Magn Reson Mater Phy DOI 10.1007/s10334-013-0417-0

RESEARCH ARTICLE

Spatial phase encoding exploiting the Bloch–Siegert shift effect Ralf Karta¨usch • Toni Driessle • Thomas Kampf • Thomas Christian Basse-Lu¨sebrink • Uvo Christoph Hoelscher Peter Michael Jakob • Florian Fidler • Xavier Helluy

•

Received: 24 May 2013 / Revised: 28 October 2013 / Accepted: 29 October 2013 Ó ESMRMB 2013

Abstract Objective The present work introduces an alternative to the conventional B0 -gradient spatial phase encoding technique. By applying far off-resonant radiofrequency (RF) pulses, a spatially dependent phase shift is introduced to the on-resonant transverse magnetization. This so-called Bloch–Siegert (BS) phase shift has been recently used for Bþ 1 -mapping. The current work presents the theoretical background for the BS spatial encoding technique (BSSET) using RF-gradients. Materials and methods Since the BS-gradient leads to nonlinear encoding, an adapted reconstruction method was developed to obtain undistorted images. To replace conventional phase encoding gradients, BS-SET was implemented in a two-dimensional (2D) spin echo sequence on a 0.5 T portable MR scanner. Results A 2D spin echo (SE) measurement imaged along a single dimension using the BS-SET was compared to a conventional SE 2D measurement. The proposed reconstruction method yielded undistorted images. Conclusions BS-gradients were demonstrated as a feasible option for spatial phase encoding. Furthermore,

R. Karta¨usch (&)  T. Kampf  T. C. Basse-Lu¨sebrink  U. C. Hoelscher  P. M. Jakob  X. Helluy Department of Experimental Physics 5, University of Wu¨rzburg, Am Hubland, 97074 Wu¨rzburg, Germany e-mail: [email protected] R. Karta¨usch  T. C. Basse-Lu¨sebrink  U. C. Hoelscher  P. M. Jakob  F. Fidler  X. Helluy Research Center for Magnetic Resonance Bavaria e.V., Wu¨rzburg, Germany R. Karta¨usch  T. Driessle Pure Devices GmbH, Wu¨rzburg, Germany

undistorted BS-SET images could be obtained using the proposed reconstruction method. Keywords Magnetic resonance imaging  RF gradients  Spatial encoding  Bloch–Siegert

Introduction Conventional magnetic resonance (MR) images are spatially encoded using a gradient coil set that creates three orthogonal, constant B0 -gradients within the volume of interest. In general, the gradient coils are self-shielded, water cooled and driven by high-power amplifiers. However, the application of conventional gradient coils is subject to certain limitations. For example, quickly switching the B0 gradients can result in severe vibrations and acoustic noise [1]. Furthermore, since B0 -gradients are pulsed, eddy currents are induced in nearby conductors. Moreover, peripheral nerve stimulation is caused by high slew rates of B0 -gradients, and thus imposes a lower limit on ramp times. To avoid some of the problems introduced by B0 -gradients, radiofrequency (RF)-gradients have long been used to spatially encode MR images. This method allows fast switching to high amplitudes without inducing eddy currents [2], acoustic vibrations, or causing nerve stimulation. Nevertheless, high RF power is necessary to obtain a sufficient phase shift [2]. Therefore, particularly with high magnetic field strengths, the relative high specific absorption rate (SAR) must be considered. In 1979, Hoult et al. introduced rotating frame imaging, a RF-gradient–based imaging method [3–5]. Since then, the technique has seen many improvements [2], such as fast

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imaging using RF Echo Planar imaging [6] and phase sensitive encoding, which mimics the effect of B0 -gradients. More recently, the phase-encoding potential of transmit/receive phased arrays was investigated [7, 8]. However, as phase shifts in phased arrays are limited, the technique [7] has not yet reached the efficiency of Fourier encoding. This paper investigates an alternative to B0 -gradient– based phase encoding. Our approach exploits the so-called Bloch–Siegert (BS) effect, originally described by Ramsey et al., that is generated by off-resonant RF-pulses [9, 10]. The BS effect creates a phase shift in the transverse magnetization and does not significantly tip the magnetization vector [11]. Using adapted RF-coils with a linear shaped B1 -field, this phase shift can be used for spatial phase encoding. Thus, the BS spatial encoding technique (BSSET) was implemented into a standard spin echo (SE) imaging sequence for phase encoding. In the present work, two-dimensional (2D) images were acquired on a 0.5 T system and an adapted reconstruction technique was applied to obtain undistorted images.

