of theULF-MRI system and the sum-

of-squares (SoS) image using theFourier transform, f–x methodusing only the z-component of

the remanence field and the f–xmethod using both the rema-

nence and concomitant fields.The added blue grid may facili-tate observing the distortion. b:A photo of the phantom.

Concomitant and Remanence Field Artifacts in ULF MRI 9

field, concomitant field, and remanence field, it is possi-ble that the magnitude of the total magnetic field duringfrequency encoding at some locations in the FOV israther smooth. Thus, the spatial resolution becomesrather inhomogeneous over the FOV. Second, the con-comitant and remanence fields can disturb the desiredincremental phase-encoding gradient moments requiredfor spatial encoding. Aliasing-like and blurring artifactscan respectively occur at locations with a stronger orweaker magnetic field than required in the ideal phaseencoding.

Based on Eq. [2], the situation of multiple frequencysurfaces and spatial aliasing occurs when the gradientfield is so strong that the e>2. In such a situation, ourmethod using only a single receiver coil cannot recon-struct an image unambiguously. However, it is conceiva-ble that when there are multiple receivers with distinctsensitivities, our method may be able to resolve thisdifficulty.

This f–x method is similar to the method of recon-structing high-field MRI encoded by nonlinear spatiallyencoded magnetic fields (SEMs) (23). In this high-fieldacquisition, both phase and frequency encodings stillwork reasonably well such that two-dimensional Fouriertransform can be applied to generate an image over anon-Cartesian coordinate system. Subsequent interpola-tion is needed to restore the image over the Cartesiancoordinate system. However, when disturbed by concom-itant and remanence fields, we cannot apply the Fouriertransform along the phase-encoding direction, becausethe incremental gradient moment for each phase-encod-ing step is no longer constant. Thus we have to solve aset of linear equations (Eq. [8]) instead of using the Fou-rier transform.

While there are numerous methods (24,25) for correct-ing distortion artifacts in high-field MRI, the principaldifference is that these methods only account for the fielddisturbance along the main magnetic field direction.Magnetization is still considered precessing only aroundthe z axis during phase and frequency encoding. Differ-ently, in ULF MRI, magnetization can precess around anydirection of the axis during phase and frequency encod-ing. Figure 10 clearly demonstrates this difference: whenonly considering the disturbance on magnetization pre-cession over a constant plane like in high-field MRI, thereconstruction still shows prominent distortion.

The f–x method is indeed a distortion correction ontop of the Fourier transform. Importantly, the “Fouriertransform” in this context incorporates all physical dis-turbances generated by concomitant and remanencefields in order to generate a correct spatial distributionof magnetization. Our method is certainly not the tradi-tional fast Fourier transform (FFT) method and thus ittakes longer time than applying the FFT along bothphase and frequency encoding axes. To our understand-ing, there is no method of correcting distortion in ULFMRI generated by fields in both phase and frequencyaxes after sum-of-squares image reconstruction. Ourmethod is one approach aiming at mitigating this techni-cal challenge by integrating the Fourier transform anddistortion correct in the phase encoding direction in onestep.

The proposed method is not only limited to the spinecho pulse sequence, for example we can achieve phaseencoding by modulation the duration of the phase-encoding gradient with fixed amplitude. Although theaccrued phase is linearly increased in different phaseencoding steps when using a phase encoding gradientwith a fixed amplitude and linearly-stepped durations,two issues different from the conventional MRI preventus from using the Fourier transform to obtain distortion-free spatial information directly. First, because of con-comitant fields, the magnetization at the beginning ofphase encoding may not be orthogonal to the magneticfield. This means that only the orthogonal component ofmagnetization with respect to the magnetic field will beencoded among phase encoding steps, while the magnet-ization component in parallel to the magnetic field can-not be encoded. This parallel magnetization componentwill appear as signal at zero frequency after Fouriertransform along the phase encoding direction. The sec-ond issue is, because of concomitant fields, the magnet-ization precession axis in phase encoding is not inparallel with the magnetization precession axis in fre-quency encoding. In an exemplary problematic case,suppose at one particular location where these two pre-cession axes are perpendicular to each other and themagnetization aligns with the frequency encoding mag-netic field after phase encoding. Since the magnetizationhas no perpendicular component with respect to the fre-quency encoding field, we can no longer detect such DCsignal originating from that particular location. However,the above-mentioned two challenges can be mitigated byour method, because these physical phenomena have ex-plicitly modeled in the encoding matrix (Eq. [7]).

Although we did not demonstrate our method toachieve spatiotemporal resolution enhancement or theoptimal combination across channels using the SENSEmethod (26), we expect that these can be achieved if thesensitivity information is available in Eq. [7]. In ULFMRI, the SNR is typically too low to be traded-off forspatiotemporal resolution enhancement. We also expectthat our method can be used to reconstruct images usingdata collected using any k-space trajectory, in which themagnetic field during the read-out is constant. Thisincludes rectilinear Cartesian k-space sampling and a ra-dial k-space trajectory. Further studies are required todemonstrate the feasibility of our method in such gener-alized cases.

REFERENCES

1. Clarke J, Hatridge M, M€oßle M. SQUID-detected magnetic resonance

imaging in microtesla fields. Ann Rev Biomed Eng 2007;9:389–413.

