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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Magn Reson Med. Author manuscript; available in PMC 2013 July 1.
Published in final edited form as:
Published online 2011 December 2. doi:  10.1002/mrm.23199
PMCID: PMC3295910

Simultaneous Variable Flip Angle – Actual Flip Angle Imaging (VAFI) Method for Improved Accuracy and Precision of Three-dimensional T1 and B1 Measurements


A new time-efficient and accurate technique for simultaneous mapping of T1 and B1 is proposed based on a combination of the Actual Flip angle Imaging (AFI) and Variable Flip Angle (VFA) methods: VAFI. VAFI utilizes a single AFI and one or more spoiled gradient-echo (SPGR) acquisitions with a simultaneous non-linear fitting procedure to yield accurate T1/B1 maps. The advantage of VAFI is high accuracy at either short T1 times or long TR in the AFI sequence. Simulations show this method is accurate to 0.03% in FA and 0.07% in T1 for TR/T1 times over the range of 0.01 to 0.45. We show for the case of brain imaging that it is sufficient to use only one small flip angle SPGR acquisition, which results in reduced spoiling requirements and a significant scan time reduction compared to the original VFA. In-vivo validation yielded high-quality 3D T1 maps and T1 measurements within 10% of previously published values, and within a clinically acceptable scan time. The VAFI method will increase the accuracy and clinical feasibility of many quantitative MRI methods requiring T1/B1 mapping such as DCE perfusion and quantitative MTI.

Keywords: T1 Relaxometry, B1 Mapping, Actual Flip Angle Imaging (AFI), Spoiled Gradient Echo


Accurate measurements of longitudinal relaxation time (T1) are essential to many quantitative MRI techniques and clinical applications. Traditional inversion recovery (IR) T1 mapping methods often result in prohibitively long acquisitions due to a long repetition time (TR) (1). The Look-Locker method (2) may be used to accelerate the IR experiment by frequent sampling of the entire recovery curve. This two-dimensional (2D) approach can be modified to allow multi-slice volumetric coverage (3,4). A three dimensional Look-Locker method (5) has also been proposed to improve the SNR and slice direction resolution of 2D techniques at the cost of blurring in phase encode directions due to segmented k-space acquisition.

The variable flip angle (VFA) method (6,7), also known as driven-equilibrium single-pulse observation of T1 (DESPOT1) (7), uses several short TR spoiled gradient echo (SPGR) acquisitions with varying flip angle (FA) to measure T1. It has gained significant popularity over the past decades due to its superior time efficiency, allowing rapid and accurate three-dimensional (3D) T1 mapping with high resolution and large spatial coverage (810). While the acceleration of any T1 mapping method is possible by the use of fast readout sequences (e.g. echo-planar or spiral) (11,12), applications of such techniques are limited to a large extent due to image artifacts inherent to long gradient-echo signal readouts. In contrast, VFA is the only fast T1 mapping method to date which affords clinically reasonable scan times and 3D spatial coverage without a multi-echo readout, and therefore provides an optimal solution for high-resolution anatomic applications. In spite of these benefits, the VFA method suffers from a strong dependence on an accurate knowledge of excitation flip angle, which leads to significant errors in the presence of a non-uniform radiofrequency excitation field (B1), slab excitation pulse profiles, or FA miscalibration (9,10,13). This is particularly problematic at high field strengths, where additional wave effects in tissue may cause B1 variations up to 30% at 3T and in situ FA calibration and correction is necessary on a per-subject basis (14).

It is therefore commonly accepted that VFA T1 mapping at 3T or higher field strengths needs to be combined with an appropriate B1 correction method (9,10), and many methods to measure flip angle have been developed and applied to the correction of VFA T1 measurements. Similar to IR-based T1 mapping techniques, the double angle method (DAM) (15) uses a 2D multislice acquisition with a long TR to avoid T1 weighting of the acquired maps. Scan time can be reduced by minimizing T1 dependence with specialized radiofrequency pre-saturation pulses (16,17) to achieve a uniform saturation of magnetization over a large area prior to readout. However, the 2D implementation imposes restrictions on the use of DAM as a correction technique for 3D VFA T1 measurements due to slice profile effects.

