MRI image Reconstruction Under Linear andHyperbolic gradients

XU Ya-jie, TIAN Hao-ran, ZHU Zhen-wei, ZHANG Guang-cai,CHANG Yan, JIANG Rui-rui, WANG Xiao-guan, YANG Xiao-dong*

Medical Imaging Department,SIBET, CAS,

SuZhou, the People’s Republic of ChinaEmail: xiaodong.yang@sibet.ac.cn

Abstract—Nonlinear gradient imaging has advantages of lowerperipheral nerve stimulation (PNS), fast imaging rate, and highedge resolution, but the blurring exists in the center part. Toresolve this problem, linear gradient x, y is employed together withnonlinear gradient, such as 4-Dimensional Radial In/Out (4DRIO),O-Space, etc. In this paper, we propose a hyperbola method, usingx, y gradient and hyperbolic nonlinear gradient x2 − y2. Numericalsimulation reveals that hyperbola sampling scheme and iterativealgorithm leads to high efficiency in image reconstruction, andhyperbola method shows better reconstruction compare with O-Space technology.

Index Terms—nonlinear gradient imaging, hyperbola method,sampling scheme, iterative algorithm

I. INTRODUCTION

Along with the continuous improvement of MRI equipmentfield strength, human subjects are suffering more PeripheralNerve Stimulation (PNS). However, human endurance is limited,it is very urgent to accelerate MRI imaging rate and improvehuman subjects comfort.

Hennig et al[3] proposed the Patloc (Parallel imaging tech-nique using Local gradients) technology, using nonlinear spatialencoding magnetic fields (SEMs), and multi-channel parallelacquisition to position image space. Patloc technology, integratingthe advantages of nonlinear gradient and phased array coilacquisition, is of lower PNS and faster imaging rate, comparingwith the conventional linear gradient. Nonlinear gradient parallelimaging has the following characteristics: 1) high edge resolution;2) blurring in the center part imaging, called “hole”. Resolutiondegradation in the center of the Field Of View (FOV) is onaccount of the low spatial derivative of the encoding fields[8].In addition, parallel phased array coils sensitivity is weak in thecenter for the symmetric structure, so that they do not provideadequate localization to make up for the defect of nonlineargradient. To resolve this problem, researchers have made variousattempts, mostly about the simultaneous usage of linear andnonlinear gradient.

Gallichan simultaneously applied the linear encoding andquadratic PatLoc SEM (x2 − y2, xy) following the radial tra-

jectories with the same angle of advance between spokes, andobtained an image improvement in the center part imaging,named 4-Dimensional Radial In/Out (4DRIO)[2]. Stockmann’sO-Space[8], [9], [7] technology employed the spherical har-monics Z2 order magnetic field component (z2 − 1

2 (x2 + y2))

and linear gradient to obtain axis planar imaging; by changingthe position of the Z2 center in the FOV, they resolved theblurring flaw in nonlinear gradient imaging center. But as theZ2 function has denoted, the magnetic field varies in threecoordinates, so that dephasing exists in three-dimensional (3D)encoding due to the through-plane evolution imparted by the z2

field variation[7]. However, nonlinear gradient x2−y2 only relatesto x, y coordinate, so it is more convenient to apply this gradientexperimentally than Z2 order magnetic gradient in O-space.

Furthermore, in the earlier research, shim coils were used togenerate the nonlinear gradient, now high performance PatLocgradient coils are available[10], [1], to generate “hyperbolicparaboloid”nonlinear gradient x2 − y2 by strong coil current,which has higher intensity than of the gradient of shim coils.So here we choose hyperbolic gradient field x2 − y2 to studyMRI nonlinear gradient imaging technology.

