Abstract

Compressed sensing has reinvigorated the field of non-Cartesian sampling in magnetic resonance imaging (MRI).

Until now there has been no 3D radial view-order whichmeets all the desired characteristics for simultaneousdynamic/high-resolution imaging, such as for self-navigatedmotion-corrected high resolution neuroimaging.

In this work, we examine the use of Hierarchical Equal Areaiso-Latitude Pixelization (HEALPix) for generation of three-dimensional (3D) radial view-orders for MRI, and compare to aselection of commonly used 3D view-orders.

The resulting trajectories were evaluated through simulationof the point spread function and slanted surface object suitablefor modulation transfer function, contrast ratio, and SNRmeasurement. Results from the HEALPix view-order werecompared to Generalized Spiral, 2D Golden Means, and Randomview-orders.

Finally, we show the first use of the HEALPix view-order toacquire in-vivo brain images.

1. Introduction

Compressed sensing with multi-core and GPU-basedreconstruction has renewed and increased interest in 2D, hybrid,and 3D radial non-Cartesian MRI.

View-orders in 3D radial MRI typically cover a spherical (orellipsoidal) volume of k-space with center-out (spokes) for FIDreadout as in ZTE, UTE and SWIFT [5] or echo readout throughthe center (diameters). There are several properties desirable fora 3D radial view-order scheme, not all of which are compatible.

An ideal 3D radial isotropic view-order scheme has all theproperties shown in Table 1. Efficiency embodies the favorableproperties of equi-distribution, separation, covering, quasi-uniformity and minimal Riesz potential energy described in [3].Computability is defined as the ability to generate view orders ina deterministic repeatable manner without excessive computationtime (such as required for energy minimization). Smooth naturalordering is defined as having a gradient trajectory without largearbitrary jumps or the need for any additional sorting.

Simultaneous self-navigated anatomical/dynamic MRIrequires one of the most difficult properties of the view-orderscheme. Each subset of views used for a navigator segment musthave enough coverage to reconstruct a low resolution image (orprovide isotropic spatial information to other estimators) as wellas combine into an efficient fully sampled set of view data foranatomical reconstruction.

The Generalized Spiral view-order has been the convenientchoice for non-dynamic 3D radial imaging when additionalspoiling is not required and is useful for low sound pressurelevels in quiet MRI . It posses all the desired characteristics for a

3D radial MRI view-order except the ability to divide intoefficient subsets.

The 2D Golden Means view-order [1] has most of the desiredcharacteristics excepting smooth natural order. It is efficient if thenumber of points in a subset is not too small.

2. HEALPix

As seen from Table 1, HEALPix (Hierarchical Equal Area iso-Latitude Pixelization) [2; 3] is a good candidate for a 3D radialdynamic view-order. It possesses all the desirable characteristicsof the Generalized Spiral in addition to being divisible intoefficient subsets. HEALPix also has the desirable characteristicsof the 2D Golden Means view-order, but has smooth naturalordering for the subsets and the whole.

HEALPix was devised by K. M. Górski at the TheoreticalAstrophysics Center (TAC) in Copenhagen, Denmark anddeveloped by a dedicated team of collaborators [4]. It wasdesigned with three properties essential for computationalefficiency in discretization functions on the unit sphere (S2) andprocessing large amounts of data:

1. The sphere is hierarchically tessellated into curvilinearquadrilaterals. 2. The pixelization is an equal area partition ofS2. 3. The point sets are distributed along fixed lines of latitude.

The last allows for utilization of fast spherical functiontransforms, and rapid search and addressing. This property mayfurther enable k-space domain estimation of motion.

HEALPix initially divides the sphere into 12 node points withVoronoi regions of equal areas. Each area is then recursivelysubdivided into smaller equal areas, which for maximalconvenience are powers of 2. In the following N is the totalnumber of nodes covering the sphere, which we use as theendpoints of the center-out radial “spoke” views. N isconstrained to be N=12M2 where M is a non zeropositive integer. For computational efficiency and scaling Mis further constrained to be M=2K where K is a positiveinteger. The example of K=2 is shown in Figure 2. Ingeneral, we have the number of nodes: N=12×4K

3. Methods and Implementation

All generated view-order tables, image data and analysis scriptsare deposited at [6] in the interest of reproducible research.

3.1. Sequence Modifications

We modified the 3D Radial Silent sequence [7; 8; 8]to accepttable input for view-ordering.

3.2. View-order Generation

We utilize the specific implementation of HEALPix, “MEALPix”which is Matlab™ based [9]. Additional useful code forHEALPix and other sphere coverage is available from [10]. All

HEALPix View-order for3D Radial Self-Navigated Motion-Corrected ZTE MRI

Curtis A. Corum1, Stanley J. Kruger2 and Vincent A. Magnotta2

1Champaign Imaging LLC, 2University of Iowa, Department of Radiologycurt.corum@champaignimaging.com

radial views were center-out (spokes). Each consisted of usingthe unit sphere surface point generated by the MEALPix packageand scaled to the gradient discretization value of 32676. Theendpoint components were rounded via floor to an integer value.

In addition we generate Spiral [3; 7; 11], 2D Golden andRandom – view-order sets [1].

