Anisotropic field-of-views in radial imaging
TTRANSACTIONS ON MEDICAL IMAGING 1
Anisotropic Field-of-Views in Radial Imaging
Peder E. Z. Larson, Paul T. Gurney, and Dwight G. Nishimura,
Member, IEEE
Abstract — Radial imaging techniques, such as projection-reconstruction (PR), are used in MRI for dynamic imaging,angiography, and short- T imaging. They are robust to flow andmotion, have diffuse aliasing patterns, and support short readoutsand echo times. One drawback is that standard implementationsdo not support anisotropic field-of-view shapes, which are usedto match the imaging parameters to the object or region-of-interest. A set of fast, simple algorithms for 2D and 3D PR, and3D cones acquisitions are introduced that match the samplingdensity in frequency space to the desired field-of-view shape.Tailoring the acquisitions allows for reduction of aliasing artifactsin undersampled applications or scan time reductions withoutintroducing aliasing in fully-sampled applications. It also makespossible new radial imaging applications that were previouslyunsuitable, such as imaging elongated regions or thin slabs. 2DPR longitudinal leg images and thin-slab, single breath-hold 3DPR abdomen images, both with isotropic resolution, demonstratethese new possibilities. No scan volume efficiency is lost by usinganisotropic field-of-views. The acquisition trajectories can becomputed on a scan by scan basis. Index Terms — Radial imaging, projection-reconstruction, PR,3D Cones, anisotropic field-of-view
I. I
NTRODUCTION R ADIAL medical imaging methods were first used inX-ray computerized tomography (CT) where data isacquired on radial projections. These projections correspondto radial lines in frequency space, a fact that inspired the firstmagnetic resonance imaging (MRI) acquisitions to also occuron radial lines [1]. These 2D imaging methods are also knownas projection-reconstruction (PR) and projection acquisition(PA). Radial imaging also refers to 3D acquisition techniquessuch as 3D PR, as well as 3D cones [2], [3].Radial MRI is used for dynamic imaging applicationsbecause it is inherently robust to motion [4] and flow [5]. Itis used in angiography [6], with applications such as contrast-enhanced angiography [7], [8] and time-resolved angiography[9]. Radial MRI readouts can support very short repetitiontimes (TRs) and echo times (TEs). Short TRs are usefulfor steady-state free precession (SSFP) imaging, particularlyif high resolution is required or for quadrature fat/waterseparation. Both the short TRs and robustness to motion andflow have been taken advantage of in balanced SSFP coronaryartery imaging [10], [11]. Short TEs are required for ultra-short echo time (UTE) MRI, which can image collagen-richtissues such as tendons, ligaments and menisci, as well as
This work was supported by NIH grants 2R01-HL39297 and 1R01-EB002524, and GE Medical Systems.The corresponding author can be reached at [email protected] authors are with the Magnetic Resonance Systems Research Labora-tory, Stanford University, Stanford, CA 94305.Copyright (c) 2007 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected]. calcifications, myelin, periosteum and cortical bone [12], [13].In addition to 2D acquisitions, 3D radial imaging techniques,such as 3D PR [14] and 3D cones [15], have been applied toUTE.3D PR has also been used for the Vastly undersampledIsotropic Projection Reconstruction (VIPR) imaging method[7] for MR angiography and efficient phase-contrast flowimaging [16]. The isotropic resolution of 3D PR is advan-tageous for multiplanar and 3D reformats. VIPR uses un-dersampled acquisitions because the number of projectionsrequired for fully-sampled 3D PR is prohibitively large. Theundersampling is tolerated because of the diffuse aliasingproperties of radial trajectories [7]. Another solution to theoften prohibitively large number of projections required by3D PR is to use conical trajectories [2], [3]. These samplein a spiral pattern along cones with various projection angles,allowing for significantly faster volumetric coverage than 3DPR.In 2D and 3D PR trajectories, as well as 3D cones, a criticalnumber of projection lines must be acquired to support agiven field-of-view (FOV) – also referred to as a region-of-support or region-of-interest. The lines are normally acquiredwith equiangular spacing, resulting in only circular and spher-ical FOV shapes. Many imaging applications, however, haveanisotropic dimensions and would benefit from anisotropicFOVs.In this paper we introduce a new method to simply andeasily design radial imaging trajectories for anisotropic FOVs.One previous method applied several varying angular densityfunctions to obtain anisotropic 2D FOVs [17], [18]. Ourmethod also utilizes non-uniform angular spacing, and is ableto exactly match the sampling to the desired FOV shape.Design algorithms for 2D and 3D PR, as well as 3D conesare presented. Tailoring the FOV for non-circular objects orregions-of-interest allows for scan time reductions withoutintroducing aliasing artifacts. In undersampled applications,this tailoring will reduce the occurrence of aliasing artifacts.Additionally, our algorithms support anisotropic projectionlengths, and, equivalently, resolution. This is useful for reduc-ing gradient slew rate demands in 3D cones, and could alsobe used for reducing gradient distortion artifacts by favoringthe more homogeneous or powerful axes. It can also be usedto trade image resolution in certain dimensions for acquisitiontime. II. T
HEORY
In radial imaging, the raw data is acquired on radial lines infrequency space, referred to as “k-space” in MRI. A discretenumber of these lines, also known as projections or spokes,are sampled during data acquisition. Sampling theory tells a r X i v : . [ ee ss . I V ] J a n TRANSACTIONS ON MEDICAL IMAGING us that the sample separation determines the FOV. Cartesiansamples are generally equally spaced, and the FOV is equalto the inverse of the sample spacing, k . In radial imaging,the spacing between adjacent projections in k-space variesradially, and is most sparse at the end of the projections. Thesample spacing along the projections must also be considered,although in MRI this usually does not limit the FOV as muchas the distance between projections. We will investigate howthis projection separation determines the FOV and resolution.Further discussion of radial sampling theory can be foundin [19]. A. FOV vs. PSF