II. THEORY

According to Schultz[5], in a parallel imaging system MRIsignal can be written as

Sα(k) =

∫v

m(x)cα(x)eikTψ(x) dx (1)

in which α denotes the parallel coil, m(x) is the magnetizationover the region of interest v, cα denotes the αth coil sensitivityprofile, which is a vector of the same length as the number ofgradient encoding fields, k describes the net gradient moment ofeach field, and ψ(x) is a multi-dimensional function representingall the gradient encoding fields. Using x2 − y2 and x, y as theSEMs and B0 offset to complete the square[6], ψ(x) can bewritten as

ψ(x) = (x− xl)2 − (y − yl)

2 (2)

978-1-4673-5936-8/13/$31.00 ©2013 IEEE 139

Unlike the conventional encoding strategy, k only has readoutdirection, while the phase encoding is substituted by variedxl, yl values, center moving of x2 − y2 along with x, y gradientchanging.

III. METHODS

As mentioned above, center blurring defect exists in nonlineargradient imaging, due to the low gradient in symmetric center(refer to the symmetrical gradient), so x, y gradients are mainlyused to move nonlinear gradient center around. Aim of simulta-neous usage of nonlinear and linear gradients is to retain theedge advantage of x2 − y2 gradient, and to compensate thecenter blurring with x, y gradients. Fig.1 shows x2 − y2 centerdistributes along a circle curve, adjusted by linear gradients, inwhich composited SEMs are assigned evenly across the FOV, toeliminate gradient encoding “vacuum”.

Sampling scheme should be chosen properly to promote theimage reconstruction, including sampling curve, and parameterof curve determined by linear gradients intensity. In numericalphantom simulation, hyperbola curve, as well as circle, spiral,diagonal sampling curves are studied, with consideration ofstructure feature of gradient x2 − y2 . After an appropriatesampling curve is chosen, gradients intensity effects are alsoillustrated. Theoretically, strong linear gradient will amelioratethe image reconstruction, however the increasing of gradient willalso cause the risk of PNS. To avoid usage of strong lineargradient is a significant standard in our gradient design. Weimplement data simulation of diverse linear gradient intensities,which are expressed as sampling curve distributions, to studygradients intensity effects for image reconstruction.

Comparison of O-space and hyperbola method is simulated

Fig. 1. 64×64 SEMs of linear and nonlinear gradient composition

with iterative algorithm while the acceleration factor R = 4,8, which is achieved by reducing the center points with Rtimes. In O-Space simulation, the circle scheme shows the meansquared error (MSE) reconstruction[8], so in our simulation,circle scheme is employed to perform O-Space, and appropriatesampling scheme we here select is chosen according to samplingscheme experiment. Eight phased array receive coils are used inour simulation, and the coil profiles are shown in Fig.2.

IV. EXPERIMENT

Software simulations and image reconstruction are carried outusing Matlab (The Mathworks, Natick, MA) with computer con-figuration 32bit, 2.93 GHz dual-core processor, and Shepp-loganphantom is used in our numerical simulation. Image reconstruc-tion is performed by solving the linear equation, Ax=b, and LSQRand Karcmarc methods are used in earlier paper[8] to solve theasymmetric linear equation. In our simulation, sampling schemesand gradient intensity experiments directly use pseudo-inverseof A (pinv(A) in Matlab) to obtain the image reconstruction,and in comparison experiment between O-Space and hyperbolamethod, simulation is performed with CG iterative algorithm[4]with iteration time = 8. 512×512 encoding matrix size is used inthe pseudo-inverse of A reconstruction, and 128×128 in iterativealgorithm reconstruction. In experiment involved with acceleratedreconstruction, the matrix size is reduced corresponding to the Rvalue .

V. RESULT

A. Sampling Scheme

xl, yl, in equation (2) determine the center placement ofthe superimposed gradient. Comparison of different samplingcurve is performed while the nonlinear gradient intensity isset to 1 (the same below), and the linear gradient intensity isdetermined by xl, yl . In detail, sampling curves, manifested asxl, yl placement, can be respectively written as

Fig. 2. phased array coils profiles

140

1) Circle:x2l + y2l = 1 (3)

2) Hyperbola:x2l − y2l = 1 (4)

3) Diagonal:xl = ±yl (5)

4) Spiral:x2l + y2l = r(t) (6)

r(t) in spiral scheme is changed along with sampling time t andhere it is set to [0, 1].

