Characterization of Phase-Based Methods Used for Transmission Field Uniformity Mapping: A Magnetic Resonance Study at 3.0 T and 7.0 T

Knowledge of the transmission field (B1 +) of radio-frequency coils is crucial for high field (B0 = 3.0 T) and ultrahigh field (B0≥7.0 T) magnetic resonance applications to overcome constraints dictated by electrodynamics in the short wavelength regime with the ultimate goal to improve the image quality. For this purpose B1 + mapping methods are used, which are commonly magnitude-based. In this study an analysis of five phase-based methods for three-dimensional mapping of the B1 + field is presented. The five methods are implemented in a 3D gradient-echo technique. Each method makes use of different RF-pulses (composite or off-resonance pulses) to encode the effective intensity of the B1 + field into the phase of the magnetization. The different RF-pulses result in different trajectories of the magnetization, different use of the transverse magnetization and different sensitivities to B1 + inhomogeneities and frequency offsets, as demonstrated by numerical simulations. The characterization of the five methods also includes phantom experiments and in vivo studies of the human brain at 3.0 T and at 7.0 T. It is shown how the characteristics of each method affect the quality of the B1 + maps. Implications for in vivo B1 + mapping at 3.0 T and 7.0 T are discussed.


Introduction
Non-uniformities of the transmission radio-frequency (RF) field (B 1 + ) constitute an adverse factor for high field (B 0 = 3.0 T) and ultrahigh field (B 0 $7.0 T) magnetic resonance (MR), which may render diagnostics challenging. This practical impediment is pronounced when imaging techniques sensitive to the excitation flip angle (FA) are applied. The knowledge of the B 1 + field distribution is essential to correct for B 1 + non-uniformities of single channel or multi-channel transmit (TX) RF-coils. To trim or shim the B 1 + field, multiple channel transmission has been pioneered [1][2][3]. For this purpose, multi transmit arrays are used, which require B 1 + mapping routines to calibrate each individual RF coil element. This procedure can be time consuming when using TX arrays comprising many transmit elements. Consequently accurate and fast B 1 + distribution mapping is the key for ultrahigh field clinical applications. B 1 + mapping approaches commonly used are mainly magnitude-based and are generally confined to the ratios or the fit of signal intensity images [4][5][6][7][8][9][10][11]. For this purpose sets of images are acquired using either two flip angles [4][5][6], identical flip angles but different repetition times (TR) [7], variable flip angles [8,9] or also signals from spin-echoes and stimulated-echoes [10], as well as signals from gradient-echoes and stimulated-echoes [11]. For most of these magnitude-based approaches the quantitative B 1 + evaluation may be influenced by saturation effects given by T 1 relaxation. This problem can be overcome with the use of long repetition times (TR), which, however, would result in prolonged acquisition times. Alternatively, phase-based methods have been proposed as they are insensitive to T 1 relaxation. They were also found to be more accurate than magnitude-based methods, especially at low flip angle regimes [12].
Realizing the advantages of phase-based methods for B 1 + mapping, this work characterizes five of these methods: A) an optimized version for high field proton MRI [13] of the low flip angle method proposed by Mugler [14,15] here named ''Optimized low-flip-angle method'', B) the phase sensitive method of Morrell [16] here named ''Phase-sensitive method'', C) the phase-based method of Santoro [17,18], applied to high field proton MRI [19] here named ''FFA-CUP method'', D) the Bloch-Siegert shift method of Sacolick [20][21][22] here named ''Bloch-Siegert method'' and E) the orthogonal pulses method proposed by Chang [23] here named ''Orthogonal-pulses method''. These phase-based methods share in common the use of a composite or off-resonance RF-pulse to encode the spatial B 1 + magnitude information into the phase of the magnetization vector (M). Each method uses a different scheme of the RF-phases, generating a different evolution of M. The sensitivity to B 1 + variations and frequency offsets is examined using numerical simulations of the Bloch equations. Phantom experiments and human brain imaging studies are conducted at 3.0 T and 7.0 T to scrutinize each method. This includes the assessment of repetition times achievable, according to specific absorption rate (SAR) levels, as well as the susceptibility to offresonance effects. For a balanced comparison, all methods are used in conjunction with the same reading module.