The FOV of a given sampling pattern is characterizedby its Fourier transform ( FT ), known as the point spreadfunction (PSF). Since the sampling pattern is multiplied bythe frequency data, the PSF is convolved with the image data.The central lobe of the PSF introduces a blurring, and thusdetermines the resolution of the resulting image. Aliasing lobesin the PSF are outside this central lobe and determine theFOV of the resulting image. In radial imaging, the distinctshape of the aliasing lobe determines the FOV, but they are notalways the same shape. Convexity of the aliasing lobe shapein the PSF will ensure that the FOV shape is exactly the same.Polygons, rectangles, cuboids, ellipsoids, and ovals are someexamples of convex shapes useful for medical imaging. Whilethere are concave PSF shapes that will support concave FOVs,such as a “plus”-sign shape, we will assume convexity sincethis includes most objects and regions-of-interest in medicalimaging. B. 2D Sampling
The FOV of a 2D radial imaging trajectory can be analyzedby decomposition into adjacent spokes. A pair of adjacentspokes can be approximated as two parallel lines, shown inFig. 1a and described mathematically as P ( k x , k y ) = [ δ ( k y − ∆ k φ δ ( k y + ∆ k φ · rect( k x k max ) , (1)where ∆ k φ is the separation at the end of the projectionsand k max is the length of the projections. The PSF of Eq. 1,illustrated in Fig. 1b, is: FT { P ( k x , k y ) } = C cos( π ∆ k φ y ) · sinc(2 k max x ) , (2)where C includes all scaling factors. The peaks of | cos( π ∆ k φ y ) | will introduce aliasing when convolved with theobject, thus limiting the FOV perpendicularly to the directionof the lines to k φ . The actual PSF is much more peakedthan a cosine because the sample spacings in k y from otherprojections introduce phase variations that cause the aliasingto cancel out along y between and ± k φ , as is shown inthe “Results” section. This is analogous to Cartesian samplingwhere the PSF of two parallel lines is also a cosine but phasevariations from the other sampling lines cause sharp aliasingpeaks. The resolution is limited to a minimum of k max duethe sinc(2 k max x ) term. k x k y FT xy FTFT a bc de f k max ( φ ) k max ( φ ) 12 k max k max k max k max k x k y xyk x k y xy φφ ∆ k φ ( φ ) φ + π ∆ k φ ( φ ) 1 ∆ k φ ∆ k φ ∆ k φ ∆ k φ Fig. 1. Sampling in 2D radial imaging. (a) Approximation of adjacentprojections by parallel sampled lines. (b) Parallel lines PSF illustration. Thegray lines show the spatial variation derived in Eq. 2. Aliasing lobes appearat k φ . With additional projections, the aliasing lobes become much sharperthat the cosine shown. (c) Isotropic PR trajectory with constant angularspacing and extent. (d) Isotropic PR PSF illustration. (e) PR trajectory withvariable angular spacing and extent. (f) Angularly varying PR PSF illustration. If all projections are equally spaced, as shown in Fig. 1c,the alias-free FOV and resolution, res , are:
F OV = 1∆ k φ ≈ k max ∆Θ = Nπk max , (3) res = 12 k max , (4)where ∆Θ is the angular spacing between spokes in radians, N is the number of full projections acquired, and k max is theextent of the spokes in k-space (Fig. 1d). Here we have usedthe approximation that sin(∆Θ) ≈ ∆Θ for small ∆Θ .From Fourier theory, rotation of the parallel lines rotatestheir FT , thus the angular spacing between spokes can bevaried to produce anisotropic FOV shapes. Similarly, if k max is varied, the resolution size changes angularly. Equation 2tells us that the sample spacing at angle φ determines theperpendicular FOV, while k max at φ determines the resolutionat φ . This is shown in Fig. 1e and f, and is formally expressed ARSON et al. : FOVS IN RADIAL IMAGING 3 as,
F OV ( φ + π k φ ( φ ) ≈ k max ( φ )∆Θ( φ ) , (5) res ( φ ) = 12 k max ( φ ) . (6)Discrete sampling along the parallel lines results in repeti-tion of the FT pattern (Eq. 2) in this sampling direction ( x inEq. 2 and Fig. 1a). Radial sample spacing of ∆ k r results inrepetitions at k r . These aliasing lobes are often eliminatedin MRI because of low-pass filters applied during the readout.These filters will, however, limit the FOV in the projectiondirection to k r because they are matched to the samplespacing. Therefore, the maximum radial sampling spacingmust be ∆ k r ≤ F OV ( φ )) (7)to ensure there is no aliasing or FOV restrictions due to thesampling along the projections. C. 3D Sampling
Radial sampling in 3D can be analyzed with the sameprinciples as in 2D. Again, approximating adjacent spokes asa set of parallel lines, we now have: P ( k x , k y , k z ) = [ δ ( k y − ∆ k φ δ ( k y + ∆ k φ · rect( k x k max ) · δ ( k z ) . (8)The PSF is then the same as Eq. 2: FT { P ( k x , k y , k z ) } = K cos( π ∆ k φ y ) · sinc(2 k max x ) . (9)This aliasing pattern is the same as illustrated in Fig. 1b, butwith an infinite extent in z .A set of adjacent cones can be modeled by rotating a pairof parallel lines about the k z -axis, as shown in Fig. 2a, whichrotates the PSF of these parallel lines about the z -axis. Theresulting PSF pattern is circularly symmetric about the z -axisand with polar deflection of the aliasing lobe of θ + π fora cone deflection of θ , shown in Fig. 2b. We are using thespherical coordinate notation where θ , the polar angle, is thedeflection from the positive z -axis and φ , the azimuthal angle,is the rotation from the positive x -axis.Adjacent projections on a cone produce aliasing patternsthat are rotations of the PSF in Eq. 9, as illustrated in Fig. 2cand d. The aliasing lobes are the nearest to the origin in the x - y plane at an azimuthal angle φ + π and at a distance of / ∆ k φ .They extend in a direction that is determined by thecone deflection angle, as shown in Fig. 2d.Combining these results, we find that the sampling alonga cone limits the FOV azimuthally in the x - y plane whilethe sampling of different cones limits the FOV in the polardirection. Mathematically, F