2. Packard M, Varian R. Free nuclear induction in the earths magnetic

field. Phys Rev 1954;93:941–941.

3. Fischer HW, Rinck PA, Vanhaverbeke Y, Muller RN. Nuclear-

relaxation of human brain gray and white matter - analysis of field-

dependence and implications for MRI. Magnet Reson Med 1990;16:

317–334.

4. Busch S, Hatridge M, M€oßle M, Myers W, Wong T, M€uck M, Chew

K, Kuchinsky K, Simko J, Clarke J. Measurements of T1-relaxation

in ex vivo prostate tissue at 132 mT. Magn Reson Med 2012;67:

1138–1145.

5. Vesanen PT, Nieminen JO, Zevenhoven KC, et al. Hybrid ultra-low-

field MRI and MEG system based on a commercial whole-head neuro-

magnetometer. Magn Reson Med 2012. doi: 10.1002/mrm.24413.

10 Hsu et al.

6. Norris DG, Hutchison JMS. Concomitant magnetic-field gradients and

their effects on imaging at low magnetic-field strengths. Magn Reson

Imaging 1990;8:33–37.

7. Volegov PL, Mosher JC, Espy MA, Kraus RH Jr. On concomitant gra-

dients in low-field MRI. J Magn Reson 2005;175:103–113.

8. Yablonskiy DA, Sukstanskii AL, Ackerman JJH. Image artifacts in

very low magnetic field MRI: The role of concomitant gradients.

J Magn Reson 2005;174:279–286.

9. Bernstein MA, Zhou XJ, Polzin JA, King KF, Ganin A, Pelc NJ, Glover

GH. Concomitant gradient terms in phase contrast MR: analysis and

correction. Magn Reson Med 1998;39:300–308.

10. Zhou XJ, Du YP, Bernstein MA, Reynolds HG, Maier JK, Polzin JA.

Concomitant magnetic-field-induced artifacts in axial echo planar

imaging. Magn Reson Med 1998;39:596–605.

11. Du YP, Joe Zhou X, Bernstein MA. Correction of concomitant mag-

netic field-induced image artifacts in nonaxial echo-planar imaging.

Magn Reson Med 2002;48:509–515.

12. King KF, Ganin A, Zhou XJ, Bernstein MA. Concomitant gradient

field effects in spiral scans. Magn Reson Med 1999;41:103–112.

13. Sica CT, Meyer CH. Concomitant gradient field effects in balanced

steady-state free precession. Magn Reson Med 2007;57:721–730.

14. Zhou XJ, Tan SG, Bernstein MA. Artifacts induced by concomitant

magnetic field in fast spin-echo imaging. Magn Reson Med 1998;

40:582–591.

15. Meriles CA, Sakellariou D, Trabesinger AH, Demas V, Pines A. Zero-

to low-field MRI with averaging of concomitant gradient fields. Proc

Natl Acad Sci USA 2005;102:1840–1842.

16. Meriles CA, Sakellariou D, Trabesinger AH. Theory of MRI in the

presence of zero to low magnetic fields and tensor imaging field gra-

dients. J Magn Reson 2006;182:106–114.

17. Bouchard L-S. Unidirectional magnetic-field gradients and geometric-

phase errors during Fourier encoding using orthogonal ac fields.

Phys Rev B 2006;74:054103.

18. Kelso N, Lee S-K, Bouchard L-S, Demas V, M€uck M, Pines A, Clarke

J. Distortion-free magnetic resonance imaging in the zero-field limit.

J Magn Reson 2009;200:285–290.

19. Zotev VS, Volegov PL, Matlashov AN, Espy MA, Mosher JC, Kraus

RH. Parallel MRI at microtesla fields. J Magn Reson 2008;192:

197–208.

20. Myers WR, Mossle M, Clarke J. Correction of concomitant gradient

artifacts in experimental microtesla MRI. J Magn Reson 2005;177:

274–284.

21. Nieminen JO, Ilmoniemi RJ. Solving the problem of concomitant gra-

dients in ultra-low-field MRI. J Magn Reson 2010;207:213–219.

22. Magnelind PE, Gomez JJ, Matlashov AN, Owens T, Sandin JH, Vole-

gov PL, Espy MA. Co-registration of interleaved MEG and ULF MRI

using a 7 channel low-Tc SQUID system. IEEE Trans Appl Supercond

2011;21:456–460.

23. Schultz G, Ullmann P, Lehr H, Welz AM, Hennig J, Zaitsev M.

Reconstruction of MRI data encoded with arbitrarily shaped, curvilin-

ear, nonbijective magnetic fields. Magn Reson Med 2010;64:

1390–1404.

24. Doran SJ, Charles-Edwards L, Reinsberg SA, Leach MO. A complete

distortion correction for MR images: I. Gradient warp correction.

Phys Med Biol 2005;50:1343–1361.

25. O’Donnell M, Edelstein WA. NMR imaging in the presence of mag-

netic field inhomogeneities and gradient field nonlinearities. Med

Phys 1985;12:20–26.

26. Pruessmann KP, Weiger M, Scheidegger MB, Boesiger P. SENSE: sen-

sitivity encoding for fast MRI. Magn Reson Med 1999;42:952–962.

Concomitant and Remanence Field Artifacts in ULF MRI 11