Theory
The B 1 + mapping methods analyzed in this work make use of a complex RF-pulse envelope (a rectangular composite pulse or an off-resonance Fermi pulse) for excitation, with separately controlled amplitude and phase (Fig. 1). Each pulse achieves a different trajectory of the magnetization M, depending on the combination of amplitude and phase of the RF-pulse. The trajectories of M for the five pulses are depicted in Fig. 2 for the ideal case where DB 0 = 0. The presence of B 0 inhomogeneities, or other sources of frequency offsets, results in deviations from the ideal trajectory.
All the five trajectories can be represented by a different polygon lying on the surface of a unitary sphere. Each of them is characterized by a different number of sides and it is traversed a different number of times: A -Optimized low-flip-angle method) a squared trajectory which is traversed for one and a half turns [13][14][15], B -Phase-sensitive method) a rectangular trajectory which is traversed for a half turn [16], C -FFA-CUP method) an off-origin loop trajectory which is traversed for a single turn [17][18][19], D -Bloch-Siegert method) an initial excitation followed by an off-resonance pulse (which is equivalent to traversing a small circular trajectory for several turns) [20][21][22] and E -Orthogonal-pulses method) a square trajectory which is traversed for a half turn [23].
At the end of each RF-pulse, the local magnetization presents a phase accrual depending on the local B 1 + intensity and frequency offset experienced, as shown in the curves of Fig. 3. The theoretical description of this effect has been already reported in [16,18,20,23], and is briefly resumed here.
For methods A, B, C and E the RF-pulse can be divided into sub-pulses of flip-angle a and RF-phase F, denoted with a F . Each sub-pulse a F represents a rotation about a different axis, due to their different RF-phase. The magnetization accumulates a phase shift which is proportional to the local B 1 + field intensity because of the non-commutativity of rotations about different axes.
Method D, after the initial excitation, uses an off-resonance RFpulse. In this case the RF-phase varies linearly within the pulse and the magnetization accumulates a phase shift proportional to the local B 1 + field intensity, due to the well-known Bloch-Siegert shift effect [24]. To cancel phase contributions due to sources other than B 1 + , such as the receive coil sensitivity (B 1 2 ), the acquisition of two phase images, obtained with opposite senses of rotation of the magnetization (opposite RF-phase schemes), is required for all methods. The subtraction of the images preserves the B 1 + information, while removing all other time-independent phase contributions.
A -Optimized low-flip-angle method. Excitation is performed by the application of the non-selective composite pulse: [a 2135 a 245 a 45 a 135 a 2135 a 245 ] (Fig. 1A). The pulse moves the magnetization vector about a square, of side length a, through 1.5 turns ( Fig. 2A) [13]. A second image must be acquired using a corresponding pulse that moves the magnetization in the opposite sense of rotation: [a 245 a 2135 a 135 a 45 a 245 a 2135 ].
B -Phase-sensitive method. Excitation is performed by the application of the non-selective composite pulse: [2a 0 a 90 ] (Fig. 1B). For a small flip angle a the pulse moves the magnetization vector along a rectangular trajectory, with one side of length 2a and the other of length a, through 0.5 turns (Fig. 2B blue line). The method is originally proposed using a nominal flip angle a = 90u (which performs the trajectory in Fig. 2B -red line) [16]. A second image must be acquired with the first sub-pulse reversed in sign: [2a 180 a 90 ].
C -FFA-CUP method. Excitation is performed by the application of the non-selective composite pulse: [p0 290 a 2157.5 a 2112.5 a 267.5 a 222.5 a +22.5 a +67.5 a +112.5 a +157.5 ] (Fig. 1C). The magnetization vector is moved away from the origin by the first sub-pulse, named p0. The phase accrual is achieved by traversing for 1.0 turn an octagonal trajectory of side a shifted from the origin. The use of the first pulse p0 separates the excitation from the phase accrual in order to optimize the sensitivity to B 1 + variations [17,19]. A second image must be acquired with the composite pulse: [p0 +90 a +157.5 a +112.5 a +67.5 a +22.5 a 222.5 a 267.5 a 2112.5 a 2157.5 ].
D -Bloch-Siegert method. This method makes use of an off-resonance pulse of frequency shift Dv RF applied immediately after an excitation: [p0 290 a 290,Dv ] (Fig. 1D). The off-resonance pulse moves the magnetization about a circular trajectory traversed several times (Fig. 2D). The number of loops is given by the duration of the pulse multiplied by the off-resonance frequency. The off-resonance pulse can be seen as a pulse in which the RF phase F is continuously varied during its duration t, according to F = Dv RF? t. A second image must be acquired using the opposite frequency shift: -Dv RF . The method is originally proposed using an offresonance Fermi pulse of frequency shift Dv RF = 4 kHz and duration of 8 ms [20]. However it has been widely shown [21,22]  that different values of the pulse duration and frequency shift, as well as different pulse shapes, can be used to optimize this method. Here we used a Fermi pulse of frequency shift Dv RF = 4 kHz and duration of 4 ms.
E -Orthogonal-pulses method. Excitation is performed by the application of the non-selective composite pulse: [a 0 a 90 ] (Fig. 1E). For a small flip angle a the pulse moves the magnetization vector along a square trajectory, with side of length a, through 0.5 turns (Fig. 2E -blue line). The method is originally proposed using a nominal flip angle a = 60u (which performs the trajectory in Fig. 2E -red line) [23]. A second image must be acquired with the phases of the two sub-pulses swapped: [a 90 a 0 ].