OV θ ( θ + π , φ ) = 1∆ k θ ( θ ) (10) F OV φ ( θ = π , φ + π k φ ( φ ) (11) res ( θ, φ ) = 12 k max ( θ, φ ) , (12) k x FT xk z zk x FT k z a bc d D k q D k q D k f D k q q + p q + p q fq k y yk y D k f xy f D k f x'y' z x'y' = +- z = q q q q k y k x k z D k f f q Fig. 2. Sampling in 3D radial imaging. (a) Approximation of adjacent conesby rotated parallel lines. (b) PSF illustration of adjacent cones model, whichis the parallel lines PSF (Fig. 1b) rotated around the z -axis. (c) Adjacentprojections on two different cones. (d) PSF illustrations for the adjacentprojections. The top illustration shows the intersection of the PSF with the x - y plane, which is approximately identical for the projections on both cones.The bottom illustration shows the planes which contain the aliasing lobes.These planes are perpendicular to the y (cid:48) axis at y (cid:48) = ± / ∆ k φ , and thelobes extend and different angles depending on the cone angle. where the variables are all illustrated in Fig. 2. The resolutionis determined by the projection extents. The samples along theprojection must again satisfy Eq. 7.III. M ETHODS
A. 2D Projection-Reconstruction
Two-dimensional radial imaging trajectories can be definedby the projection angles, projection lengths, and the samplingpattern along the projections. Our algorithm designs a setof projection angles, Θ[ n ] , and projection lengths, K max [ n ] ,for a desired FOV pattern. The sampling pattern along theprojections is not designed, and should be chosen to meet thecondition in Eq. 7.The algorithm is shown in Fig. 3. The desired FOV mustbe specified as a function of angle, F OV ( φ ) , and must be π -periodic ( F OV ( φ ) = F OV ( φ + π ) ). The desired projectionlengths can be specified as k max ( φ ) , and also must be π -periodic. An initial angle of φ may be specified, althoughthe default of φ = 0 is appropriate for most applications. TRANSACTIONS ON MEDICAL IMAGING
1. Initialize
Θ[1] = φ and n = 1
2. Calculate: ∆Θ est = 1 k max (Θ[ n ]) F OV (Θ[ n ] + π )∆Θ = 1 k max (Θ[ n ] + ∆Θ est ) F OV (Θ[ n ] + ∆Θ est + π )Θ[ n + 1] = Θ[ n ] + ∆Θ and increment n
3. Repeat previous step until Θ[ n ] > φ + φ width
4. Choose a scaling factor, S , and the number ofprojections returned, N , as follows:If Θ[ n ] − ( φ + φ width ) < ( φ + φ width ) − Θ[ n − choose S = φ width Θ[ n ] − φ and N = n − Else S = φ width Θ[ n − − φ and N = n −
5. Scale the set of angles
Θ[1 , . . . , N ] = S · (Θ[1 , . . . , N ] − φ ) + φ
6. Calculate K max [ n ] = k max (Θ[ n ]) Fig. 3. Generalized 2D anisotropic FOV radial imaging algorithm
The angular width of the resulting angles, φ width , may alsobe specified, and will be π for most applications, includingfull-projection PR. One application of varying this width ishalf-projection PR, where φ width = 2 π could be used. Theseparameters are also useful for designing 3D PR trajectories -see Section III-C.After initialization, the relationship derived in Eq. 5 isused to sequentially calculate a set of projection angles instep 2 of the algorithm. The projection separation is firstestimated, ∆Θ est , and then the actual projection separation isthen calculated using the angle Θ[ n ] + ∆Θ est . Without thisestimation, the resulting FOV will be very slightly rotatedand distorted because the projection separation is centeredbetween Θ[ n ] and Θ[ n + 1] . Since Θ[ n + 1] is unknown, Θ[ n ] + ∆Θ est provides a good estimate of the middle angle,especially compared to using Θ[ n ] . Additional iterations forincreased accuracy are possible but provide little additionalbenefit.The sequential nature of the algorithm and the requiredperiodicity of the projection angles results in an undesiredangle spacing at the end of the design. Steps 4 and 5 correctfor this by scaling the set of projection angles to the chosen φ width . The scaling slightly distorts the resulting FOV shapeso the scaling factor, S , is chosen by step 4 to be as close to1 as possible.This step could also be modified to require thatthe number of projections be even, odd, or a scalar multiplefor acquisition strategies such as multi-echo PR [8].The computation cost of this algorithm is small since itinvolves simple calculations and requires no iterations or largematrix operations. B. 3D Cones
The 2D design algorithm can directly be applied to design-ing 3D cones imaging trajectories. The 3D FOV shapes thatare achievable are circularly symmetric about one axis, whichwe define to be the z -axis, corresponding to cones wrappingaround the k z -axis as shown in Fig. 2a and b. These shapescan be described by F OV ( θ, φ ) = F OV θ ( θ ) , as is illustratedin Fig. 4a and b. For example, a rectangular-shaped F OV θ ( θ ) will yield a cylindrical FOV, while an elliptical F OV ( θ ) yieldsan ellipsoid.For anisotropic 3D cones design, the 2D PR algorithm isused with inputs of F OV ( θ ) and k max ( θ ) (optional), φ width = π , and φ = k max (0) F OV ( π ) . This φ ensures that no conesare just a single projection. The resulting set of projectionangles and extents describe the set of cones to be used, wherethe angles, Θ[ n ] , represent the deflection from the k z -axis.Sampling within each cone and generation of appropriategradient waveforms are described in [3]. k x FT k z Q [n] xz FT F [m n ] xz f FOV q FOV f FT xz FOV a bc de f k x k z k x k z Fig. 4. 3D radial trajectory design (cones-based). (a) Cones designed with2D algorithm. (b) Sampling of cones introduces
F OV θ , which is circularlysymmetric about the z -axis. (c) Projections on each cone designed with2D algorithm using a random starting angle φ . (d) Sampling along conesintroduces F OV φ , which is approximately independent of z . (e) 3D PRtrajectory resulting from sampling in (a) and (c). (f) The 3D PR FOV isthe minimum of F OV θ (b) and F OV φ (d). ARSON et al. : FOVS IN RADIAL IMAGING 5
C. 3D Projection-Reconstruction
We have created two methods for designing 3D PR trajec-tories with anisotropic FOVs. One samples a set of cones, the“cones-based” method, while the other designs and samplesa spiraling path, the “spiral-based” method. Figure 5 showstwo resulting sampling patterns for both the cones-based andspiral-based design methods.