UBS ¼

Zs 0

DxRF ¼a1 xB1

ð1Þ

with xB1 ¼ cBþ 1 being the nutation frequency associated with the Bþ -magnitude of the RF-field, and DxRF the dif1 ference between the Larmor frequency and the off-resonant-frequency of the circular RF-field xRF ðDxRF ¼ x0  xRF Þ. Ramsey [11] has shown that the BS-shift takes the following form when constraint 1 applies: xBS ¼

x2B1 xB ¼ 1: 2DxRF 2a

ð2Þ

BS-induced phase shifts If the frequency shift induced by magnetic field inhomogeneities Dx0 is negligible compared to the off-resonance frequency Dx0  DxRF

ð3Þ

an off-resonant RF-pulse applied with a duration of s results in the phase shift [11]:

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x2B1 ðtÞ dt: 2DxRF ðtÞ

ð4Þ

0

x2 s

B1 . Under thus inducing a phase shift of UBS ¼ 2Dx RF constraint 1, the phase shift can be calculated by xB1 s . The induced phase shift combined with UBS ¼ 2a imaging can then be used to map the spatial variation of the Bþ 1 -field [11–13], as first described by Emsley et al. to explain phase artifacts in NMR spectra [14].

BS-gradient As indicated above, a spatially varying Bþ 1 -field leads to a x2 ð xÞs

B1 spatially dependent phase shift UBS ðxÞ ¼ 2Dx . ConseRF þ quently, a B1 -field with a constant gradient of the magnitude

dBþ ðxÞ

( d1 x ¼ G) results in a spatially dependent phase shift of

Theory

The BS-shift xBS , commonly known in the MR community as the frequency shift of the Larmor frequency x0 , is induced by a far off-resonant circular RF-field fulfilling the following constraint [10]:

Zs

The following assumes a constant frequency offset and a constant Bþ 1 magnitude over its duration s (hard pulse),

UBS ðxÞ ¼

BS basics

xBS ðtÞdt ¼

ðcGxÞ2 s: 2DxRF

ð5Þ

Applying a BS-pulse with a certain duration and fixed magnitude (Fig. 1b) is similar to applying a conventional B0 -gradient (Fig. 1a). Assuming a constant B0 -gradient, a linear phase shift is introduced (Fig. 1a). However, since the phase shift is 2 proportional to Bþ with the BS-SET, quadratic phase 1 encoding is introduced by a constant off-resonant RF-fieldgradient (Fig. 1b). Thus, to obtain an undistorted image, an adapted reconstruction based on the spatial dependence of the Bþ 1 -field is needed. An alternative method is to apply the Bþ 1 -gradient on-resonant, and thus achieve spatial encoding by variations of the magnetization magnitude (rotating frame imaging Fig. 1c). For example, using composite pulses [15], it is possible to convert variations of the magnetization magnitude into a phase distribution in the transverse plane. Contrary to BSSET, a linear phase shift is achieved with the rotating frame technique (Fig. 1c). Since the BS-SET method only weakly interacts with the magnetization magnitude, it is clearly different from the rotating frame approach [3] which strongly modulates the on-resonant magnetization.

Materials and methods Hardware A 0.5 T home-built scanner equipped with a home-built three-dimensional (3D) B0 -gradient system was used for

Magn Reson Mater Phy Fig. 1 Comparison of the static field gradient g0 and the RFgradients g1 : a the static field gradient g0 induces linearly encoded dephasing in the transverse plane, while b produces quadratic spatial phase modulation in the transverse plane (BS-SET) when constraint 1 is valid. The rotating frame gradient c produces an on-resonant dephasing in the vertical plane. For further details on rotating frame imaging, please refer to [2]

Fig. 2 a Sketch of the 2D SE BS-SET sequence. The first half of the k-space is acquired with BS-pulses placed before the 180° pulse and the second half b with the BS-pulses placed after the 180° pulse. The variable length of the BS-pulse indicates different k-space lines. The read encoding is achieved using a conventional B0 -gradient. c Overview of a phase diagram and the k-space. The induced phase shifts are colored corresponding to the BS-pulses of a, b. d Sketch of the coil system. A coil with a 12 mm diameter and 10 loops tuned to 600 kHz off resonance was used with an on-resonant receive(rx)/ transmit(tx) coil

the imaging experiments. The magnet had an homogeneity of approximately 50 ppm, covering a volume of 1 cm3. Figure 2d provides a sketch of the setup. A solenoid coil with a diameter of 1 cm and a height of 2 mm was used for transmission and reception of the MR signal. Additionally, a BS transmit coil was inserted, providing the RF-field for the BS-SET measurements. The BS-coil consisted of ten loops with an inner diameter of 12 mm. For geometrical decoupling, the BS-coil was set at a 90° angle to the transmit/receive coil and was placed next to the sample. Additionally, the coils were further decoupled through active switching [16], in which both coils were driven by independent transmitters and controlled by specific blanking signals.