It is important to note that a 3D B1 mapping technique with matched geometry is preferable in the context of VFA T1 mapping to simplify registration between data volumes. One 3D approach encodes the flip angle in the phase of the MR signal (18). This reduces the T1 dependence due to previous magnetization history and can image over a large dynamic range of FA, but is sensitive to main field inhomogeneity and requires a long TR relative to T1 to avoid phase oscillations. Other fast techniques have recently been proposed based on a stimulated-echo echo-planar acquisition (19) and a modification of DAM using a catalyzed preparative sequence (20). Methods which use a steady-state gradient echo readout are particularly well suited as these pulse sequences are closely related to the SPGR sequence used in VFA, and can therefore be acquired with the same matrix size and readout scheme, greatly increasing the consistency between the FA and T1 mapping steps. Such methods include a multi-point search of the signal null corresponding to the 180° flip angle (21,22), a B1-dependant shift in the spin resonance frequency (23), an inversion-recovery SPGR termed driven equilibrium single pulse observation of T1 with high-speed incorporation of RF field inhomogeneities (DESPOT1-HIFI) (24), and a dual TR SPGR acquisition termed actual flip-angle imaging (AFI) (25).

The accurate combination of a B1 mapping technique with 3D VFA T1 mapping requires consideration of additional factors affecting flip angle values, including excitation slab/slice profile and RF pulse properties (26). Two recently proposed methods, DESPOT1-HIFI (24) and AFI (25), measure FA using the exact same excitation pulse envelope and slab geometry as the VFA experiment, allowing measurements with a very similar FA distribution present in both sequences. DESPOT1-HIFI requires an additional B1-insensitive adiabatic inversion preparation, which, with proper tuning of the inversion time to the expected T1 range, enables fast and accurate FA mapping. To avoid T1 weighting, the AFI method estimates FA maps using the ratio of two signals from two different TRs (TRAFI1,2) using a linearized form of the signal equation (see next section) which holds under a main assumption that

TRAFI1,2  T1

The calculated flip angle map can then be applied as a pre-calibration step to a traditional VFA T1 measurement. AFI is well suited for the correction of VFA measurements because of the identical readout scheme to the SPGR pulse sequence, the use of an identical RF excitation pulse, and lack of a need for additional magnetization preparation with specialized RF pulses (20,23,24).

In spite of the many benefits of AFI, recent studies have shown that large spoiler gradients are required for effective suppression (spoiling) of transverse magnetization and thus accurate FA measurements (27). Such large gradients require a considerable increase in the TR compared to a standard short TR SPGR sequence, and therefore an overall increase in scan time. While proper spoiling is essential for accurate FA quantification, increasing the TR of the AFI sequence may lead to a violation of the main assumption (Eq. [1]) (25), posing another significant problem with the accuracy and overall efficiency of the technique. Similarly, strong spoiler gradients are also necessary in VFA SPGR measurements for accurate T1 mapping, especially at larger flip angles (27,28). This further increases the overall time of the technique.

An independent mechanism for violation of the AFI assumption (Eq. [1]) is T1 shortening due to contrast agents, which may render post-contrast FA measurements highly inaccurate. Post-contrast T1 measurements should therefore be corrected with a pre-contrast B1 map. However this is not feasible in some applications, such as manganese-enhanced MRI of axon pathways, which requires injection of a contrast agent approximately 24 hours before the scan (29). A wide range of applications would greatly benefit from a rapid and accurate method to measure T1 and FA in the presence of T1 shortening contrast agents (30,31).

AFI and VFA are highly complementary methods; AFI suffers inaccuracies due to an assumption about T1, while VFA suffers inaccuracies due to an assumption about FA. In this paper, we propose an efficient method to tackle the quantification of T1 and B1 in the problematic AFI regime (TRAFI1,2T1) arising from a long AFI TR due to spoiling requirements and/or short T1 times. In our method, Variable flip angle – Actual Flip angle Imaging (VAFI), T1 and B1 are mapped simultaneously using an optimized combination of AFI and VFA SPGR acquisitions. By exploiting the close relationship of the SPGR and AFI signal equations, we demonstrate that the TRAFI1,2 << T1 assumption is not necessary when T1 and B1 are mapped simultaneously, resulting in a high flip angle mapping accuracy over a broad range of T1 (32), an improvement in the precision of T1 and B1 measurements, and increased time efficiency of such measurements. We also show that VAFI reduces the spoiling requirements of the SPGR portion of the method, allowing a shorter TR and overall decrease in acquisition time.