Generally if superimposed SEMs are not properly cover theimage space, aliasing will generate. Fig.3 shows image recon-structions under sampling curves above, in which the hyperbolascheme shows the best reconstruction, and the less one is spiral.Diagonal scheme has obvious artifacts in the diagonal direction ofthe image, due to the characteristic of multi-gradient compositionthat low gradients concentrate in diagonal position. Circle schemehas comparable resolution in the middle of image space comparedwith hyperbola sample curve, while obvious aliasing exists in thetop part of the image, and this may be attributed to the settingof center placement of x2 − y2 gradient (r of circle samplingcurve), related to x, y gradient intensity, does not large enoughto cover the whole image space. In the following part, we willperform more simulations of varied x, y gradient intensity withhyperbola scheme to study the linear gradients intensity effect ofimage reconstruction.

Numerical simulation is implemented with the same encodingmatrix size under continuously varying linear gradient intensity,

Fig. 3. Comparison of Circle, Hyperbola, Diagonal, Radial sampling schemereconstruction with Reference image

written asx2l − y2l = r (7)

MSE and peak signal to noise ratio (PSNR) is compared in Fig.4.At r range [0, 1.8], enhancement of x, y gradient intensity

leads to remarkable promotion of PSNR; When r is 1.8, PSNRalmost reach the highest point, implying the best coverage of thehyperbola scheme has been achieved under this linear gradientintensity, and further increasing of r yields almost no differences,seen from r values greater than 1.8, so we could draw aconclusion that infinite increasing of linear gradient intensity isunnecessary for the improvement of reconstruction, and also riskyof causing PNS, a proper sample curve should be chosen basedon the actual needs and experiment.

B. Comparison between Hyperbola method and O-Space

To improve the image SNR, iterative algorithm is used toperform the comparison experiment between hyperbola methodand O-Space while R is set to 4, 8. In the simulation, rangesof x2 − y2, Z2 (z2 − 1

2 (x2 + y2)) intensity are identical, with

z2 item set 0, and linear gradients kept the same amplitude.According to above simulation and Stockmann[8], hyperbola andcircle schemes are chosen for hyperbola method and O-Space innumerical simulation respectively. For the 64×64 Shepp-loganphantom, the convergence of CG iterative algorithm is achievedwhen iteration time is beyond 8, so it is set to 8 for all thesimulation.

Result of hyperbola method in Fig.5 shows that CG itera-tive algorithm reconstruction obtains much better performancecompared with directly pseudo-inverse of encoding matrix re-construction in Fig.4, according to MSE with reference image,but time consumption in iterative algorithm also increases. This

Fig. 4. MSE and PSNR under varied r of hyperbola sampling scheme

141

comparison shows the availability of hyperbola method andpractical application potential.

Along with increase of R, both hyperbola method and O-Spaceperformance degrade gradually. Comparison shows hyperbolamethod reconstruction achieves smaller MSE compare with O-Space in all acceleration factor, especially in center part imaging.At R = 4, hyperbola method superiority is smaller, yet still higherresolution , and R = 8, both results deteriorate. Further experimentshould be carried out in future work.

VI. CONCLUSION

In this paper we studies hyperbola method employing nonlineargradient x2 − y2 and linear gradients x, y simultaneously.

With numerical simulation, we discuss reconstruction of hyper-bola method with hyperbola, spiral, circle, and diagonal samplingcurve, and the result shows hyperbola scheme can obtain thebest reconstruction. We also study linear gradient intensitieseffect of reconstruction with hyperbola scheme, and the resultshows when r in equation (7) is 1.8, the best performance withhyperbola sampling scheme is achieved; Furthermore, we useiterative algorithm to get image reconstruction, and it showsprominent advantage than directly pseudo-inverse result; Finally,in comparison experiment with O-Space reconstruction followingiterative algorithm of increasing R, hyperbola method with hy-perbola sampling scheme achieves more outstanding performanceat all the R value reconstruction, implying the potential of thismethod.

