The actual flip angle imaging (AFI) B1 mapping method (1) has found wide use because it is both efficient and close to being independent of T1 effects (2–6). In its standard form, AFI uses nonselective RF excitation pulses leading to the requirement of 3D spatial encoding. This is problematic for typical clinical or research examinations involving parallel transmission as the necessity of sequentially mapping B1 fields from multiple RF coils leads to long calibration scan durations. The use of AFI with slice selective pulses would lead to very large savings in acquisition time for situations in which B1 information is only required within a single slice; however, it is now recognized that slice selection leads to systematic error in the estimated B1 (7).
AFI uses a steady-state spoiled gradient echo sequence with a constant flip angle θ but alternating interleaved repetition times TR1 and TR2. Separate images are formed from the signals obtained from each TR period; S1 refers to the signal resulting from TR1 and S2 from TR2. If it may be assumed that transverse coherences are perfectly spoiled, then expressions for S1 and S2 are given by Eqs. 1–3 in Ref.1. Under the assumption that both repetition times are short with respect to T1, the following expressions are obtained [Eqs. 5 and 6 in Ref.1] relating the ratio of signals from the two images (r = S2/S1) to the flip angle
where n = TR2/TR1. In practice, a pair of images is acquired, and the flip angle is computed from the ratio of signals.
When discussing B1 mapping, the terms “flip angle map” and “B1 map” are often used interchangeably, while the end product is usually a map of the transmit sensitivity σ(x). Transmit sensitivity is defined as a dimensionless value such that the physical B1 field produced by a given coil is
where b(t) is the nominal B1 field produced by the coil (defined as being spatially constant) expressed in physical units; x is a general spatial coordinate. The flip angle produced by a nonselective rectangular RF pulse of duration τ and amplitude β is θ(x) = γσ(x)βτ (neglecting off-resonance effects). A transmit sensitivity map is straightforwardly obtained from a flip angle map (from AFI or any other method) by dividing by the nominal flip angle θnom = γβτ. For slice selective pulses, the situation is more complicated because a range of flip angles exists through the slice; here, we define the nominal flip angle as the flip angle achieved at the centre of the slice when σ = 1, i.e., θnom = γ∫b(t)dt. Assuming that σ(x) is constant over the width of the slice profile, such that σ(x) = σ(x,y) where x and y are in-plane spatial coordinates, we can write θ(x) = θ(z,σ(x,y),b(t),G(t)). In other words, we attribute all z variation of θ to the slice selection process involving the gradient waveform G(t) and the RF waveform b(t), and all in-plane variation to transmit sensitivity σ(x,y). When slice selective pulses are used in the AFI sequence, a distribution of steady-state signals is present in each voxel, and the total signal is given by the integral through the slice. As was noted by Wu et al. (7), the result is a systematic error in the inferred flip angle and hence transmit sensitivity. The ratio of signals is, however, predictable from the slice selection pulse waveform, and so a map of σ(x,y) can still be inferred from slice selective measurements. In this article, we present a practical procedure for transmit sensitivity mapping using AFI with arbitrary slice selective RF pulses.
Simulation and σ-Map Reconstruction
At equilibrium, the magnetization vector M = (Mx,My,Mz) is (0,0,1)—i.e., of unit length and aligned with z. After the application of a slice selective RF pulse, M can be obtained by numerical integration of the Bloch equations with the relevant pulse and gradient waveforms. By repeating the simulation for a range of values of σ and z, we obtain M(z,σ,b(t),G(t)). From this, we obtain the flip angle distribution
and relative phase
where Mxy = Mx + iMy. The waveforms b(t) and G(t) are fixed for a specific RF pulse and so may be dropped from the expressions for clarity. Under the assumption of perfect transverse spoiling, the steady-state signal intensities are given by
where E1 = exp(−TR1/T1), E2 = exp(−TR2/T1) and terms which are both common to S1 and S2 and independent of θ have been dropped. The signal received when using 2D spatial encoding can then be estimated by integration with respect to z:
and therefore, the ratio of signals is
The resulting expression relates the transmit sensitivity to the ratio of the slice selective images. It is, however, also a function of T1, with a sensitivity that depends on the RF pulse itself.
