Excitation and geometrically matched local encoding of curved slices

FULL PAPER Magnetic Resonance in Medicine 000:000–000 (2012) Excitation and Geometrically Matched Local Encoding of Curved Slices Hans Weber,1* Daniel Gallichan,1 Gerrit Schultz,1 Chris A. Cocosco,1 Sebastian Littin,1 Wilfried Reichardt,1 Anna Welz,1 Walter Witschey,2 Jürgen Hennig,1 and Maxim Zaitsev1 have involved the acquisition of multiple planar slices to cover the entire volume of interest followed by curved planar reformations to create images in the coordinate system of the curved 3D anatomical structure (1,2). Although increased automation of these reformation techniques has reduced the required manual interaction, data acquisition is still time consuming and often inefficient, as a larger volume than actually required has to be acquired. The need for 3D acquisitions and complex reformation can be avoided by adaptation of the slice shape to the specific region of interest. This can be of particular interest for dynamic investigations in nonplanar anatomical regions like in fMRI of specific parts of the cortical surface or dynamic measurements in curved vessels. One way to perform curved-slice imaging in combination with linear SEMs is the application of multidimensional radiofrequency (RF)-pulses, as demonstrated by B€ ornert and Sch€affter (3). In principle, tailored RF-pulses allow selection of arbitrarily shaped slices and require no extra hardware. The undesirably long pulse duration can be shortened partially by use of parallel transmit approaches (4,5), but there remain open issues with respect to the sensitivity to off-resonance effects and in particular specific absorption rate constraints which continue to limit their applicability. Besides the selection process, curved-slice imaging also requires a dedicated in-plane encoding strategy. Conventional linear encoding represents a projection of the selected magnetization onto a plane. Depending on the degree of nonorthogonality between the slice-selection and the in-plane encoding gradients, a variety of complications may need to be considered, including nonrectangular voxel shapes with geometric distortions, encoding ambiguities and through-plane dephasing. For a curved slice with curvature restricted to a single dimension, Jochimsen and Norris (6) proposed a projection of the magnetization onto a set of planes. Within a single-shot technique, the curved dimension is approximated by plane segments, which define the planes to be acquired in a 3D k-space. To avoid aliasing, every plane has to be fully sampled even though a part of the information is excluded in the final image. Compared to standard encoding, this leads to a loss in efficiency which is proportional to the number of segments. Consequently, only a rough approximation of the curved slice with a low number of segments is feasible. As an alternative, B€ ornert presented In this work, the concept of excitation and geometrically matched local in-plane encoding of curved slices (ExLoc) is introduced. ExLoc is based on a set of locally near-orthogonal spatial encoding magnetic fields, thus maintaining a local rectangular shape of the individual voxels and avoiding potential problems arising due to highly irregular voxel shapes. Unlike existing methods for exciting curved slices based on multidimensional radiofrequency-pulses, excitation and geometrically matched local encoding of curved slices does not require long duration or computationally expensive radiofrequency-pulses. As each encoding field consists of a superposition of potentially arbitrary (spatially linear or nonlinear) magnetic field components, the resulting field shape can be adapted with high flexibility to the specific region of interest. For extended nonplanar structures, this results in improved relevant volume coverage for fewer excited slices and thus increased efficiency. In addition to the mathematical description for the generation of dedicated encoding fields and data reconstruction, a verification of the ExLoc concept in phantom experiments and examples for in vivo curved single and multislice imaging are presented. Magn Reson Med 000:000– C 2012 Wiley Periodicals, Inc. 000, 2012. V Key words: MRI; curved slice; functional MRI; nonlinear encoding fields; spatial encoding; image reconstruction; PatLoc; ExLoc INTRODUCTION MRI typically allows visualization of thin slices with arbitrary orientation and a rectangular voxel shape. The spatial linearity of the applied spatial encoding magnetic fields (SEMs) is conventionally set as a hardware design requirement. However, this restricts the technique to the selection and in-plane encoding of planar slices. In particular for structures exhibiting a curved three-dimensional (3D) morphology, such as the spine or the cortex, a description with planes is not optimal as it is not possible to show all important details simultaneously in one image. Previous approaches to deal with this problem 1 Department of Radiology, Medical Physics, University Medical Centre Freiburg, Freiburg, Germany. 