Numerical Simulations
MATLAB (MathWorks Inc, Natick, USA) software was used to calculate the dynamics of the magnetization during the excitation pulses, by means of numerical simulations of the Bloch equations. A range of values of the frequency offset (21 kHz # DB 0 #1 kHz, with an increment of 50 Hz) and of the B 1 + intensity (rescaling the flip angle from 0 to 2 times the nominal value, with an increment of 0.05) was used. The sensitivity of the different methods to the local variations of the B 1 + field and of the frequency offset is expressed by the variable Y (Fig. 3), which is defined as the subtraction of the phase accruals obtained from the two complementary scans required by each method. The intensity of the B 1 + is expressed in terms of the total flip angle (TotalFA) used by each pulse, which results from the total duration and amplitude of the RF applied, regardless of its RF-phase scheme. At high field strengths the TotalFA represents a crucial parameter, as the SAR levels limit the lowest achievable TR. This is especially the case for the methods used in this work, which require values of TotalFA of the order of several tens to a few hundred degrees. In order to quantify the efficiency (e) of each method to convert the employed RF-power into a phase accrual Y the following variable was defined and calculated: where t RF is the total duration of the pulse.

MR Hardware
Phantom studies and in vivo experiments of the human brain were performed at magnetic field strengths of 3.0 T and 7.0 T. For this purpose, methods A-E were implemented on a clinical 3.0 T MR-scanner (TIM Verio, Siemens Healthcare, Erlangen, Germany) and a whole body 7.0 T MR-scanner (Magnetom, Siemens Healthcare, Erlangen Germany), using a dedicated sequence development environment (IDEA, Siemens Healthcare, Erlangen, Germany). At 3.0 T a transmit/receive (TX/RX) birdcage coil (Siemens Healthcare, Erlangen, Germany) operating in the circular polarized (CP) mode was used (diameter = 27 cm,

Implementation of the B 1 + Mapping Techniques
The implementation comprises a standard 3D gradient-echo sequence, where the excitation is performed for each method by the non-selective RF-pulses sketched in Fig. 1 and described in the Theory section.
To reduce bulk motion effects the two images required by each method were acquired interleavedly. To examine and correct for variations in the main magnetic field (B 0 ) across the object DB 0 maps were acquired. For this purpose a secondary gradient-echo readout was added to the sequence; the DB 0 maps (Fig. 4) were obtained from the subtraction of the two phase images acquired at different echo times (TE) [25]. B 1 + maps (Figs. 5-8) were calculated for each method from the measured phase accrual Y and the DB 0 map, using the corresponding curve of sensitivity of Fig. 3 as a lookup table and performing a linear 2D interpolation.
For comparison, a standard 3D gradient-echo technique was used to acquire three-dimensional B 1 + maps using the doubleangle method (DAM) [4]. This required the acquisition of two magnitude images with nominal flip angles of a = 60u and 2a = 120u together with repetition times of TR.5T 1 [5].