Variable Flip Angle (VFA) T1 Mapping

T1 measurements based on VFA SPGR acquisitions have been previously described in (68,33). Given ideal spoiling of transverse magnetization, the SPGR signal can be described as follows:


where TR and excitation flip angle α are control parameters prescribed from the operator console and the equilibrium magnetization M0 and T1 are free parameters to be determined. Here, we assume that M0 absorbs additional factors due to T2* relaxation, receiver gains, and coil sensitivity profile. In VFA T1 mapping, two or more acquisitions with varying α and fixed TR are acquired and fitted to yield M0 and T1. The problem of data fitting can be conveniently casted into a linear form and fitted with least squares regression (8), or the curve can be fitted directly using nonlinear methods, with the latter approach in general providing better noise performance (34). In practice, two “ideal” flip angles are enough to provide optimized VFA T1 mapping for a given T1 value (33), the lower flip angle yielding an image with proton-density like contrast and higher flip angle with T1-like contrast.

Actual Flip Angle (AFI) Imaging

The AFI sequence is a spoiled gradient-recalled echo acquisition with two identical excitation RF pulses with flip angle α separated by two interleaved repetitions times TRAFI1,2 (25). Given complete spoiling of transverse magnetization at the end of each repetition time (25), the steady-state signal from the AFI sequence is described by the following equations:


It was shown in (25) that under the condition of Eq. [1] there exists an efficient first-order exponential approximation without an explicit dependence on M0 or T1. The signal then can be solved for the flip angle as follows:


Variable Flip Angle – Actual Flip Angle Imaging (VAFI)

In order to avoid the main AFI assumption (Eq. [1]), we propose a novel approach, VAFI, which takes as an input one AFI dataset and one or more VFA SPGR datasets. In VAFI, the dependence of the AFI flip angle on T1 is not ignored, but instead coupled in a single nonlinear optimization procedure for an unbiased simultaneous estimation of both parameters. We assume that M0 is consistent between SPGR and AFI acquisitions (Eq. [2] and Eq. [3] respectively), which requires that the RF excitation pulse, slab selection gradient, and echo times be identical for both the SPGR and AFI pulse sequences, and that the signal from both pulse sequences is perfectly spoiled (25,27). Additionally, we assume that error in α scales linearly with transmit field across the range of flip angles used in the VFA and AFI measurements. This assumption has been made in many previous FA mapping techniques (1517) and recent Bloch equation simulation results have shown less than 4% deviation from linearity over a range of 2° to 70° (35). For a set of SPGR flip angles αi and an AFI flip angle αAFI with corresponding observed signals sSPGR,i and sAFI1,2, the unknown parameters can be determined on a per pixel basis by simultaneously fitting both the SPGR and AFI signal models using nonlinear least squares:

[M0   T1   κ]=arg minM0   T1   κ(i=12(SAFI,iSAFI,i(αAFI·κ))2+i=1NSPGR(sSPGR,iSSPGR(αi·κ))2)

where argminx(f(x)) is the set of values x for which f (x) attains its smallest value.

This procedure simultaneously yields a solution for three unknown parameters: M0, T1, and κ, where κ is a scaling factor between actual and prescribed flip angle. Theoretically, this sets the lower limit on the total number of required measurements to three. As a single AFI sequence yields two independent measurements, we anticipate the minimum number of SPGR measurements to be one. As AFI requires a large flip angle (50°–60°) to optimize the estimation of FA maps and yields images with T1-like contrast, we anticipate that the SPGR measurement with the larger flip angle may be removed compared to the standard VFA experiment design (8,33).