Four different slice selective RF pulses were used in this study: a gaussian pulse, an asymmetric 2-lobe sinc pulse, and symmetric 3-lobe and 5-lobe sinc pulses. Each one had θnom = 80°, a nominal slice thickness of 10 mm, and peak b(t) amplitude of 5 μT; further details are given in Table 1. For each pulse, the excited magnetization M(z,σ) was computed for a grid of points from −40 mm to +40 mm in z with spacing 0.5 mm, and 60 points ranging from 0 to 1.6 in σ. Relaxation effects were neglected, and the waveforms b(t) and G(t) were sampled with a time step of 10 μsec. It is important to include the slice rephase gradient in this simulation to accurately estimate ϕ(z,σ). Computation took ∼8 min per pulse using Matlab 2007a (The Mathworks, Natick, MA) on a desktop PC (Windows XP, 2.66 GHz Quad core CPU, 4GB RAM). M(z,σ) need only be computed once for each RF pulse and saved in a library; subsequently, the relevant result may be loaded and used to derive a lookup table r(σ, T1) for any given instance of the AFI sequence with arbitrary TR1/TR2 and for any T1 by use of Eqs. 4–8.
Table 1. Slice Selective RF Pulse Parameters
Peak b(t) amplitude (μT)
Gradient strength (mT m−1)
Gradient refocusing area (mT m−1 msec)
In most situations, the T1 distribution within the object being imaged is unknown and cannot be used to fully correct the transmit sensitivity map. Instead, we take a single reference T1 value within the approximate range of T1s expected in the object and compute the lookup table r(σ) for this specific T1 value; we have investigated the effect of making this assumption for the four pulses considered.
Experiments were performed on phantoms and in vivo, using the AFI sequence with both selective and nonselective RF pulses. In all cases, the nominal flip angle was 80° with TR1 = 30 msec and TR2 = 150 msec. The modified phase cycling scheme proposed by Nehrke in (8) was used with RF phase increment angle 129.3° and strong spoiling gradients. The effect of the gradients can be quantified by the dimensionless “diffusion damping factor” d, related to the square of the time averaged gradient [see Eqs. 17 and 18 in (8)]. It was shown by Nehrke that d ≥ 0.3 is sufficient to achieve a high degree of spoiling in the brain, a necessary condition for the validity of Eqs. 1, 2, and 6. For all presented data d = 1.0 except where otherwise indicated. RF power calibration was performed using the scanner's standard preparation phase. For all data, maps of θ(x) were obtained by use of Eq. 2 and converted to σ(x) by dividing by θnom. For the experiments using 2D encoding these were compared with maps calculated using lookup tables. All experiments involving 2D spatial encoding involved the acquisition of one single slice.
Two phantoms were used. The first was a spherical flask of diameter 16 cm containing a solution of water and CuSO4 with T1 = 290 msec. This phantom was studied at 3 T where it produces substantial local variations in B1. It was imaged on a Philips Achieva MRI system fitted with an eight-channel parallel transmission body coil (9) using an eight-channel head coil for signal reception. For all experiments on this 3T system, the parallel transmission coil was configured to produce a quadrature excitation at the centre of the bore but without using load specific compensation such as RF shimming (10). For this phantom, maps were obtained using a nonselective RF pulse (duration 0.65 msec) and the asymmetric 2-lobe sinc pulse. The imaged voxel size was 4 × 4 × 10 mm3 (matrix size 45 × 45 × 19 for 3D and 45 × 45 for 2D); total acquisition times were 2 min 35 sec and 49 sec, respectively, including six averages for the slice selective map. Data were also acquired using the slice selective pulse with a 3D readout to image the slice profiles in the steady-state sequence (i.e., S1,2(z)). Voxel size was 5 × 5 × 0.5 mm3 (matrix size 40 × 40 × 100); imaging time was 12 min 00 sec.
The second phantom was used to investigate T1 dependence. It consisted of two rows of six 15 mL centrifuge tubes containing solutions of MnCl2 in distilled water with approximate concentrations of 0, 0.01, 0.05, 0.10, 0.15, and 0.20 mM. The rows of tubes were arranged with solutions in opposing concentration order and were imaged on a Philips 1.5T Achieva MRI system (Philips Healthcare, Best, The Netherlands) with an eight-channel head coil for signal reception. Imaging was performed at 1.5 T because at this field strength, the whole body birdcage resonator used for transmission produces a uniform B1 field (i.e., σ(x) = constant) over the ∼120 mm field of view. T1 relaxation times (at 1.5 T) measured using an inversion recovery prepared fast spin echo sequence (inversion delay = 50 − 2600 msec, TR = 15 sec) were 2680 ± 10, 2230 ± 20, 1330 ± 10, 910 ± 10, 600 ± 10, and 530 ± 10 msec. The tubes were imaged while placed at the centre of the bore, aligned with the main magnetic field. In addition to 3D transmit sensitivity maps, 2D maps were obtained using all four RF pulses detailed above. The 3D sequence used a nonselective RF pulse of duration 0.26 msec. Acquired voxel sizes were 1 × 1 × 1 mm3 (matrix size 120 × 120 × 50) and 1 ×1 × 10 mm3 (matrix size 120 × 50); total acquisition times were 14 min 07 sec and 2 min 22 sec for the 3D and 2D sequences, respectively, where the latter included 16 signal averages. Gradient spoiling resulted in a diffusion damping factor of 1.1 for these experiments. The required lookup tables were created with reference T1 value 1000 msec.