Specific Absorption Rate Adjustment
Each method uses a different RF-power level. Since the SAR represents the limiting factor for the minimum achievable TR at high field strengths for all methods, the RF-pulse amplitudes (i.e. the nominal B 1 + ) were individually adjusted for each method in order to accomplish identical SAR levels, given a common repetition time. This corresponds to truncating the sensitivity curves of Fig. 3 to a TotalFA value which guarantees identical SAR levels for all methods. The TRs were adjusted in in vivo experiments -according to the volunteer weight -to achieve a nominal SAR level of 2.4 W/kg. This value corresponds to 75% of the SAR limit for the normal and first level operating modes for head imaging, as given by the IEC guidelines [26].
The nominal values of a, TotalFA and B 1 + for the five methods are reported in Table 1, together with the duration of the RFpulses and the repetition times. The nominal B 1 + values are calculated starting from the reference voltage necessary to obtain a 1 ms rectangular p-pulse, and adjusted according to the duration t and the TotalFA of the pulses of the five methods. The reported B 1 + intensity represents the average value within the pulse. Identical parameters were used for both phantom and in vivo experiments.

Phantom Studies
A synopsis of the imaging parameters used for phantom studies at 3.0 T and 7.0 T is shown in Table 1. The basic imaging parameters were kept constant for all methods, including: field of view FOV = (20062006200) mm 3 , matrix size of 32632616 (plus zero-filling interpolation) and receiver bandwidth BW = 800 Hz/ pixel. The echo times were set to the minimum possible value,  + maps obtained with methods A-E in phantom at 3.0 T using a birdcage TX/RX coil. Identical repetition times (TR = 30 ms) and SAR levels were used for all methods. The same partition of the B 1 + map acquired for comparison with the DAM with TR = 500 ms is also shown (F). All maps are normalized to their nominal B 1 + given in Table 1  A spherical phantom (18 cm diameter), filled with water and doped with 50 mM Na and 20 mM CuSO 4 , was prepared. This setup provides sufficient RF-loading and short T 1 relaxation time Table 1. Nominal values of the initial excitation angle (p0), flip angle of each sub-pulse (a), total flip angle (TotalFA), B 1 + intensity, total duration of the RF-pulses, echo times (TE) and repetition times (TR) for the experiments performed in phantom and in vivo, at 3.0 T and 7.0 T, with methods A-E.  Figure 6. B 1 + maps in phantom at 7.0 T. Central axial partition of the 3D B 1 + maps obtained with methods A-E in phantom at 7.0 T using a birdcage TX/RX coil. Identical repetition times (TR = 110 ms) and SAR levels were used for all methods. The same partition of the B 1 + map acquired for comparison with the DAM with TR = 500 ms is also shown (F). All maps are normalized to their nominal B 1 + given in Table 1

Ethics Statement
For the in vivo feasibility study, two healthy subjects without any known history of neurovascular disease were included after due approval by the local ethical committee (registration number DE/ CA73/5550/09, Landesamt für Arbeitsschutz, Gesundheitsschutz und technische Sicherheit, Berlin, Germany). Informed written consent was obtained from each volunteer prior to the study.

In vivo Studies
Human brain imaging was performed at 3.0 T and 7.0 T in healthy subjects using the five phase-based methods (A-E) and the DAM. FOV was adjusted to (23062306176) mm 3 at 3.0 T and (21062106160) mm 3 at 7.0 T, in order to cover the whole brain with 16 sagittal partitions using a matrix size of 32632616 (plus zero-fill interpolation). The inter echo time for the DB 0 map was set to DTE = 2.46 ms at 3.0 T and DTE = 3.06 ms at 7.0 T, to make sure that fat and water are in phase for both TEs.
For the DAM approach only a central partition of the brain was acquired, since covering the whole brain would have required several hours of scan time: a constraint that is dictated by the T 1 of the brain (gray matter: T 1 < 1800 ms, white matter T 1 < 1000 ms at 3.0 T [27]), so that the repetition time was set to TR = 6000 ms.