1) Cones-based Design:
This method designs 3D projec-tions for an anisotropic FOV by first designing a set ofcones and then appropriately sampling on each cone. Isotropicsampling in these two dimensions will result in a sphericalFOV, with the number of projections per cone proportional to sin( θ ) [20], where θ is the polar angle of the cone.Equations 10 and 11, illustrated in Fig. 2, tell us that thecones sampled will define a limiting FOV that is circularlysymmetric about the z -axis, F OV θ ( θ, φ ) = F OV θ ( θ ) , whilethe samples on each cone define a maximum FOV that isapproximately invariant in z , F OV φ ( θ, φ ) ≈ F OV φ ( x, y ) (Fig. 4a-d). We will describe F OV φ using the deflection anglein cylindrical coordinates, φ c , resulting in the simple notation F OV φ ( x, y ) = F OV φ ( φ c ) . Both sampling patterns limit theFOV, resulting in F OV = min(
F OV θ ( θ ) , F OV φ ( φ c )) , (13)as illustrated in Fig. 4e and f.This method takes inputs of F OV θ and F OV φ , with F OV φ ( φ c ) ≤ F OV θ ( π (14)to ensure that the cone spacing does not introduce aliasinginside F OV φ in the x - y plane. k max ,θ ( θ ) can also be used.Varying k max ,φ has not been incorporated into our algorithmbecause it causes the polar angle spacing to vary azimuthally,resulting in distortion of the supported FOV. The cones are de-signed using the 2D PR algorithm with F OV θ ( θ ) , k max ,θ ( θ ) , a bc d Fig. 5. 3D PR trajectories. (a) Cones-based design, and (b) Spiral-baseddesign for uniform sampling, resulting in a spherical FOV. (c) Cones-baseddesign, and (d) Spiral-based design of an ellipsoid FOV with, non-uniformsampling in both the azimuthal and polar directions. θ = 0 , and θ width = π . The resulting Θ[ n ] and K max [ n ] ,where n = 1 , . . . , N cones , are then sampled, also using the2D PR design algorithm.The spokes covering each cone are designed with F OV φ ( φ c ) and φ width = 2 π . For cone n , k max ,φ = K max [ n ] · sin(Θ[ n ]) to adjust for the circumference of the cone. Theinitial angle φ on each cone is chosen at random uniformly inthe interval [0 , K max [ n ] · sin(Θ[ n ]) · F OV φ ( π ) ] . This randomizationreduces coherent aliasing artifacts that are introduced when φ = 0 for each cone (see Fig. 10). Note that the first cone( n = 1 ) is just a single projection along the k z -axis. The resultis a set of Φ[ m n ] for m n = 1 , . . . , M n and n = 1 , . . . , N cones ,where M n is the number of projections within cone n .To complete the full-projection design, an additional coneis added in the k x - k y plane ( θ = π ) that is only azimuthallysampled along half of the circumference. The 2D PR designalgorithm is used, with F OV φ ( φ c ) , k max ,φ = k max ,θ ( π ) , arandomized φ , and φ width = π . For half-projection designs,the additional half-sampled cone is not needed and θ width = π is used for the cone design.