For the BS-SET measurements, the BS-coil was tuned to an offset frequency of ?600 kHz instead of the Larmor frequency (approximately 20 MHz). Furthermore, a RFamplifier (Bruker BioSpin GmbH, Rheinstetten, Germany) was used to amplify the BS-pulses. For measurement operation, a conventional DriveL console (Pure Devices GmbH, Wu¨rzburg, Germany) with two separate transmit channels was used. Therein, one of the transmit channels was used to operate the transmit/receive coil and one to operate the BS-coil. B? 1 mapping Prior to the BS-SET experiments, the Bþ 1 -profile of the BScoil was characterized using an homogeneous 4 mm

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diameter phantom filled with oil. For the BS-Bþ 1 mapping, a transmit/receive solenoid coil was used for signal excitation and reception and the conventional 3D B0 -gradient system was used for spatial encoding. A SE -based BS-Bþ 1 mapping technique was implemented [11] utilizing a ?600 kHz off-resonant hard BS-pulse applied with the BScoil. The echo time (TE) was set to 18 ms and the BS-pulse length was set to 0.5 ms. Furthermore, a 64 9 64 matrix was acquired. Please note that the BS-Bþ 1 mapping technique from [11] uses ±DxRF to remove the setup dependent phase. Only positive off-resonances for the BS-pulses can be used with the current setup. Hence, a scan containing opposite phase shifts can be achieved by applying the BS-pulse after the refocusing pulse. Unfortunately, due to local B0-inhomogeneities, using only positive off-resonances for the BS-pulse does not cancel the error of the BS-Phase [17]. However, since our setup used a high off-resonance frequency of 600 kHz, this deviation was negligible. Furthermore, the high B1 magnitude enabled a very high encoding strength compared to standard B1 mapping [11]. Therefore, the maximum encoding strength in this setup was mainly limited by the fact that intra-pixel phase shifts larger than 2 p cannot be unwrapped.

0.021 ms difference of the pulse length per phase step. Furthermore, no BS-pulse was applied for acquisition of the k-space center. For read encoding, conventional B0 -gradients were used. A 140 9 140 matrix was acquired and averaged 40 times with a TE of 12.5 ms. Moreover, as the transmit/receive coil had a height less than 2 mm, slice gradients were not applied. For comparison, an image using a standard SE 2D sequence without BS-encoding was acquired with the same parameters and gradient settings as similar as possible to the BS-SET measurements. Both measurements were averaged 40 times. 2 Since the phase shift is proportional to Bþ 1 , only positive or only negative phase values are induced across the imaging region. For conventional experiments, the gradient isocenter is usually close to the center of the imaging object. In contrast, using BS-SET, the Bþ 1 -field is not null within the sample and corresponds to an object that is shifted away from the isocenter in the presence of B0 gradients. Thus, a field of view correction is required. Additionally, as explained in the following subsection, a reconstruction based on a B1 -map is needed to acquire nondistorted images. Reconstruction

BS-SET measurements 2

In phantom experiments (4 mm diameter oil sample with three stacks of glass pads with different thicknesses: 70, 140 and 210 lm) and imaging of plant stems (4 mm diameter), 2D BS-SET images were acquired using the BScoil for spatial encoding. The conventional phase encoding gradient was replaced by ?600 kHz off-resonant hard pulses with a fixed amplitude (Fig. 2a, b), resulting in ratio a (Eq. 1) of approximately 5. To achieve different phase encoding values, the BS-pulse duration was systematically increased. Due to the quadratic dependence on the B1 magnitude, a positive and a negative B1 magnitude introduces a phase shift in the same direction. To obtain an opposite phase shift, a negative frequency offset can be used similar to BS-based Bþ 1 -mapping [11]. However, our setup did not allow the BS pulses to be applied at -600 kHz due to the 146 kHz bandwidth of the BS-coil. Therefore, the k-space was symmetrically sampled by changing the BS-pulse position within the sequence (Fig. 2a, b). Thus, to encode the first half of the k-space, the BS-pulses were placed before the 180° pulse, inverting all prior phases (Fig. 2a, c). Consequently, the second k-space half was acquired from the BS-pulses placed after the 180° pulse using the same BS parameters (Fig. 2b, c). The pulses were set to a maximum length of 3 ms. Since 140 phase steps were acquired, this resulted in a