Numerical Simulations
The results derived from the simulations are shown in Figs. 2 and 3. The trajectories of M during excitation (Fig. 2) are used to qualitatively estimate the use of transverse magnetization. The curves displayed in Fig. 3 represent the sensitivities to B 1 + variations (expressed as the TotalFA) and frequency offsets (DB 0 in Hz). The frequency offset range is identical for all the curves, while the total flip angle ranges vary (TotalFA axis), as well as the phase accrual ranges (Y axis). Efficiency e is used to combine the flip angle range and phase accrual characteristics in a single variable that supports a balanced comparison. The values of e were calculated for each method at the center of the sensitivity curves using Eq. 1.
A -Optimized low-flip-angle method. This method revealed the lowest B 1 + sensitivity among all methods, with e = 0.44 ms/deg. Its sensitivity curve presents a rather flat dependency upon frequency offsets. A discontinuity is observed for some combinations of DB 0 and TotalFA (Fig. 3A). In terms of usage of the transverse magnetization its composite pulse is equivalent to an excitation of ffiffi ffi 2 p a ( Fig. 2A). B -Phase-sensitive method. This method shows the highest B 1 + sensitivity, with e = 1.91 ms/deg. For frequency offsets exceeding a range of approximately 6500 Hz the phase accrual Y experiences a discontinuity (Fig. 3B). A folding, leading to nonunique phase information which could not be decoded into the B 1 + value, can also be observed outside of this range. This method employs the highest transverse magnetization, as the composite pulse is equivalent to a 90u excitation, when a = 90u (Fig. 2B).
C -FFA-CUP method. This method has a high B 1 + sensitivity, with e = 1.52 ms/deg. The dependency upon frequency offsets is more pronounced than in method A. This method does not present the discontinuities observed for methods A and B (Fig. 3C). The use of transverse magnetization is equal to p0 (Fig. 2C). Its value can be chosen equal to the Ernst angle in order to optimize the signal, without affecting the B 1 + sensitivity. This is not possible for methods A and B.
D -Bloch-Siegert method. This method exhibits a low B 1 + sensitivity, with e = 0.66 ms/deg, because it requires a much larger TotalFA compared to the other methods. This results also in a longer pulse duration, which manifests itself in a TE prolongation. The sensitivity curve of this method presents a rather modest dependency upon frequency offsets (Fig. 3D). The use of transverse magnetization depends only on the initial excitation p0 (Fig. 2D), and can therefore be controlled, like for method C. E -Orthogonal-pulses method. This method presents an intermediate B 1 + sensitivity, with e = 0.82 ms/deg. Its sensitivity curve shows a non-negligible dependency upon frequency offsets. A reduced B 1 + sensitivity was observed for small flip angles versus the high flip angle regime (Fig. 3E). In terms of usage of the transverse magnetization the composite pulse used is equivalent to a flip angle larger than a (at small flip angles it is equal to ffiffi ffi 2 p a, like for method A) (Fig. 2E).

Phantom Studies
The results derived from phantom experiments at 3.0 T and 7.0 T are shown in Figs. 5 and 6. All maps present the typical behavior of a birdcage resonator, where B 1 + is higher at the center. At 7.0 T, due to destructive interference patterns, some areas around the center present a lower intensity. B 0 inhomogeneities can be observed at the air-water interface from the DB 0 maps shown in Fig. 4 (top), especially at 7.0 T. However, the fit performed using the sensitivity curves provides a very good estimation of B 1 + in these regions. Even for the phasesensitive method (B), the FFA-CUP method (C) and the orthogonal-pulses method (E) which are most sensitive to frequency offsets. This is confirmed by the contour plots in Figs. 5 and 6. Compared to the DAM no B 1 + distortion can be observed in these areas. All methods revealed sufficient signal, as the T 1 of the phantom was short enough (T 1 < 70 ms) to allow recovery of the magnetization.