2 Department of Radiology, University of Pennsylvania, Philadelphia, Pennsylvania, USA. Grant sponsor: German Federal Ministry of Education and Research; Grant number: #13N9298 (INUMAC). *Correspondence to: Hans Weber, Department of Radiology, Medical Physics, University Medical Centre Freiburg, Breisacher Str. 60a, 79106 Freiburg, Germany. E-mail: hans.weber@uniklinik-freiburg.de Received 1 March 2012; revised 25 April 2012; accepted 13 May 2012. DOI 10.1002/mrm.24364 Published online in Wiley Online Library (wileyonlinelibrary.com). C 2012 Wiley Periodicals, Inc. V 1 2 Weber et al. an encoding method for curved slices based on RF-pulse encoding (7). The curved dimension is described by a chain of voxels; each encoded using a set of specially designed 2D RF-pulses. Although, this allows an unwarping of the curved slices, the complexity of the multidimensional RF-pulses results in low spatial resolution along the curved dimension and a high sensitivity to off-resonances. Alternative selection of curved slices without the use of complex multidimensional RF-pulses requires spatial adaptation of the geometry of the encoding field used during RF-transmission. Already in the early 1990s, Lee and Cho (8) showed the selection of a cylindrical volume using a nonlinear SEM and a standard 1D RF-pulse. Further combination with linear SEMs has been shown to allow a shift of the selection volume (9) and additional variation of the RF-pulse offset-frequency a change of its shape (10). But, due to the availability of only one nonlinear SEM, flexibility with respect to form and orientation of the selected volume was still strongly limited. As these methods were intended for 3D volume selection, linear encoding could be applied without restraints. In this work, we present a concept (excitation and geometrically matched local encoding of curved slices (ExLoc)) for curved-slice imaging, where curvature, orientation and position are each adjusted to the object under investigation with high flexibility. Importantly, we complement curved-slice selection by geometrically matched local in-plane encoding along the curved surface to maintain a locally rectangular voxel-shape. The concept itself is based on the application of a set of locally near-orthogonal SEMs, which can have linear and nonlinear spatial variation, in combination with standard RF-excitation. In terms of the encoding fields, ExLoc represents an extension of PatLoc imaging (11), which uses nonlinear SEMs for in-plane encoding of planar slices. As for standard imaging using conventional linear field gradients, each SEM can be directly allocated to a slice selection, readout and phase encoding field. The near orthogonality of the fields used for in-plane encoding to each other and to the slice-selection field guarantees a projection of the selected magnetization approximately perpendicular to the plane of the curved slice. Unlike linear in-plane encoding of the curved slice, this approach is able to maintain a close-to-rectangular shape for all voxels. Besides a mathematical description for the generation of dedicated SEMs and data reconstruction, we present a concept verification using phantom experiments and examples of in vivo (multi-)curved-slice imaging. THEORY For arbitrary SEMs, the signal s from an RF receive coil with homogenous sensitivity is described by a generalized signal-equation, neglecting relaxation effects, as previously shown for PatLoc imaging (12): Z T sðkÞ ¼ rðxÞeik wðxÞ dx ½1
V The quantity r(x) describes the magnetization excited in the volume V at position x. The components of the multidimensional function w(x) correspond to the SEM sensitivities. The variable k describes the PatLoc k-space; its components can simply be defined as the effective currents driving the individual SEMs integrated over time: ki ¼ $Ii(t)dt. The dimension of k and w(x) represents the number of SEMs applied for spatial localization. Within this study, we assume that only two SEMs are applied after RF excitation (read and phase encoding), thus w(x) and k have only two components, w(x) ¼ (wR(x), wp(x))T and kT ¼ (kR,kp). The slice-selection field, BS(x), is then given by BS(x) ¼ ISwS(x). Excitation with an Arbitrary SEM For a standard slice-selective RF-pulse, the excitation volume V and thus shape, orientation and position of the selected slice, is defined by both the frequency properties of the RF-pulse and the geometry of the constant SEM BS(x) applied during transmission: for an ideal frequency-selective pulse all spins within a certain frequency range Dv are excited and all spins precessing at a different frequency are not excited at all. The precession frequency v of a spin at location x is given by v ¼ gBS(x), with g being the gyromagnetic ratio. Thus, the excited volume can be described as V ¼ fx 2 R2 jgBS ðxÞ 2 Dvg: ½2