In vivo experiments were carried out on the 3T Achieva system. Research ethics committee approval was obtained for all human scanning as part of this study, and all participants gave written informed consent. Low resolution σ-maps suitable for RF shimming applications were acquired in the brain of a healthy volunteer; in 3D using a nonselective pulse of duration 0.40 msec and in 2D using each of the four slice selective pulses detailed above. The acquired voxel size was 5 × 5 × 10 mm3 giving a total imaging time of 7.2 sec for 2D and 2 min 26 sec for 3D with only one signal average used in each case (matrix size 44 × 40 in 2D and 44 × 40 × 21 in 3D). The minimum echo time with water and fat in phase was used for all in vivo experiments; this was 2.3 msec for all except the 3 and 5-lobed sinc pulses which are longer and so required an echo time of 4.6 msec. Slice selective maps were reconstructed using a lookup table calculated for each pulse, all with reference T1 = 1000 msec, chosen to approximate the average T1 value of brain tissue at 3T. Image reconstruction was repeated 200 times to assess the average time taken.
Figure 1 compares simulated slice profiles with those measured in the spherical flask phantom at 3 T. The strongly nonuniform B1 field produced within this phantom resulted in a measured σ of 1.14 at the centre of the phantom (see Fig. 2). Figure 1a shows the simulated flip angle profile θ(z) for σ = 1.14; the profile is not rectangular but the excitation outside the intended slice (nominal slice thickness 10 mm) dies away quickly. Figure 1b shows slice profiles S1,2(z) computed from θ(z) for the relevant T1 (290 msec) and TR1/TR2 = 30/150 msec; S1 and S2 are very different and contain significant contributions from outside the intended slice. The measured slice profiles from the centre of the phantom (Fig. 1c) compare very well qualitatively with the simulations. Simulated and imaged signal ratios (Fig. 1d) agree very well, indicating good quantitative agreement between simulation and experiment. Figure 2 contains line profiles through acquired and corrected maps for this phantom. There is a large discrepancy between 3D and 2D acquisitions when using the standard AFI formula (Fig. 2a); however, once corrected there is close agreement (Fig. 2b). Homogenous phantoms with a known T1 (here T1 = 290 msec) are a special case for which a very high quality correction can be achieved using this method. In this case, T1 was not sufficiently long in comparison with the TR1 (30 msec) and TR2 (150 msec) for Eq. 1 to be a good approximation even for the 3D B1 map. Thus, the expression for the exact ratio of S1 and S2 [Eq. 4 in Ref.1] was used in the form of a lookup table to correct the 3D measurement.
Figure 3 shows region of interest analysis of the MnCl2 phantom σ-maps plotted alongside theoretical predictions (curves) derived from simulation. These specific predictions are valid only for TR1/TR2 = 30/150 msec; different sequence parameters produce different slice profile sensitivities. If the system is perfectly calibrated, we expect to measure σ(x) = 1 everywhere. Errors in power calibration appear as a scaling of the measured values of σ such that the data points do not agree with the curves in Fig. 3. To compensate for this effect, all of the theoretical predictions were globally scaled to agree with the data; the necessary scaling factor was 0.996 indicating a very small error in the scanner's automated power calibration step. The nonselective method underestimates σ for very short T1 but is accurate for higher values, as expected. The apparent transmit sensitivities measured using the slice selective pulses are all lower than expected from the standard AFI equation (Fig. 3a), and each pulse produces a different relationship with T1. The theoretical predictions agree well with the measured data, suggesting that if T1 information were available then σ-maps could be reconstructed precisely for all T1 using any of these pulses. In general, however, this information is not available and hence a lookup table with a single reference T1 value is used (1000 msec in this case). The corrected 2D estimates (Fig. 3b) are only accurate for T1 = 1000 msec and vary to different degrees for other values of T1. The complete relationship r(σ,T1) for the 2-lobe asymmetric sinc pulse is given in Fig. 4a, and the lookup tables for T1 = 1000 msec used to make Fig. 3b are plotted in Fig. 4b. Figure 4c demonstrates the effect of ignoring this result and instead using the observed signal ratios from Fig. 4b to estimate σ via Eq. 2: the apparent transmit sensitivity is always less than the actual one. All of these curves are valid only for one specific TR1/TR2 combination; however, they may be scaled by considering the ratio TR1/T1.