In vivo Studies
The results of the human brain studies at 3.0 T and 7.0 T are summarized in Figs. 7 and 8. The results obtained with methods A-E (Figs. 7-8, A-E) for brain regions are in agreement with the DAM (Figs. 7-8, F). The typical B 1 + peak of a birdcage coil can be observed at the center of the brain both at 3.0 T and 7.0 T. At 7.0 T a region of signal void due to destructive interference is visible in the area of the cerebellum. It should be noted that all methods except the Bloch-Siegert method (D) present some regions where the B 1 + estimation is not correct, both at 3.0 T and at 7.0 T. This is due to the 3.5 ppm chemical shift between fat and water (corresponding to a resonance frequency difference of 150 Hz/ T). The individual phases of water and fat signals are affected by the presence of B 1 + inhomogeneities and frequency offsets, as demonstrated by the sensitivity curves (Fig. 3). Since the signal from each pixel is given by the complex sum of these two components, the resulting phase is decoded into a wrong B 1 + value during the fitting. In fact, the DB 0 maps shown in Fig. 4 (bottom) do not account for this effect, as they were acquired with fat and water in phase. Due to air-tissue interfaces, strong B 0 offsets were observed in the sphenoid sinuses area, extending into the interior of the brain. The correction fit performs correctly in this region. At 7.0 T the Bloch-Siegert method (D) shows SNR loss in the regions with short T 2 * , such as the areas nearby the bones.

Discussion
In this work five phase-based methods used for B 1 + mapping have been examined carefully at magnetic field strengths of 3.0 T and 7.0 T. The characteristics of each method were analyzed by means of numerical simulations, phantom studies and in vivo experiments.
Although all methods have in common the use of a complex RFpulse (composite or off-resonance pulse) for excitation, in conjunction with the same gradient-echo readout scheme, it is shown here that the five methods exhibit different sensitivities to B 1 + inhomogeneities and frequency offsets. Furthermore they make different use of transverse magnetization and hence reveal different SNR, depending on the TR/T 1 ratio. For these reasons, the quality of the B 1 + maps obtained from each method depends on the specific experimental conditions (T 1 , T 2 Among the methods used here, the Bloch-Siegert method (D) was found to be the least sensitive to DB 0 offsets and chemical shift effects, due to its flat sensitivity curve in the frequency offset direction.
Unlike all the other methods, the Bloch-Siegert method (D) supports also 2D mapping. This can be beneficial when time constraints do not allow for a full 3D acquisition; for example for B 1 + mapping of the heart, where scan time constraints dictated by cardiac and respiratory motion need to be managed carefully. On the other hand the Bloch-Siegert method (D) presents a smaller efficiency than the phase-sensitive method (B), the FFA-CUP method (C) and the orthogonal-pulses method (E), and requires the longest pulse duration among all the methods. This feature results in SNR degradation for short T 2 * regions, such as interfaces with strong susceptibility gradients.
As our work is focused on the excitation pulse, results are derived using a standard 3D gradient-echo sequence, a Cartesian k-space sampling scheme and a single channel TX/RX coil. However, all the methods are inherently compatible with other 3D-imaging modules and k-space sampling schemes, as long as the phase information is preserved. Therefore all the methods can be accelerated using multi-echo techniques, or k-space undersampling techniques. This can be useful for B 1 + mapping applications in other organs, including cardiac or abdominal MRI where physiological motion constraints dictate the viable window of data acquisition.

Conclusion
The B 1 + mapping techniques examined here provided characteristics which underline the capabilities of phase-based methods, including the scan time advantage over conventional magnitude-based B 1 + mapping methods. All presented methods can be adjusted to provide enough B 1 + sensitivity without exceeding the clinical SAR limits. However, some characteristics, such as the sensitivity to B 1 + inhomogeneities and frequency offsets and the consumption of longitudinal magnetization, are different for each method. This has an impact on the performance, depending on the specific experimental conditions.