2) Spiral-based Design:
This method of 3D PR design isinspired by a method that isotropically samples the unit sphereon a spiral path [21]. The design is similar to the cones-basedmethod, taking the same inputs of
F OV θ and F OV φ , with F OV θ ( π ) = max( F OV φ ( φ c )) , and, optionally, k max ,θ . Thismethod is best suited for half-projections, and an extensionto full-projection design is described in Appendix I. It alsoresults in a more diffuse aliasing pattern (see Fig. 10).First, a set of polar sampling angles is designed with the2D PR algorithm exactly as in the cones-based method forhalf-projections, with F OV θ , k max ,θ , θ = 0 , and θ width = π , yielding ˆΘ[ n ] and ˆ K max [ n ] for n = 1 , . . . , N polar . Tocreate a continuous sample sample path in the longitudinaldirection, these sets are linearly interpolated to final samplesof ˆΘ[ N polar + 1] = π and ˆ K max [ N polar + 1] = k max ,θ ( π ) .The interpolation uses the number of projections required,which is found by estimation using the required number ofazimuthal samples on a cone at θ = π , N φ, est , as well as ˆΘ[ n ] and ˆ K max [ n ] . The 2D PR algorithm with F OV φ , k max ,φ = k max ,θ ( π ) , φ = 0 , and φ width = 2 π is used to compute N φ, est . The number of projections between ˆΘ[ n ] and ˆΘ[ n + 1] will be approximately N n, est = N φ, est sin( ˆΘ[ n ] + ˆΘ[ n + 1]2 ) × ˆ K max [ n ] + ˆ K max [ n + 1]2 k max ,φ . (15)The total number of projections is chosen to be N = N polar (cid:88) n =1 N n, est . (16)The linear interpolation is done such that there are N n, est projections between ˆΘ[ n ] and ˆΘ[ n + 1] . This is done by usinga parametrization, θ ( t ) , where θ ( t n ) = ˆΘ[ n ] , t n +1 − t n = N n, est , and t = 1 . The polar projection angles are computedby sampling the linear interpolation of this parametrization as Θ[ m ] = θ ( m ) for m = 1 , . . . , N . If variable extents are used, TRANSACTIONS ON MEDICAL IMAGING the interpolation of k ( t ) , where k ( t n ) = ˆ K max [ n ] , is sampledas K max [ m ] = k ( m ) .The azimuthal sampling, Φ[ m ] , is then computed similarlyto step 2 in Fig. 3, with an additional scaling factor also used inthe cones-based design to compensate the cone circumference: ∆Φ est = 1 K max [ m ] sin(Θ[ m ]) F OV φ (Φ[ m ] + π ) (17) ∆Φ = 1 K max [ m ] sin(Θ[ m ]) F OV φ (Φ[ m ] + ∆Φ est + π ) (18) Φ[ m + 1] = Φ[ m ] + ∆Φ , (19)with the initial angle of Φ[1] = 0 . This results in a set ofprojection angles, Φ[ m ] , Θ[ m ] , and extents, K max [ m ] yieldingthe anisotropic FOV described in Eq. 13. D. Reconstruction
All images were reconstructed with the gridding algorithm[22], and PSFs were calculated by gridding data of all ones.Each data point is multiplied by a density compensation factor(dcf) to correct for the unequal sample spacing, which resultsin coloring of the noise and an intrinsic loss in the signal-to-noise ratio (SNR) efficiency [23]. The dcf can be separated intoa radial and angular components, where the angular componentaccounts for the anisotropic projection spacing. The angulardcf is chosen so that the same radial dcf can be used oneach projection, provided they all have an equal number ofidentically distributed radial samples. When the samples areequally spaced this radial component is linear for 2D PR andquadratic for 3D PR.The 2D angular dcf ( D θ ) is proportional to the projectionseparation, ∆ k φ , given from Eq. 5. For a separable dcf, D θ must also be proportional to the projection length becausethe scaling of the radial sample spacing by k max cannot bedescribed in a single radial dcf. The resulting 2D PR angulardcf is D θ (Θ[ n ]) = K max [ n ] · ∆ k φ = K max [ n ] F OV (Θ[ n ] + π ) . (20)For anisotropic 3D cones, this compensation factor is ap-plied to each cone by dividing Eq. 10 in reference [3] by D θ .In 3D PR, the density compensation for a given projection isfound by multiplying the compensation factors for the polarand azimuthal sampling: D D (Θ[ n ] , Φ[ m n ]) = D θ (Θ[ n ]) × D φ (Φ[ m n ])= K max [ n ] F OV θ (Θ[ n ] + π ) × F OV φ (Φ[ m n ] + π ) . (21)For the full-projection spiral-based design, this dcf is slightlymodified as described in Appendix I. E. Design Functions
The anisotropic FOVs were designed in Matlab 7.0 (TheMathworks, Natick, MA, USA). The design functions for 2DPR, 3D cones, and both 3D PR methods, along with accompa-nying documentation are available for general use at . F. MRI Experiments
A GE Excite 1.5T scanner with gradients capable of 40mT/m amplitude and 150 T/m/s slew rate (GE Healthcare,Milwaukee, WI) was used for all experiments. The 2D PRimages were acquired with a UTE sequence using half-projection acquisitions, 5 mm slice thickness, TE = 500 µ s,TR = 100 ms, 30 ◦ flip angle, 512 samples per projection, and1 mm resolution with a transmit/receive extremity coil. The3D PR images were acquired with a spoiled gradient-recalledecho (SPGR) sequence using full-projection acquisitions withTE = 3 ms and TR = 10 ms. The bottle phantom imagesused a 15 ◦ flip angle, 256 samples per projection, and 1mm isotropic resolution with a transmit/receive head coil. The -100 -50 0 50 10010 -1 -2 -3 -4 -100 -50 0 50 10010 -1 -2 -3 -4 -100 -50 0 50 10010 -1 -2 -3 -4 -100 -50 0 50 10010 -1 -2 -3 -4 -100 -50 0 50 10010 -1 -2 -3 -4 abcde Fig. 6. Projection sampling patterns (left column) and PSFs (middle column),with plots along x (black, dashed line) and y (gray line) axes in the rightcolumn. FOV shapes: (a) Circle, (b) Ellipse, (c) Rectangle, (d) Oval, and (e)Diamond. The inset PSF images are windowed narrowly to show the low-levelaliasing (arrows) within the desired FOV (dashed lines). The plots show thealiasing peaks and the isotropic resolution in the central lobes. Small FOVsare used for visualization of the variable angular density. ARSON et al. : FOVS IN RADIAL IMAGING 7 in vivo images used a 30 ◦ flip angle, 3 cm slab-selectiveRF excitation, 192 samples per projection, a body coil, andwere acquired in a single 25 second breath-hold. Both fully-sampled and undersampled cylindrical FOVs with 3 and 2 mmisotropic resolution, respectively, are compared to isotropicFOVs requiring the same number of projections.IV. R ESULTS
A. 2D PSFs
Figure 6 shows some sampling patterns designed with the2D anisotropic FOV algorithm and their corresponding PSFs,showing that the desired FOV shapes are achieved. They haveisotropic resolution, as shown in the PSF central lobes. Thereis some low-level aliasing introduced inside the desired FOVfor the anisotropic shapes, shown in the inset PSF images. SeeSection V for a full discussion.Figure 7 shows the relationship between the number ofprojections and the FOV area for elliptical and rectangularFOV shapes. The relationship is quadratic for a circular FOV,and the other shapes are also approximately quadratic. Theanisotropic shapes are slightly more efficient, with efficiencyincreasing as the shapes become narrower. This is due to thelow-level aliasing seen in Fig. 6, which is larger for narrowershapes (see Section V).Some sampling patterns with variable k max patterns andtheir corresponding PSFs are shown in Figure 8. The angularlyvarying projection lengths results in anisotropic resolution,seen in the main lobe of the PSFs. There is again some low-level aliasing introduced inside the desired FOV in Fig. 8aand e, but there is none of this aliasing in b, c, and d. N projections F O V a r ea ( p i x e l s )