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Due to the Bþ 1 dependency of the phase shift, a constant þ B1 -gradient results in quadratic spatial encoding when using BS-SET. Therefore, a linear reconstruction using a Fourier transform yields a distorted image. Hence, a reconstruction considering this displacement was applied to BS-SET images. In a first step, signal was simulated using a 1D numerical phantom (cf. Fig. 3b). For BS phase encoding, a Bþ 1field with a constant gradient increasing from left to right was assumed (cf. Fig. 3a). The BS phase encoding was realized through a phase shift calculated by the matrix /E : 0 1 2 þ2 ðNÞBþ ð x Þ . . . ð N ÞB ð x Þ 0 end 1 1 2 2 B C ðN  1ÞBþ ðx0 Þ . . . ðN  1ÞBþ c2 Ds B 1 1 ðxend Þ C B C /E ¼ .. .. C 2DxRF B @ A . . 2

ð0ÞBþ 1 ðx 0 Þ

2

. . . ð0ÞBþ 1 ðxend Þ

Here N is the number of phase steps, x is the location and s is the maximum pulse length decreased in every phase step by Ds ¼ s=N. The magnitude of B1 is kept constant within all phase steps. Hence, the signal for all phase steps can be simulated by: ~ S ¼ ei/E ~ qðxÞ with the spin density function as qð xÞ. For the simulation, random noise was added to the k-space resulting in a

Magn Reson Mater Phy Fig. 3 a The simulated B1 gradient. b The simulated rectangular probe. c The BSencoded signal shows the distortion when reconstructed only by a Fourier transformation (red dashes: no noise added; black line: noise added). d Reconstruction by an analytic Fourier method (red dashes: no noise added; black line: noise added). Equation 6 was used for evaluation

signal-to-noise ratio (SNR) of 50. The noise was added independently to the real and imaginary parts. In Fig. 3c, the simulated BS-signal was zero filled by a factor of two and the profile was reconstructed using a standard fast Fourier transform (FFT). Two distortion effects are visible: the quadratic displacement of the rectangles with a higher resolution closer to the fictive coil located on the right, and consequently, the increasing SNR with increasing distance from the coil (Fig. 3c). A nonlinear reconstruction technique based on the spatial dependency of the Bþ 1 -field is required for the procural of undistorted images. There are several approaches to reconstruct nonlinear encoded images [18–20]. The present study applied a reconstruction technique based on an analytic Fourier method. This reconstruction rearranges the encoding equation, containing a nonlinear arbitrary field f(x), to a form in which the inverse Fourier transformation can be used [21]. In the context of BS-SET, f(x) is xBS ðxÞand the time dependent signal reads: Z1 qð xÞeif ðxÞt dx Sð t Þ ¼ 1

invertible), a formula can be found to reconstruct the spin density qð xÞ from the intermediate density function qð yÞ:   ð6Þ qð xÞ ¼ qð yÞf 0 f 1 ð yÞ with the help of a Bþ 1 -map and the measured intermediate density function qð yÞ, Equation (6) enables calculation of the undistorted qð xÞ. To reconstruct qð xÞ; qð yÞ is linearly regridded and weighted by f 0 ðf 1 ð yÞÞ: The reconstruction of the simulated signal is shown in (Fig. 3d).

Results B? 1 mapping Figure 4a shows the Bþ 1 mapping resulting from the described BS-SE sequence. The Bþ 1 mapping revealed a þ linear B1 profile of the BS-coil (Fig. 4b). The values reached a maximum of approximately 2.4 mT and a minimum of 0.9 mT. Hence, an approximately 375 mT/m B1 gradient could be achieved. Measurements for several samples showed that the B1 -values did not significantly depend on the loading of the coil.