The frequency range Dv ¼ [v0  BW/2,v0 þ BW/2] results from the RF-pulse bandwidth BW and its frequency offset v0. For a small bandwidth, the excited volume will resemble a thin slice whose borders are formed by the isosurfaces where the SEM has the field strength BSB ¼ (v0 6 BW/2)/g. Where a simple linear gradient would result in a conventional planar slice, application of a nonlinear SEM yields a non-cubical volume V, representing a curved slice. To excite a slice with the desired shape, the geometry of the slice-selection SEM Bs(x) and the RF-pulse frequency property v0 have to be optimized to match Eq. 2. In case of multiple possible solutions further boundary parameters such as bandwidth and field strength can be taken into account. Efficient In-Plane Encoding of Curved Slices In principle, in-plane encoding of a curved slice can be performed with conventional linear SEMs. However, this is not an optimal choice as it can lead to a variety of complications and inefficiencies. For efficient encoding, the frequency dispersion generated by the SEMs BR(x) and BP(x) inside the curved slice has to be as large as possible. No dispersion should occur along the direction perpendicular to the isosurfaces of the slice (¼ through the slice), as this does not contribute to in-plane encoding but may results in signal loss due to phase cancellations across the direction normal to the curved slice. As a consequence, the most efficient encoding is achieved with the local gradients of BR(x) and BP(x) being oriented as perpendicular as possible to each other and to the local gradient of the SEM BS(x) applied for slice selection. Including also the aim for maximum gradient strength, the design criterion for the dedicated SEMs can be expressed as maxfðGR ðxÞ  GP ðxÞÞ  GS ðxÞg ½3
with Gi(x) ¼ !Bi(x) for all points x of the excited volume V. ExLoc 3 FIG. 1. Cross sections of possible ExLoc slices using first (x, y) and second order (x2  y2, 2xy) (a), or first and fourth order (x4  6x2y2 þ y4, 4xy(x2  y2)) SEM components (b). Frequency offset and shape of the encoding fields used for slice selection (dotted line) define the slice to be selected (black area). Geometrically matched local in-plane encoding (solid line) along the curved dimension results in rectangular voxel shapes as shown by the intersections of the isocontours. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] Realization of Dedicated SEMs n X Image Reconstruction Because of the curvature of the ExLoc slice, the signal equation (Eq. 1) requires analysis in 3D. However, based on a 3D variable transformation a ¼ u(x), the problem can be reduced to 2D (for details see appendix): Z ~ bÞeik R aþk P b dadb: sðk R ; k P Þ ¼ ^rða; bÞdða; ½7
U In practice, freedom of choice regarding the geometry of the SEMs is not only constrained by Maxwell’s equations but also by the practical feasibility. To increase flexibility, all ExLoc SEMs Bi(x) are created by a superposition of n spatially linear and/or nonlinear varying field components (termed SEM components in this article). These SEM components are characterized by the fact that they are physically realizable. Then, the SEMs used for imaging are described according to Bi ðIi ; xÞ ¼ Ii ci ðxÞ ¼ Ii 2xy) (a), or first and fourth order (x4  6x2y2 þ y4, 4xy (x2  y2)) SEM components (b). In combination with the offset frequency the isocontours of the slice-selection SEM BS(x) define the selected slice; the additional superposition with one of the in-plane SEMs BP(x) yields the voxel shapes. The difference in voxel shape for linear and ExLoc encoding is demonstrated in Fig. 2 for a curved slice with shape comparable to the one shown in Fig. 1a. For geometrically matched local in-plane encoding as proposed by the ExLoc concept, the locally orthogonal encoding fields result in a more favorable close-torectangular voxel shape. vi;l jl ðxÞ ¼ Ii xTi nðxÞ ¼ wTi nðxÞ; l¼1 Thus, the ExLoc data correspond to the Fourier transform of a distorted and intensity modulated representa~ bÞ) of the image in the so-called (PatLoc) tion (^rða; bÞdða; encoding space. The magnetization within the curved slice can therefore be retrieved by (a) performing an inverse FFT to the signal data, followed by (b) division by d̃(a,b) and by (c) transforming the distorted image from encoding space coordinates a ¼ (a,b,c) back to Cartesian object space coordinates x ¼ (x,y,z) according to x ¼ u21(a). Within encoding space, the excited volume U represents a plane slice with equidistant voxel spacing. As for in-plane PatLoc encoding [c.f. (12)], the in- ½4
wTi IixTi . with The spatial variation of the SEM sensi¼ tivity used for encoding, ci(x), is itself defined by the current weightings vi,l of the SEM components and their individual sensitivities jl. Thus, in practice, the final field geometry can be adjusted by varying the currents driving the individual SEM components. As a consequence, the ideal SEM B*S(x) required for the selection of the desired slice is approximated with the technically feasible SEM BS(x). The total weighting vector wS (composed of elements ISvS,l) can be determined by minimizing minw |BS ðxÞ  wTS nðxÞ| ½5
for all points x within the slice volume V. The SEM design criterion (Eq. 3) yields the spatial derivatives of the ideal SEMs for in-plane encoding (G*R(x), G*P(x)). Analogous to Eq. 5, the individual sensitivity weightings vR,l and vP,l for the feasible SEMs BR(x) and BP(x) are determined by minimizing minw |Gi ðxÞ  wTi rnðxÞ| ½6