Differing from pre-existing O-Space method, x2 − y2 gradientonly changes in the x, y coordinate, providing the convenience

Fig. 5. Comparison of O-Space and hyperbola method reconstruction withR=1,4,8

of multi-gradient calibration in actual experiment. In addition,Patloc team has created appropriate gradient coils to generate x2−y2 magnetic field, consequently make the high power gradient,and field investigation possible. The next step we will performmore experiments of hyperbola method with parallel imaging,non-cartesian sampling, the reconstruction algorithm, and in-vivoexperiment as well.

ACKNOWLEDGMENT

The authors thank Jason Stockmann at MRRC, Yale University,Gerrit Schultz and Daniel Gallichan in the PatLoc team led byJuergen Hennig, for their helps on rf coil profiles providingand useful discussions. We are also grateful to National Natu-ral Science Foundation of China, grant no. 11105096, appliedbasic science and technology projects of SuZhou, grant no.SYG201125, Scientific equipment research of CAS, grant no.YZ201253. Finally, we thank the anonymous reviewers of thismanuscript.

REFERENCES

[1] C. A. Cocosco, D. Gallichan, A. J. Dewdney, G. Schultz, A. M. Welz, W. R.Witschey, H. Weber, J. Hennig, and M. Zaitsev, “First in-vivo results witha patloc gradient insert coil for human head imaging,” in Proc. Intl. Soc.Mag. Reson. Med. 19, p. 714.

[2] D. Gallichan, C. A. Cocosco, A. Dewdney, G. Schultz, A. Welz, J. r. Hennig,and M. Zaitsev, “Simultaneously driven linear and nonlinear spatial encodingfields in mri,” Magn Reson Med, vol. 65, no. 3, pp. 702–714, 2011.

[3] J. Hennig, A. M. Welz, G. Schultz, J. Korvink, Z. Liu, O. Speck, andM. Zaitsev, “Parallel imaging in non-bijective, curvilinear magnetic fieldgradients: a concept study,” MAGMA, vol. 21, no. 1-2, pp. 5–14, 2008.

[4] K. P. Pruessmann, M. Weiger, P. B. rnert, and P. Boesiger, “Advances in sen-sitivity encoding with arbitrary k-space trajectories,” Magnetic Resonancein Medicine, vol. 46, pp. 638–651, 2001.

[5] G. Schultz, P. Ullmann, H. Lehr, A. M. Welz, J. Hennig, Zaitsev, and Maxim,“Reconstruction of mri data encoded with arbitrarily shaped, curvilinear,nonbijective magnetic fields,” Magn Reson Med, vol. 64, no. 5, pp. 1390–1403, 2010.

[6] J. P. Stockmann, P. A. Ciris, and R. T. Constable, “Efficient o-space parallelimaging with higher-order encoding gradients and no phase encoding,” inProc. Intl. Soc. Mag. Reson. Med, p. 762.

[7] J. P. Stockmann, G. Galiana, L. Tam, C. Juchem, T. W. Nixon, and R. T.Constable, “In vivo o-space imaging with a dedicated 12 cm z2 insert coilon a human 3t scanner using phase map calibration,” Magn Reson Med,2012.

[8] J. P. Stockmann, P. A. Ciris, G. Galiana, L. Tam, and R. T. Constable,“O-space imaging: Highly efficient parallel imaging using second-ordernonlinear fields as encoding gradients with no phase encoding,” Magn ResonMed, vol. 64, no. 2, pp. 447–456, 2010.

[9] L. K. Tam, J. P. Stockmann, G. Galiana, and R. T. Constable, “Null spaceimaging: Nonlinear magnetic encoding fields designed complementary toreceiver coil sensitivities for improved acceleration in parallel imaging,”Magn Reson Med, 2011.

[10] A. M. Welz, M. Zaitsev, F. Jia, Z. Liu, J. Korvink, H. Schmidt, H. Lehr,H. Post, A. Dewney, and J. Hennig, “Development of a non-shielded patlocgradient insert for human head imaging,” in Proc. Intl. Soc. Mag. Reson.Med., vol. 17, p. 3073.

142