2

Furthermore, it is possible to define y ¼ f ð xÞ  Bþ 1 and dy dx

¼ dfdxðxÞ ¼ f 0 ðxÞ and then substitute the differential dx by

dx ¼

dy . f 0 ðxÞ

Sð t Þ ¼

Z1 1

qð xÞ iyt e dy f 0 ðxÞ

with qð yÞ ¼ qð xÞ=f 0 ð xÞ the equation reads Sð t Þ ¼

yZð1Þ

qð yÞeiyt dy

yð1Þ

Here, qð yÞ can be calculated by the conventional inverse Fourier transform. Assuming x ¼ f 1 ð yÞ exists (Bþ 1 is

BS-SET experiments The 2D BS-SET measurements from the phantom experiments are shown in Fig. 5. The vertical direction was encoded using the BS-shift, and the horizontal direction was encoded by a conventional B0 read gradient. Figure 5a shows a distorted version of the Fourier reconstructed BS-image. The same data were reconstructed using the nonlinear Fourier reconstruction method (Eq. 6), which acknowledges the spatial dependence of the measured B1 -field (Fig. 5a). This method results in an undistorted BS-image, as shown in Fig. 5b. For comparison, the result of a conventional 2D SE sequence is shown in Fig. 5c. Due 2 to the Bþ 1 dependency, with increasing distance from the

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Magn Reson Mater Phy Fig. 4 a Bþ 1 -map of the BS-coil acquired with a standard BS-B1 mapping sequence (coil at top). b 1D B1 -profile across the sample. The origin is indicated by the blue dotted line in a) (coil right)

Fig. 5 2D BS-SET image of 4 mm diameter oil sample with glass pads. The 70 lm thick pads are in stacks of one, two and three pads. a 2D BS-SET image with only a Fourier transform applied to the raw data. Please note the low resolution at the bottom of the image opposite to the BS-coil and the increasing resolution towards the BS-

coil (top). b Nonlinear reconstruction using Eq. 6 based on the extrapolated Bþ 1 -map shown in Fig. 4a. c Result of a standard SE 2D sequence without BS-encoding

BS-coil, the resolution and the SNR increase. The green arrow on the right in Fig. 5b indicates the oil between the glass pads. In the lower area of the BS-SET image, the resolution is too low to differentiate the single pads or clearly show the edges of the sample. The smallest stack of 70 lm (left red arrow) has sharper edges compared to the image in Fig. 5c without BS-encoding. While the BS-SET images nicely reproduce most of the phantom features and accord well with standard SE images, the glass plates nevertheless appear slightly deformed. The origin of these spatial deformations is most likely due to projecting the 3D volume B1 -field on a 2D map. This might lead to inaccuracies in the reconstruction, since it assumes a constant B1 along the slice direction. The low SNR introduced by very high resolution close to the BS-coil is highlighted by the red circle. Both measurements were supposed to be as similar as possible; therefore, spoilers and phase cycling were not used. This resulted in the artifact shown in Fig. 5c (yellow arrow). The artifact is also present in Fig. 5b, although far from the object due to FOV shift introduced by the B21 dependency. The BS-SET images of the plant stems and the photos of the cross-section are shown in Fig. 6. The outside form of the plant samples was well reproduced in both cases.

Furthermore, the Galium images revealed both brighter and darker inner ring structures. The images demonstrate that accurate imaging remains feasible in real world measurement situations.

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Discussion The presented results demonstrated the feasibility of onedimensional (1D) spatial phase encoding based on the BSeffect. A relatively simple setup with a single BS-coil allowed spatial encoding of a 70 lm thick glass plate embedded in an oil phantom, as shown in (Fig. 5). Figure 5b shows the nonlinear BS-encoding results in a resolution dependent on the pixel location. The BS-encoding strength is highest close to the coil and declines with increasing distance to the BS-coil. Since the resolution depends on the BS phase encoding strength, the resolution has a similar pattern in a BS-SET image. Consequently, the encoding, and thus the resolution, decreases with increasing distance from the BS-coil (cf. Fig. 5b). Therefore, fine details of the imaged object could not be resolved. Due to the spatially dependent resolution, however, the SNR increased with increasing distance to the coil. Similar

Magn Reson Mater Phy Fig. 6 These images resulted from the application of BS-SET to measure plant stems; the parameters were changed to a 100 9 100 matrix, a 4 ms pulse at 600 kHz and a TE of 18 ms. a Photos of a cross-section slice from Galium plant stems. b The BS-SET measurements revealed the inner structure of the plants and accurately reproduced the characteristic outer shape of the stem cross section