700 8001.82 x 10 Fig. 7. Number of projections for elliptical and rectangular FOV shapesversus the resulting FOV (shape) area. All the FOVs have 1 pixel isotropicresolution. There is no loss in efficiency by using anisotropic shapes, and infact they are slightly more efficient than the isotropic case. This increase inefficiency comes at the cost of low-level aliasing within the FOV, seen inFig. 6. -1 -2 -3 -4 -1 -2 -3 -4 -1 -2 -3 -4 -1 -2 -3 -4 -1 -2 -3 -4 abcde -80 -40 0 40 80 -1 -4 -2 0 2 4 -80 -40 0 40 80 -1 -4 -2 0 2 4 -80 -40 0 40 80 -1 -4 -2 0 2 4 -80 -40 0 40 80 -1 -4 -2 0 2 4 -80 -40 0 40 80 -1 -4 -2 0 2 4 Fig. 8. Variable k max sampling patterns (left column) and PSFs (middlecolumn, plots in right column), with enlarged views of the central lobe inset.The black, dashed line plot is along the x -axis, and the gray line is alongthe y -axis in (b), (c), and (e), and along x = y in (a) and (d). FOV/ k max shapes: (a) Circle/Star, (b) Ellipse/Ellipse, (c) Rectangle/Rectangle, (d) Dia-mond/Diamond, and (e) Oval/Ellipse. All the combinations have a minimumresolution size of 1 pixel, with maximum resolutions 2 pixels in (a), (b), and(e), 1.72 pixels in (c), and 1.33 pixels in (d). The shape combinations in (b),(c), and (d) have no low-level aliasing inside the FOV shape, unlike (a) and(e), and the isotropic resolution projections in Fig. 6. Small FOVs are usedfor visibility of the variable angular density. For these three sampling patterns, k max is the dual of theFOV shape, or k max ( φ ) ∝ F OV ( φ + π ) . In this case, thewidth of the aliasing lobes is inversely proportional to theFOV and the angular density compensation, D θ , is uniform(Eq. 20). See Section V for more discussion of the dualshapes. Reducing the resolution also reduces the number ofprojections required. For the same FOV shapes with isotropicresolution, the trajectories in Fig. 8 require between 13 and36% less projections, demonstrating the trade-off that can bemade between resolution in a given dimension and acquisitiontime. TRANSACTIONS ON MEDICAL IMAGING a bc d
Fig. 9. PSFs for 3D PR trajectories with anisotropic FOVs using the spiral-based design. FOVs: (a) Oval
F OV θ and ellipse F OV φ , (b) Cylinder, (c)Cuboid, and (d) Ellipsoid with an ellipsoid k max ,θ shape, shrunk in the k z direction. The top row for each shape shows the PSF in the y = 0 and x = 0 planes, and the bottom row shows the z = 0 plane and a 3-plane view.Similarly to the 2D case, there is some low-level aliasing (arrows) inside thedesired FOV shapes (dashed lines), which is particularly visible in (b). Animations of how the PSF evolves as the algorithm pro-gresses are available at . The principles of the sam-pling approximations used (Fig. 1) are visible in the movies.
B. 3D PSFs
The PSFs for 3D cones trajectories are 2D PR PSFs rotatedabout one axis, as is illustrated in Fig. 4a and b.Figure 9 shows some PSFs for spiral-based 3D PR samplingpatterns. For the variety of shapes and dimensions, the desiredFOV is achieved, and using an oval for
F OV θ and ellipse for F OV φ (Fig. 9a) results in three different FOV dimensions.There are low-level aliasing artifacts present in these PSFs(arrows in Fig. 9a and b). The cuboid (Fig. 9c) has sharpcorners in all three dimensions, and is also well-defined atnon-zero z -values, demonstrating that the azimuthal samplingaliasing shape, F OV φ , is invariant in z , as assumed in Eq. 13.The PSF of an ellipsoid FOV with an ellipsoid k max ,θ is shownin Fig. 9d, demonstrating that variable k max ,θ is feasible.This trajectory has a 50% reduction in Z-gradient strengthrequirements compared to the other trajectories because k max ,θ limits the projection extents in k z . This halves the resolutionin z , but requires 33% less projections than the same FOVshape with 1 pixel isotropic resolution.Coherent aliasing lines are introduced when using the cones-based design, as shown in Fig. 10. The use of a random φ in the cones-based design eliminates the coherent line inthe x - y plane. A line along the z -axis remains because thepolar projection spacings around θ = π are all identical.The tilt of the spiral in k z diffuses the line in the spiral- ba c Fig. 10. PSFs for various 3D PR design methods. (a) Cones-based designwithout randomized φ . (b) Cones-based design with randomized φ . (c)Spiral-based design. The x = 0 plane for the shape in Fig. 9a is shown, andthe images are identically windowed. The arrows indicate coherent streakingartifacts, all of which are eliminated when using the spiral-based design. F O V v o l u m e ( p i x e l s ) N projections sphere1:1:2 cylinder1:2:2 cylinder1:1:2 ellipsoid1:2:2 ellipsoid Fig. 11. Number of projections for 3D anistropic FOV shapes, with ratiosof x : y : z lengths, versus the shape volume. All FOVs have 1 pixel isotropicresolution, and the curves shown are for both the cones-based and spiral-baseddesign methods. Most of the anisotropic shapes are actually more efficient thanthe sphere, but at the cost of low-level aliasing within the FOV, seen in Fig. 9. based design. These coherent lines would not affect fully-sampled acquisitions. In undersampled applications, whichtake advantage of the diffuse 3D PR aliasing, this could causeartifacts, making the spiral-based design advantageous.The scan volume efficiency for some 3D anisotropic FOVshapes is shown in Fig. 11. The anisotropic shapes aregenerally more efficient, although there is more variation inefficiency than in the 2D case (Fig. 7). The increased efficiencyis at the expense of some low-level aliasing inside the FOV.The 1:1:2 ( x : y : z ) shapes only have anisotropy in the polarsampling, and they are more efficient than the 1:2:2 shapesthat also have azimuthal sampling anisotropy. This differenceis caused by the anisotropic F OV φ cutting off portions of F OV θ , which can be seen in Fig. 4. ARSON et al. : FOVS IN RADIAL IMAGING 9 a bc d
Fig. 12. 2D PR Lower leg images with various FOV shapes, shown by thedashed lines. (a) 25 cm circular FOV requiring 393 projections, (b) 7.5 x 25cm elliptical FOV requiring 197 projections, (c) 6.5 x 24 cm rectangular FOVrequiring 195 projections, (d) 12.5 cm circular FOV requiring 196 projections.The non-circular FOV images (b,c) show no increase in aliasing artifacts orloss of resolution but were acquired with half the projections. The circularFOV acquired with half the projections (d) results in significant aliasingartifacts.