within the volume V as Gi(x) ¼ !Bi(x) ¼ !(wTi n(x)) wTi !=n(x). As an example, Fig. 1 shows cross-sections for feasible slice shapes using first (x,y) and second order (x2  y2, FIG. 2. Blow up of a slice cross section as shown in Fig. 1a. The intersection with the isocontours represents the resulting voxel shapes for linear (dotted line) and ExLoc (solid line) in-plane encoding. Whereas for linear encoding the voxels are distorted, a local rectangular shape is maintained for ExLoc encoding along the curved dimension. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] 4 Weber et al. plane voxel dimensions, and subsequently the position (ai,bi) for each voxel i can be determined from kR,max, kP,max according to the Nyquist criterion. The remaining coordinate c along the slice selection dimension is defined by the excitation frequency v0 and the encoding field BS applied during RF-transmission according to ~ S ðaÞ ¼ gIS c v0 ¼ gBS ðxÞ ¼ gIS cS ðxÞ ¼ gIS c ½8
where IS is the effective current driving the slice-selection SEM. Multi-Slice Imaging As in conventional imaging, ExLoc also offers increased volume coverage using multislice acquisitions by repetitive slice selection with varying RF-pulse frequency offsets v0. However, for standard multislice selection with constant RF-pulse bandwidth and constant frequencyoffset increment, the nonlinearity of the slice-selection SEM would not only result in varying thickness along the curved slice dimensions (varying intraslice thickness) but also varying interslice thickness and spacing between adjacent slices as shown in Fig. 3a. One possible solution for the selection of multiple slices with identical thickness [at the field of view (FOV) center] and equidistant spacing might be to spatially shift the nonlinear SEM for every selection, resulting in shifted copies of the original slice. But, as slice shapes are no longer complementary, adjacent slices would no longer allow an efficient coverage (Fig. 3b). As an alternative, individual adaptation of the RF-pulse bandwidth (BWi) and frequency offset (vi) for each individual slice allows a defined slice position and thickness in combination with complementary slice shapes (Fig. 3c). METHODS The ExLoc concept was implemented on 3 T MAGNETOM Trio Tim system (Siemens, Erlangen, Germany) equipped with a PatLoc gradient insert (13,14). Although the system offers only two second order (x2  y2, 2xy) SEM components in addition to the three first order ones (x, y, z), creation of encoding fields for selection and geometrically matched local in-plane encoding of slices with one curved dimension as the one shown in Fig. 1a is possible. The curved dimension lies within the xyplane where the curvature, orientation and position can be adapted. The conventional linear z-gradient is retained for encoding along the z-axis. Prior to imaging experiments, field maps of the nonlinear SEM components were fitted with their analytical expressions including correction terms for amplitude, off-center and orientation and served as basis for calculation of the dedicated SEMs and image reconstruction. Fitting with a third order polynomial was also applied to the slope of the slice-selection SEM along the slice normal nV(x) at the FOV center to allow automatic calculation of the individual RF-Pulse bandwidths and frequency offset increments for multislice imaging. An adapted spin echo (SE) sequence with phase encoding along the curved and read encoding along the noncurved dimension of the slice was used for ExLoc data acquisition. For mapping FIG. 3. Simulated cross-sections of a stack of five curved slices. a: Conventional multislice selection with fixed bandwidth and constant frequency-offset increment results in varying thickness and spacing among the individual slices. b: Slice thickness at the FOV center and spacing can be kept constant by shifting the slice-selection field for every slice. However, as the shapes of the individual slices are no longer complementary, efficient filling of the gap between the slices is not possible. c: With individual adaptation of bandwidth and frequency offset for every slice as proposed for ExLoc multislice imaging, constant thickness at the FOV center and spacing by maintaining complementary slice shapes can be achieved. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] of the slice cross-section, the spatial encoding part was replaced with conventional linear encoding with an appropriately selected orientation. In this first implementation of the ExLoc concept, slice curvature, orientation and position were adjusted manually to the object under investigation. To guide positioning, slice cross-sections overlaid on a localizer image were simulated in situ at the scanner console based on the field-map data (see, e.g., Fig. 7a). As the available SEM components represented two sets of fields with local orthogonal gradients in the xy-plane [(x, y) and (x2  y2, 2xy)], the ideal weightings vP,l for the effective phase encoding field could be determined directly by applying the corresponding rotations. Image reconstruction from raw data was performed with MATLAB (The MathWork, Natick, MA, USA) as outlined in the theory section. The transformation x ¼ u1(a) of the image from encoding space (a,b,c) into the undistorted object space (x,y,z) was achieved by solving the system cP ðx; y; zÞ  ai ¼ 0 cR ðx; y; zÞ  bi ¼ 0 ½9