Fig. 7 a Plot of the optimal theoretical B1 -field required for BS-SET spatial encoding resulting in an isotropic resolution of 2 mm over 10 cm. The dotted lines represent the maximum possible B1 -field strengths corresponding to the SAR limits of a human brain at the indicated static B0 -fields. b Plot of the maximum achievable phase

steps per second for the SAR limits of a human brain as a function of the static magnetic field for different BS-SET pulse lengths. Dotted blue line: pulse duration s ¼ 1 ms, red dotted line s ¼ 3 ms, black line: s ¼ 5 ms. The ratio a ¼ DxRF =xB1 (Eq. 1) is set to 5 in a and b. In a, TR = 200 ms (Phasesteps/s = 5), s ¼ 4 ms

effects have been observed with B0 -gradients. For example, the concept of PatLoc (Parallel Imaging Technique using Localized Gradients) enables MR imaging of spatially ambiguous encoding fields [18–20, 22]. The presented setup had a maximum sample size limited to 4.5 mm tubes; however, it provided a high B1 -gradient

strength. Covering larger objects would require a higher gradient strength than surface coils can usually supply. Furthermore, special coil designs such as in [23] would be necessary for larger volumes. Figure 7a shows an example of a B1 -field achieving linear encoding with a 2 mm pixel size over 10 cm. The simulation uses the sequence

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described in Fig. 2. The simulated pulses duration was 4 ms at 55 kHz off resonance and the repetition time was set to 200 ms. The quadratic encoding property inherent in the BS-SET required an adapted reconstruction to yield an undistorted image. This was due to the linear Bþ 1 -profile of the BS-coil. The square root shape of the simulated B1 -field thus would supply linear phase encoding, removing the need for any reconstruction other than a simple Fourier transformation. However, it remains to be shown whether or not such geometry can be realized. The maximum field strength was set to 246 lT. The SAR limits at different B0 -fields are plotted in Fig. 7a. The SAR was calculated based on the results of [24], and the limit was 3.2 W/kG according to the limits from IEC60601-2-33 head SAR for a human brain. At 0.15 T, the SAR of the simulated setup was close to this limit. Therefore, a reasonable resolution for large objects does not appear to be possible for human application at 1.5 T. Since the BS-gradient requires high RF power, SAR limitations arise at higher field strengths. The simulation in Fig. 7b shows that one phase step per 10 s at 1.5 T is possible. However, repetition times below 100 ms are possible at scanner fields below 200 mT. Consequently, application of BS-SET in a clinical setup is more feasible for low field MR [25, 26]. Samples where almost no SAR limits apply can be acquired with much higher BS-gradient strengths. Hence, hardware, such as the additional RFamplifiers needed for every BS-SET encoding direction, and the coil design are the only encoding limitations. The plotted image illustrates the ideal B1 gradient shape. Hence, it represents the minimum B1 -field needed to encode the 50 pixels required to cover 10 cm with a resolution of 2 mm when using hard pulses and a minimum a of 5. With the current coil design, a maximum RF-gradient strength of 400 mT/m with a peak power of 40 W was achieved. When compared against standard B0 -gradients, the BS-SET encoding strength is reduced by a factor of at least 10 (Eq. 2). This is due to the 600 kHz off-resonance, which limits the spatial phase shift induced by the BSpulse. Therefore, long BS-pulse durations must be used for sufficient encoding. However, the required BS-pulse duration can be reduced by increasing the Bþ 1 -field strength using more complex coil designs. For example, Ref. [23] shows a coil design that allows a gradient strength of 320 mT/m at 400 MHz. In a low field application, such a setup would achieve a much higher gradient strength [27, 28]. Furthermore, a multi resonant setup with two symmetrical frequency offsets from the Larmor frequency would allow positive and negative frequency offsets, and therefore positive and negative phase shifts. This would

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allow acquisition of the complete k-space using a gradient echo experiment with BS-SET. This work only applied hard pulses to BS-SET due to software limitations. Future approaches to increasing the BS-SET encoding strength could include optimizing RFpulses as has already been demonstrated with Bþ 1 -mapping [11].

Conclusions In this proof-of-principle work, a new spatial phase encoding technique using a B1 -gradient in combination with far off-resonant BS-pulses was demonstrated. Thus, using a low field MR-setup, BS-SET 2D MR images were obtained in reasonable measurement time. The BS-SET approach for spatial phase encoding provides an alternative to other encoding strategies, and thus has the potential to avoid problems inherent to existing strategies. Acknowledgments We thank Ashley Basse-Lu¨sebrink for tidying up English grammar and expression in this manuscript.

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