C. MRI Experiments
In vivo 2D PR leg images acquired with isotropic andanisotropic FOVs are shown in Fig. 12. The reduced FOVimages (b-d) were acquired with half the number of projectionsas the full FOV image (a). The isotropic reduced FOV (d)results in significant streaking aliasing artifacts, while usinganisotropic reduced FOVs tailored to the shape of the leg (b,c)results in no increase in artifact compared to the full isotropicFOV image. The images without aliasing (a-c) are all slightlyundersampled, as shown by the overlaid FOV shapes, but noaliasing is visible due to the relatively diffuse aliasing patternof PR.Figure 13 shows a representative slice from 3D PR acqui-sitions of a water bottle phantom using different FOVs, eachwith the same number of projections. In the isotropic FOVacquisitions, streaking artifacts are visible within the bottle andemanate from the edge of the bottle (arrows in Fig. 13a andb) because the FOV is not large enough. These artifacts havea peak amplitude of 23.2% of the signal for the cones-based a bc d
Fig. 13. 3D PR phantom images, all acquired with the same number ofprojections. Isotropic FOV using the (a) cones-based method and (b) spiral-based method. Cylindrical FOV using the (c) cones-based method and (d)spiral-based method. The top images are all windowed identically to showthe aliasing artifacts outside the bottle (white arrows). The bottom images arealso identically windowed to show the aliasing artifacts within the bottle forthe undersampled, isotropic FOVs (black arrows). The dashed lines indicatethe supported FOV. design and 11.5% for the spiral-based design. By using a fully-sampled cylindrical FOV that matches the bottle’s shape, thesestreaks are completely shifted outside of the bottle. (arrows inFig. 13c and d). The differences in aliasing diffusivity betweenthe two 3D PR design methods, shown in the PSFs in Fig. 10,can also be seen in the images. There is a single, prominentaliasing streak when using the cones-based design (long, thinarrows), which is diffused with the spiral-based design (short,fat arrows), and the peak aliasing amplitudes also reflect thisdifference.In vivo 3D PR images of a 3 cm abdomen slab acquired in asingle breath-hold are shown in Figs. 14 and 15. The isotropicFOV images have significant streaking artifacts within theabdomen (arrows in Fig. 14a and 15a) because of the highundersampling ratios. Using a thin, squished cylindrical FOVtailored to the anatomy and the excited slab eliminates thesestreaking artifacts, as shown in Fig. 14b. A small degreeof undersampling in this tailored FOV is also tolerated, asevidenced by the lack of streaking artifacts in Fig. 15b. Theisotropic FOV streaking also results in a higher level of signaloutside of the body in the axial images. More streaking isseen just outside the slab in the coronal images with theanisotropic FOV (arrows in Fig. 14b). This is expected becauseof the smaller superior-inferior dimension with the anisotropicFOV relative to the spherical FOV, for which the streaks inthis dimension are outside of the displayed region. Acquiringa fully-sampled 36 cm isotropic FOV at 3 mm isotropicresolution requires 22656 projections, while an undersampled19.6 cm FOV at 2 mm resolution requires 14744 projections,both of which are prohibitively large for single breath-holdimaging. a b
Fig. 14. Axial (top) and coronal (bottom) slices from a thin-slab 3D PR abdomen data-set with 3 mm isotropic resolution acquired in a single breath-hold.(a) Isotropic 11.4 cm FOV requiring 2303 projections. (b) Cylindrical 36 x 23 x 3 cm FOV requiring 2368 projections. Both trajectories were designed usingthe spiral-based method. Streaking artifacts result from the high axial undersampling of the isotropic FOV (arrows in (a)). The tailored FOV eliminates theseartifacts by reducing FOV in the slab dimension. This leads to the streaks in the coronal slice (arrows in (b)), but these are outside of the imaging volume. a b
Fig. 15. Same as Fig. 14, except with 2 mm isotropic resolution and increased undersampling. (a) Isotropic 8 cm FOV requiring 2519 projections. (b)Cylindrical 19.6 x 12.2 x 2.8 cm FOV requiring 2529 projections. The undersampled cylindrical FOV has no noticeable streaking artifacts (arrows), whichobscure the anatomy with the isotropic FOV.