cS ðx; y; zÞ  c ¼ 0 for every voxel i using the trust-region dogleg algorithm (15). To access the slice thickness, the transformation was repeated for the two isosurfaces forming the slice borders. Voxel size, position in the slice-intrinsic coordinates (u,v,w) and FOV along the curved dimension were calculated stepwise from the voxel positions determined above. For slice-thickness specification minimum and maximum thickness within the FOV are stated. Verification of the ExLoc selection, local in-plane encoding and reconstruction process was performed using a dedicated geometry phantom. It consisted of a 4-mm thick acrylic-glass plate bent along one dimension to create a curved surface up to an angle of 100 , placed in a container of doped water. The plate itself was perforated with holes arranged in three lines with ExLoc 5 FIG. 5. Cross section of a stack of curved slices acquired in conventional multislice mode (a) and ExLoc multislice mode (b). Conventional acquisition leads to varying global slice-thickness and varying interslice distance. Using individual adaptation of RF-pulse bandwidth and frequency, offset slices with comparable centerthickness and interslice distance can be selected. different hole diameter for each line and constant spacing. To demonstrate the rectangular projection orientation during spatial encoding, the holes were oriented perpendicular to the plane of the plate. For demonstration of the multislice mode, a standard bottle-phantom was used. In vivo singleslice and multislice measurements were performed on a human volunteer after approval of the local ethics-committee and informed consent was obtained. During ExLoc imaging, linear and nonlinear SEM components are driven simultaneously. For compliance with acoustic-noise limits human ear-corrected sound-pressure level was checked with a calibrated microphone for all sequences prior to in vivo experiments (16). To ensure prevention of peripheral nerve stimulation both sequences were first applied with downsized encoding-field amplitudes in order not to exceed a switching rate of 20 T/s (Normal Mode). Within successive repetitions the amplitudes were increased in 10% steps up to the desired maximum. An increment of 10% was chosen to stay below the minimum reported difference between perception and pain level for peripheral nerve stimulation in the range of 15 and 32% (17). RESULTS FIG. 4. a: Cross section of the selected ExLoc slice, whose shape, orientation and position is adapted to the geometry phantom. The arrows mark position and orientation of the equidistant holes. b: Local ExLoc in-plane encoding of the curved slice results in a geometrical correct representation of the phantom as indicated by the equidistant bars. Closer inspection of the holes—see magnification of the dashed box—reveals sharp edges, independent of their position on the phantom. The overlaid intensity profile was acquired along the dashed line. c: Linear in-plane encoding corresponds to a projection of the curved slice onto a flat plane. This results in a geometrically incorrect representation of the object as shown by the varying distance between the holes. Intensity increase within the image toward the center is due to the varying slice thickness. Increasing misalignment of hole orientation and projection direction toward the side of the phantom causes distorted voxel shapes resulting in increasing blurring of the hole edges. The cross section of an excited curved slice using the ExLoc technique is presented in Fig. 4a. Its shape, orientation and position are adapted to the geometry phantom so that the selected slice contains the entire phantom. Arrows mark the position of the equidistant holes, which are oriented perpendicular to the plane of the phantom. The corresponding ExLoc in-plane encoded, high resolution image of the curved slice is shown in Fig. 4b (SE; pulse repetition time/echo time: 100/18 ms; matrix: 436  436; FOV: 20.7  16.0 cm; slice thickness: 5.0/12.9 mm). As visualization of a curved slice in 3D is difficult, visualization in 2D is desirable. Therefore, all ExLoc images within this study are presented in the slice-intrinsic, curvilinear coordinate system with coordinates (u,v,w). This coordinate system is easily achieved by transforming the encoding space system [with coordinates (a,b,c)] into object space according to u1. Thus, it allows a distortion-free representation of the curved slice 6 Weber et al. FIG. 6. Singleslice ExLoc image of a human brain, acquired with an adapted SE sequence. a: Curvature, orientation and position of the slice are adapted to the lateral rim as shown on the localizer image. b: The reconstructed ExLoc image yields a panorama view of the left hemisphere. The area of maximum slice thickness exhibits the largest voxel sizes. Thus, lowest resolution but also lowest noise is achieved. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] on a plane. Within the ExLoc Image, the holes exhibit similar distance as indicated by the bars, demonstrating geometrical correct transformation of the image from the encoding space into the slice intrinsic coordinate system. As a result of accurate determination of the individual voxel-volumes, the reconstructed image exhibits a relatively homogeneous intensity. The overlaid line in the top part shows the intensity profile acquired along the dashed line. The resulting intensity increase toward the edges is due to uncompensated RF-coil-sensitivity inhomogeneities. The magnified area within the dashed box reveals sharp edges of all holes, independent from their position on the curved plate. They illustrate the successful projection of the excited spins globally perpendicular to the plane of the selected curved slice. For comparison, Fig. 4c shows the result for conventional spatial in-plane encoding using the linear x gradient instead of encoding along the curved dimension (FOV: 19.2  16.0 cm). The representation of the object is distorted as stressed by the comparison with the red bars of equal length. Even though this could be corrected for, the increasing misalignment of projection direction and slice normal toward the edges results in distorted voxel shapes similar to the ones sketched in Fig. 2. For the given phantom geometry this manifests as blurred edges of the holes, representing an effective loss in spatial resolution for structures of this orientation. In addition, image intensity increases not only toward the edges but also toward the center. The latter is due to the varying slice thickness, left uncompensated to demonstrate the magnitude of this effect. Figure 5 presents cross sections of stacks of curved slices within the homogenous phantom. The conventional method to achieve multislice imaging with constant RFpulse bandwidth and constant frequency-offset increment results in slices with varying center thickness and varying interslice distance due to the nonlinear nature of the SEM applied during slice selection (Fig. 5a). With ExLoc bandwidth and increment adaptation for every RF-pulse the slices exhibit constant center-thickness and interslice distance as illustrated in Fig. 5b. Figure 6a shows an axial localizer image with the calculated cross section of a single ExLoc slice. Its shape, orientation and position are adjusted to the lateral rim of the brain. The resulting ExLoc image (Fig. 6b; SE; pulse repetition time/echo time: 600/25 ms; matrix: 256  256; FOV: 23.8  21.0 cm; slice thickness: 2.4/6.8 mm) represents a panorama of the left cerebral hemisphere. The individual gyri are clearly distinguishable. Variation in resolution is due to the nonlinear nature of the in-plane (phase) SEM as already known from PatLoc imaging (12). The smallest voxels are in the area of minimum slice thickness. As visible in the background outside the object, noise is increased in these areas. A stack of 7 ExLoc slices (SE; pulse repetition time/ echo time: 600/25 ms; matrix: 256  256; FOV: 29.8  23.7  21.0 cm; slice thickness: 3.5/9.0 mm) is presented in Fig. 7. The stack is orientated along the occipital lobe (Fig. 7a), representing a stepwise progression through the brain. Compared to conventional planar multislice imaging, a greater number of spatially separated brain regions with comparable distance from the skull can now be imaged in the same slice. In multislice mode, the individual slices are encoded with different parts of the nonlinear SEMs. This causes a variation in resolution between the slices, combined with different FOV sizes along the curved dimension. DISCUSSION The presented work demonstrates the feasibility of the ExLoc concept for selection and geometrically matched local in-plane encoding of curved slices without the ExLoc 7 FIG. 7. Stack of curved SE slices, covering the occipital lobe with a few slices as shown on the localizer (a). The slices (b–h) represent a stepwise progression through the brain form posterior to anterior in slice-intrinsic, curved coordinates. The difference in FOV is due to the shape of the phase encoding field varying from slice to slice (applied along the horizontal image dimension). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] need for time-consuming and specific absorption rate intensive multidimensional RF-pulses. Because of the superposition of linear and nonlinear SEM components, slice curvature, orientation, and position can be adjusted with high flexibility. As demonstrated in phantom experiments, the proposed local in-plane encoding along the plane of the curved-slice achieves a local rectangular voxel-shape and allows unwarping of the curved slice without ambiguities. Because of its generalized formalism, the ExLoc data-reconstruction itself is easy to implement and requires no noteworthy computational power and time. From a safety perspective, the simultaneous application of linear and nonlinear SEM components showed no signs of inducing peripheral nerve stimulation for this implementation. This is in accordance with previously presented safety aspects for in vivo imaging with the PatLoc gradient insert (16). Application of nonlinear SEMs for slice selection results in slices with varying thickness along the curved dimension. For the given hardware, a maximum variation of 60% of the maximum slice thickness was observed. From a diagnostic point of view, the observed variations in slice thickness as well as the associated noise variation should be acceptable for applications such as functional MRI. In addition, the nature of the in-plane encoding fields causes spatial varying resolution within the images. However, compared to other nonlinear imaging techniques such as pure PatLoc imaging (18), the variation is comparably low, resulting in a suitable base resolution in the images. As all SEMs applied in this work exhibit strict local orthogonality, both components, variation in slice thickness and variation in-plane resolution, are mutually dependent (19). To which extend this strict dependency is relaxed for near-orthogonal nonlinear SEMs, is a topic of ongoing research. Nevertheless, both components can be compensated separately up to a certain degree: Comparable short multidimensional RF-pulses with low specific absorption rate allow selection of slices with constant thickness (20). In transmit encoding space, such a curved slice with constant thickness corresponds to a planar slice with varying thickness. The description of the thickness variation along the corresponding dimension requires low frequencies only. Therefore, less complex RF-pulses are necessary compared to selection of the same slice using conventional linear gradients only. Application of a dedicated subsampling strategy exploiting encoding properties of the nonlinear SEMs (21) allows adaptation of the variation in in-plane voxel size toward a more homogenous resolution. Intensity variations due to varying voxel volume can be compensated based on the information provided by the reconstruction model. In the case of moderately curved slices, image quality is acceptable even without intensity correction, as the corresponding variation is not significant compared to variations that result, for example, from inhomogeneous RF-coil sensitivities. Also, for multislice imaging, ExLoc keeps the advantage of high quality profiles selected with short RF-pulses compared to other curved-slice imaging techniques (6,7). With slice-specific adjustment of RF-pulse bandwidth and frequency offset, interslice thickness and spacing between adjacent slices can be kept constant, thus, simplifying image interpretation. Hence, ExLoc offers efficient coverage also for extended volumes, which cannot be described with a single curved-slice. Because of the near-orthogonality of the applied SEMs, alternative slab-selection in combination with 3D encoding while maintaining the curved geometry is also possible. Provided the gradients of the applied SEMs exhibit sufficient strength and alignment within the volume of interest as requested by the design criterion, selection and in-plane encoding of slices with more than one curved dimension is possible as well. However, for the final representation of the image on a screen, a suitable map projection technique has to be added. Minimum requirements for the value of the design criterion to achieve suitable imaging are a topic of ongoing research. In case of low values it might be advisable to favor orthogonality and to compensate for low image resolution with additional in-plane SEMs as proposed in Ref. 22. 8 Slice selection using nonlinear SEMs is typically accompanied by spatial-selection ambiguities. One pragmatic method to prevent excitation of multiple slices is to shift the undesired slices out of the object. As an alternative in case of multiple receivers, RF-sensitivities could be used to distinguish between the signals originating from different slices. In particular applications this might allow to increase acquisition efficiency, as multiple slices could be imaged simultaneously. Instead of adaptation to the structure under investigation, ExLoc might also be used to adjust the slice shape and orientation to the provided distribution of RF-receiver sensitivities for improved encoding efficiency at higher acceleration factors (23), in a similar way to the design motivation of O-space imaging (24). Free from being constrained to planar slices, ExLoc allows imaging in the coordinate system of a curved structure. This flexibility might be beneficial for a more efficient illustration of organic structures and allows a greater flexibility for choosing brain regions which are imaged in the same slice, as demonstrated by the presented in vivo brain images. With conventional imaging, gathering additional information from such a representation would require time-consuming 3D acquisitions followed