Figure 5 compares low resolution in vivo transmit sensitivity maps from 3D and 2D sequences at 3T. While the various slice selective pulses each result in a different underestimation of σ when no correction is used, when lookup tables are applied in the reconstruction all give approximately the same result which agrees with the 3D case. T1 related error in the slice selective results causes spatial structure in the maps, which would otherwise be smooth. The most obvious example is the visibility of the ventricles; the line segments plotted on Fig. 5 cut through part of the ventricles at approximately y = +15 mm and here we see a large spread in σ even in the corrected maps.
Based on repeated trials, the time for creation of the lookup tables was 29 ± 15 msec, including hard drive access to load in simulated M(z,σ). The reconstruction time for the σ-maps (matrix size 64 × 64) was 1.8 ± 0.2 msec using a lookup table compared with 0.7 ± 0.1 msec using the analytic solution from Eq. 2.
The phantom results show excellent agreement between theory and experiment, demonstrating that transmit sensitivity maps acquired using AFI with slice selective pulses can provide a precise measurement when the data is processed taking account of slice profile effects. Although the numerical simulation required can be time consuming, it need only be performed once if the same RF pulse is used for multiple studies. Subsequent steps are fast—a lookup table relevant to the sequence can be produced in 30 msec, and interpolation is practically instantaneous.
The approach as outlined makes no assumptions about pulse performance, so can be used for a wide variety of pulses. Of the four RF pulses included in this article, the 3-lobe and 5-lobe sinc pulses produce estimates of σ which are the least sensitive to T1, but perhaps counter intuitively our simulations and experiments show that the 3-lobe sinc performs slightly better than the 5-lobe sinc in this regard (see Fig. 3b). Even for these pulses, however, errors of over 10% are still present if the correction is not performed. Simply rescaling the acquired σ-maps would result in inaccuracy because there is not a simple linear relationship between the true and estimated values of σ, as is apparent from Fig. 4c.
The method is particularly suited for calibration experiments using homogenous phantoms with known T1, and in this case, any RF pulse can be used. To obtain good results in vivo without the need for detailed T1 mapping, it is important to select an RF pulse which minimizes the T1 dependence of the technique. For brain imaging at 3 T, the most relevant relaxation times are T1 ≈ 850 msec for white matter (11), T1 ≈ 1750 msec in cortical gray matter (11), and T1 ≈ 4000 msec in cerebrospinal fluid (12). Figure 3 predicts that a pulse specific reconstruction with reference T1 = 1000 msec should lead to an underestimation of σ in CSF by ∼12% when using the gaussian pulse and an overestimation of ∼5% when using the 5-lobe sinc pulse. Both of these effects are apparent in the line profiles on Fig. 5. The 3-lobe sinc pulse leads to the most spatially smooth 2D σ-map, and this is in agreement with the curve in Fig. 3b which shows this pulse to be the most T1 insensitive over the range of values seen in the brain. However, not all of the structure seen in the corrected maps in Fig. 5 can be adequately explained by considering these three T1 values alone; partly, this is because there is much heterogeneity of T1 in the brain (11). More importantly, we must note that when imaging heterogeneous samples, partial volume effects lead to inaccuracy because the simulations assume a single tissue compartment; if multiple compartments are present then the analysis as presented is no longer valid. For in vivo imaging, the use of an RF pulse, which provides constant estimates of σ over the expected range of T1, would give accurate results; T1 related errors when using the 3-lobe sinc pulse are predicted to be within ±3% according to Fig. 3b.
In practice, the choice of RF pulse depends also on other factors such as SAR, or the need to use short echo times; the advantage of the proposed method is that it is flexible enough to accommodate any pulse. In some of the phantom experiments presented, multiple signal averages were used to obtain unequivocal results. It can be seen from Fig. 5 that 2D AFI is not, however, fundamentally limited by low signal-to-noise ratio and that reasonable σ-maps can be obtained in just 7.2 sec for a single slice in the brain. This corresponds to less than 1 min of mapping for calibration of an eight-channel array.
We have presented a method for accurate reconstruction of transmit sensitivity maps obtained using the AFI sequence with slice selective pulses. The method allows flexible choice of RF pulse; detailed Bloch equation simulation is required for each new RF pulse considered; however, this need only be performed once and thereafter lookup tables can rapidly be produced for any given combination of TR1 and TR2 in any examination. Results for four different RF pulses have been used to demonstrate the approach. A notable feature of the data obtained is that although all the pulses were suitable for correction, performance in terms of T1 sensitivity is highly variable. Choice of pulse is, therefore, important for in vivo work, where the spread of T1 values is generally large. The relative performance of each pulse can be predicted by using the lookup tables derived in this method. Furthermore, similar simulation-based corrections could potentially be applied to other B1 calibration methods for which slice profile effects are an issue.
The authors thank Philips Research Europe (Hamburg) for support of the multi-channel transmit infrastructure.