V. D
ISCUSSION
The algorithms presented are designed with fully-sampledFOV parameters, but will also be beneficial for undersampledradial imaging applications, as demonstrated in Fig. 15. Forundersampled applications like VIPR [7], the desired FOVshape and size should be a scaled down version of the region-of-interest for equivalent undersampling factors in every di-mension. The anisotropic FOV projections can also be appliedto the highly constrained backprojection (HYPR) method fortime-resolved MRI [24] because they can be reconstructed byfiltered backprojection. Exam-specific FOVs can be used inall applications because the computation time of the designalgorithms is small. Using an initial localizing scan, thedesired FOV could be drawn or automatically detected andthe acquisition tailored appropriately.For some FOV and k max shapes, the resulting PSFs have low-level aliasing within the desired FOV. This is due to thefinite width of the aliasing lobes, as shown in Fig. 1b, whichis not accounted for in our design algorithms. The projectionangles are designed based on the peak at the center of thesealiasing lobes, leaving the potential for up to half of the lobeto overlap inside the FOV. Overlap is more likely to occur asthe radial separation of neighboring lobes increases, and thusis correlated with (cid:12)(cid:12)(cid:12) dF OV ( φ ) dφ (cid:12)(cid:12)(cid:12) . The online animations of thePSFs show how the origin of the low-level aliasing is primarilyalong the longer edges where this derivative is the largest. Thisaliasing does not originate from minima or maxima of the FOVwhere the angular derivative is zero.Narrower shapes, which have a larger angular derivative,have more desirable efficiency curves in Fig. 7 because theyhave more low-level aliasing within the FOV. For the shapesin Fig. 7, the total low-level aliasing power is approximately ARSON et al. : FOVS IN RADIAL IMAGING 11
F OV ( φ ) /k max ( φ + π ) , which is zero for the dualshapes. This derivative is very large for the FOV and k max shapes in Fig. 8e, which has significant aliasing inside thedesired FOV. Another observation is that dual ellipse samplingcan be formed by scaling an isotropic sampling pattern alongone axis. By Fourier theory, this transformation scales the PSFalong the corresponding axis by the inverse and introducesno low-level aliasing. It is possible that the other shapesmay be formed by some transformation of isotropic case. Wehave found no derivation confirming this suggested low-levelaliasing correlation or any general transformation starting withisotropic sampling.With our method, it is possible to use 3D radial imagingtrajectories to image thin slabs with isotropic resolution, asdemonstrated in Figs. 14 and 15. Previously, this was achievedby using stacks of projections [6], [9], which lacks some ofthe benefits of a true 3D radial acquisition, such as ultra-shortTEs. Stacking the projections also results in a coherent aliasingstreak along the stacking direction, similar to the streakresulting from the cones-based design (see Fig. 10), whichmay be related to artifacts reported in the stacking dimensionwith contrast enhancement [7]. Both methods support onedimension of variable resolution, and stacks of 2D anisotropicFOV projections are also possible.This projection design can also be applied to other centric-based k-space trajectories. It can be directly applied to twistingradial-line (TwiRL) trajectories [25] by specifying the angularspacing of the different acquisitions. Anisotropic resolutioncan also be incorporated by adjusting the twist in the radialline to match k max ( φ ) . Interleaved spiral acquisitions can alsobe adjusted for anisotropic FOVs. The angular spacing ofadjacent interleaves can be determined by our algorithm, andspirals with many interleaves will benefit the most from thisadjustment. This could also be combined with the previousanisotropic FOV spiral method [26] which describes the designof the spirals themselves.VI. C ONCLUSION
We have introduced a new method that designs projectionsfor anisotropic FOVs in radial imaging. These FOVs can beprecisely tailored to non-circular objects or regions-of-interestin 2D and 3D imaging. This allows for scan time reductionswithout introducing aliasing artifacts. For undersampled appli-cations, this method allows for reduction of aliasing artifacts.Algorithms have been presented for the design of 2D and 3DPR trajectories, as well as 3D cones. There is no loss in FOVarea efficiency when using anisotropic FOVs. The algorithmsare very simple and fast, allowing them to be computed on-the-fly. They also support variable trajectory extents which can be used in MRI to favor certain gradients or relax gradientconstraints in 3D cones. A
PPENDIX IF ULL - PROJECTION S PIRAL - BASED D ESIGN
For a spiral-based full-projection design, the spiralling pathis sampled for ≤ θ ≤ π by using θ width = π for the set ofinitial polar samples, and interpolating to ˆΘ[ N polar + 1] = π and ˆ K max [ N polar + 1] = k max ,θ ( π ) . Near θ = π , the polarsample spacing is not as desired because the opposite ends ofthe full-projections are spiralling in opposing directions. Thespacing between these opposing turns varies approximatelylinearly from 1.5 to 0.5 times the desired spacing over each ofthe final two half-turns of the spiral. This spacing discrepancyalso occurs with the isotropic 3D PR spiral design for full-projections [21]. This undersampling can be compensatedfor by adding an extra quarter-turn to the spiral. To dothis, the interpolation is carried out to additional samples of ˆΘ[ N polar + 2] = π + k max ,φ F OV θ ( π ) and ˆ K max [ N polar + 2] = k max ,θ ( ˆΘ[ N polar + 2]) , with N N polar +1 , est = N φ, est .The density compensation must also be slightly modifiedto accomodate both the opposing spiral paths and the extraquarter-turn. Based on the 3D spiral geometry, it can befound that the total spacing from adjacent turns decreasesapproximately linearly between 1 and 0.5 times the desiredspacing identically over the two last half-turns. Thus, the dcffrom Eq. 21 is weighted over each of the half-turns separatelyby a linear ramp from 1 to 0.5 as θ increases.R EFERENCES[1] P. C. Lauterbur, “Image formation by induced local interactions: Ex-amples employing nuclear magnetic resonance,”
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