by complex data reformatting while wasting a lot of the acquired data (1). In this work, the geometry restrictions of the available SEM components mean that the chosen curved slices in the brain only follow the anatomy for the particular z-positions shown in the localizing image. At other z-positions, the curved slices will be less physiologically relevant. In general, the flexibility of ExLoc regarding possible slice shapes, orientations and positions—and thus, its practicability—is dependent on the number and or the geometry of available SEM components. For instance, a higher number of individual SEM components is expected to allow generation of higher order SEM fields for selection of slices with stronger curvature. A higher number of components is also expected to enable a more local adaptation of the SEM’s spatial variation for an improved follow up of comparable smooth structures such as the external shape of a brain. Therefore, multicoil systems have the potential to massively increase the range of possible slice shapes, including those with curvature along both dimensions. For the lower power and switching rate requirements of dynamic shimming, such a system has been presented recently (25). In addition, slices with a smooth curvature variation might also serve as basis slice shape with the more detailed curvature achieved by comparably short multidimensional pulses. Hence, Exloc’s maximum achievable flexibility in slice shape is still a topic of ongoing research. Nevertheless, it is important to bear in mind that the major restrictions to the shape of the possible SEM fields in the presented work arise from the currently available hardware, and that we have demonstrated a proof of principle of some of the extended possibilities of imaging with nonlinear SEMs. CONCLUSION ExLoc allows selection of slices with flexible curvature, orientation, and position. No time consuming multidimensional and specific absorption rate intensive RF- Weber et al. pulses are necessary. Spatial encoding along the curved plane of the slice allows maintaining a local rectangular voxel-shape for maximized efficient resolution and slice unwarping. Because of the applied near-orthogonal encoding-fields, 3D encoding by maintaining the curved geometry for each partition is also possible. Combination of the ExLoc concept with higher-order gradient systems is necessary to further extend the variety of possible slice shapes. Nevertheless, the presented ExLoc concept presents a step toward an object customized MRI system. For purely morphological imaging curved volumes can be retrospectively reconstructed from a rectilinear 3Ddataset which covers the entire region of interest at the expense of some additional (but usually tolerable) acquisition time. The application of ExLoc therefore is most promising for dynamic examinations of curved structures where time constraints restrict the size of the volume which can be captured in rectilinear coordinates, that is, for specific parts of the cortex in fMRI or dynamic flow imaging of curved vessels. ACKNOWLEDGMENTS The authors like to thank Dr. Gigi Galiana and Dr. Martin Haas for fruitful discussions, Dr. Andrew Dewdney for his magnet expertise and Dr. Nico Splitthoff and Denis Kokorin for computational support. APPENDIX Reduction of the Signal Equation from 3D to 2D in ExLoc Based on a variable transformation, the 3D signal equation (Eq. 1) can be reduced to 2D. Define the 3D-variable transformation a :¼ u(x), where the first two components can be chosen freely and where the third component is given by u3(x) ¼ gBS(x); also define ^r ¼ r  u1 , ~ ¼ w  u1 , U :¼ u(V), and d̃ as the inverse of the absow lute value of the determinant of the Jacobian of u. Then, the signal equation can formally be written as: Z Z T ~ ikT wðaÞ ~ rðxÞeik wðxÞ dx ¼ ~rðaÞdðaÞe da U  Z ZV Z ~ ~ b; cÞeikT wða;b;cÞ dc dadb ¼ ~rða; b; cÞdða; Z  ZZ ~ ~ bÞeikT wða;bÞ ½A1
dadb ~rða; b; cÞdc  dða; ZZ ~ ~ bÞeikT wða;bÞ dadb; ¼ ^rða; bÞdða; Z with ^rða; bÞ :¼ ^rða; b; cÞdc sðkÞ ¼ Equation A1 simply expresses the fact that the signal can be represented as a 2D-problem by integrating along the direction normal to the curved slice. Please note that, in general, the presented 2D signal equation is only an approximation because it ignores a possible variation of both the volumetric correction d̃ and the in-plane ~ along the direction nV(x) perpendicencoding function w ular to the curved slice V. However, at least for thin slices, the volumetric correction is nearly constant along nV(x); moreover, it can be concluded that the above ExLoc 9 approximation is very good if through-slice dephasing is prevented by choosing the effective in-plane encoding functions wR ¼ BR(x)/IR, and wp ¼ BP(x)/IP such that their gradients are locally oriented orthogonal to the normal nV(x) as proposed by the design criterion. The signal equation is simplified even further by choosing (u1, u2) ~ ¼ Id and the equation reads: ¼ (WR, WP) because then w sðkR ; kP Þ ¼ Z ~ bÞeikR aþkP b dadb: rða; bÞdða; ½A2
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