Since HEALPix is constrained to N=12×4K views it isnot always directly commensurate with the GE Silent view-orderscheme of N=J2 . For visualization in the figures we use

N=192 for HEALPix and N=196 for the Spiral, 2DGolden Means and Random view-orders. For the PSF simulation,Hollow Cube simulation and in-vivo imaging we use

N=36864 .Figures 1 and 2 show the gradient waveform shapes and

sphere coverage with 192 or 196 views. Each point on the sphereshows the corresponding endpoint of the radial view. Random isnot shown.

3.3. Gridding Reconstruction

All simulated and acquired datasets were reconstructed using1.25 oversampled gridding with a radius 2.5 Kaiser-Bessel kernel[14] and 50 iterations of the Pipe-Menon algorithm for sampledensity correction [15].

3.4. Point Spread Function

Simulated point spread functions (PSFs) were generated for eachview-order by generating a dataset of k-space values consisting ofall ones. Analysis using a custom Octave script was conducted forthe properties shown in Table 2. The PSF full width halfmaximum (FWHM) radius was determined by 3D fitting of a 7-parameter Gaussian model (amplitude, 3 offsets and 3 sigmas).

The maximum value and the volume were determined directlyfrom the data by integrating in a radius 4 neighborhood.

Figure 3 shows slices and a mesh plot of the XY plane in the32x32x32 neighborhood around the center of the PSF. Scale islogarithmic base 10 to note the rapid falloff.

3.5. Modulation Transfer Function

A hollow cube possessing internal slanted sides was simulated bydirect synthesis of k-space [12] data and reconstructed bygridding (see Figure 4.) The modulation transfer function (MTF)in an orthogonal XY slice was estimated using the ImageJ pluginSlanted Edge MTF [13]. See Figure 6.

3.6. Contrast Ratio and SNR

The contrast ratio and SNR were evaluated using 32x32voxel patches in the center slice. Black level for contrastcame from centering the patch in the dark region at centerof the slice. White level consisted of centering the patch onthe right of center bright plateau. Noise for the SNR aswell as signal were evaluated on the bright patch.

3.7. In-vivo Neuroimaging

In-vivo brain images of a healthy human volunteer were takenusing the HEALPix view-order and are shown in Figure 5.Images were also taken with the Generalized Spiral view-order.

4. ResultsAs seen in Figure 3 and Table 2 the central region of the PSF forboth the Generalized Spiral and HEALPix are similar; HEALPixprovides slightly higher residual amplitude around the central

Table 1: Sphere Coverage Methods

Generalized Spiral HEALPix 2D Golden MeansFigure 1: Gradient Waveforms – The 2D Golden Means view-order does not have smooth natural ordering.

point (log10 of -4 vs log10 of -4.5). The fitted FWHM and peakvalues vary by less than 0.5% percent.

Contrast ratio, MTF performance, and SNR in Table 2 alsovary by less than 0.5% between the Generalized Spiral andHEALPix. The values of the 2D Golden Means and Randomview-orders fall below the Generalized Spiral and HEALPix.

MTF curve values for HEALPix and Generalized Spiraloverlap while 2D Golden Means and Random both fall below forall spatial frequencies in Figure 6.

HEALPix in-vivo brain images (Figure 5) areindistinguishable at automatic optimal window level from thosetaken with the Generalized Spiral view-order with same numberof views (not shown).

5. Conclusions

In this initial evaluation, the HEALPix 3D radial view-orderpossesses all the favorable properties of the Generalized Spiral inaddition to being divisible into efficient subsets for dynamic self-navigated protocols. It performs similarly on metrics of PSF,

MTF, contrast ratio and SNR as the Generalized Spiral and hasyielded initial in-vivo images indistinguishable at automaticoptimal window level to images taken using the GeneralizedSpiral.

6. Notes

This work was funded in part by the SBIR Phase I grantR43MH115885 from the NIH/NIMH. Curt Corum andChampaign Imaging LLC are developing technology related tothe topics reported in this proceeding. All in-vivo imaging wasperformed under an IRB-approved protocol at the University ofIowa, Magnetic Resonance Research Facility.

7. References

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Generalized Spiral HEALPix 2D Golden MeansFigure 2: Sphere Coverage – view endpoints on unit k-space sphere.

Generalized Spiral HEALPix 2D Golden MeansFigure 3: Log10 of Point Spread Function – orthogonal slices through center (Top), Mesh of XY slices (Bottom).

[2] Gorski, K. M.; Hivon, E.; Banday, A. J.; Wandelt, B. D.; Hansen, F. K.; Reinecke, M. & Bartelman, M. (2004). HEALPix -- a Framework for High Resolution Discretization, and Fast Analysis of Data Distributed on the Sphere, Astrophys.J.622:759-771,2005 . PMID:

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Generalized Spiral HEALPix 2D Golden Means

Figure 4: Simulated Hollow Cube – for modulation transfer function (MTF), contrast and SNR measurements.Shown at 10x intensity scale and 2.25 nominal field of view (for Nyquist radius).

Table 2: View-order Performance

T1w T2w

Figure 5: In-Vivo - T1 and T2 weighted images with HEALPix view-order. Figure 6: MTF Curves – hollow cube slanted edge,