As the transmit field varies slowly with spatial position, an additional term may be introduced to Eq. [5] to penalize the roughness of κ (r) and improve the noise properties of reconstructed parametric maps in a way similar to regularized fat/water imaging (36), which may be computationally challenging for 3D reconstructions. To reduce computational time, we developed a simplified three-step version of the regularization procedure:

  • Step 1: Solve Eq. [5] for spatial maps [M0 T1 κ]
  • Step 2: Apply spatial smoothing to κ (r) to yield κs (r).
  • Step 3. Solve [M0   T1]=arg minM0   T1(i=12(sAFI,iSAFI,i(αAFI·κs))2+i=1NSPGR(sSPGR,iSSPGR(αi·κs))2)



The proposed VAFI fitting method was implemented in MATLAB (The Mathworks; Natick, MA) using a built-in optimization algorithm. As an initial guess, we used the maximum signal intensity for M0, T1 = 1s, κ = 1. VAFI converged properly for all simulations and experiments without a need to modify this initial guess. Typical processing time was approximately 42 ms per voxel (64-bit Intel Linux workstation, 4 cores, 32 GB RAM). In Step 2 of the regularized VAFI and standard AFI approaches, a local polynomial fit was applied to generate smoothed and extrapolated FA maps (37) with a smoothing effect equivalent to a Gaussian kernel with a standard deviation of 4 voxels. For comparison, T1 and FA maps were generated using standard AFI-corrected VFA with linearized T1 estimation and iterative reweighting (34).

An AFI pulse sequence was implemented on a clinical GE 3.0T Discovery MR750 (GE Healthcare; Waukesha, WI) based on a product 3D SPGR sequence with the addition of a second TR as described in (25). Experiments were performed using a 32-element phased array receive coil (in-vivo experiments) and an eight-element torso coil (phantom experiments). Spoiler gradients were designed such that their areas AG1,2 were related as AG2 = nAG1 (Eq. [4]). RF spoiling was used with a quadratic phase increment of 34° (27). For all experiments, n = TRAFI2/TRAFI1 = 5, as was suggested in previous work (25). The specific strengths of the spoiler gradients were chosen based on the completeness of spoiling desired, herein referred to as “spoiling regime.”


Monte Carlo noise simulations were implemented to compare the accuracy (bias in the estimated values) and precision (noise in the estimated values) of standard two flip angle VFA with AFI correction (henceforth referred to as VFA) and VAFI. AFI and SPGR signals were generated using Eqs. [2],[3], with simulation parameters chosen to match the in-vivo experiments described in the next section (SPGR: TR = 10 ms, α = [3 18]°; AFI: TRAFI1 = 30 ms, αAFI = 55°). Signals were generated over a range of TR/T1 = 0.01 to 0.45, and for each T1 time 250,000 noise realizations were created by adding zero-mean Gaussian noise (standard deviation of 1e-4) to the signals. VAFI was implemented using the smallest SPGR flip angle signal and, to allow an equivalent comparison with VFA, using both SPGR flip angles. T1 and FA were then estimated to assess the accuracy and precision of all three approaches, the latter being assessed in terms of the T1-to-noise and α -to-noise ratios (T1NR and αNR).

A second simulation was performed to investigate the effect of flip angle selection on the VAFI experiment. For two-angle VFA, a pair of "ideal angles" has been previously identified as a way to optimize experiment design for a specific T1, however these angles were derived based on the linearized version of the SPGR equation (33) which is no longer valid in the VAFI technique. In order to assess the noise performance of our new method, a set of 45,000 noise realizations (standard deviation of 1e-4) were generated over a range of TR/T1 = 0.02 to 0.15. In the first set of simulations, αAFI was held constant at 55° while the SPGR flip angle was varied over a range of 1° to 20°. In the second set, SPGR flip angle was held constant at 5° while αAFI was varied over a range of 40° to 70°. All other parameters match the previous simulation.

A third simulation was performed to evaluate the effect of gradient and RF spoiling on the accuracy of VAFI quantification. AFI and SPGR signals were generated using a combined isochromat summation and diffusion propagator model (27), which takes into account transverse signal dephasing due to both gradient and diffusion effects. Signals were generated for prototypical tissues at 3T over a range of RF phase increments [var phi]o = 0 to 180° (white matter (WM): T1/T2 = 1000/70 ms, diffusion coefficient D = 0.70 × 10−3 mm2/s; gray matter (GM): T1/T2 = 1500/100 ms, D = 0.80 × 10−3 mm2/s; cerebrospinal fluid (CSF): T1/T2 = 4000/1900 ms, D = 3.00 × 10−3 mm2/s). SPGR signals were simulated using α = [3 18]° for three spoiling regimes: weak (TR = 10 ms, AG = 38 mT*ms/m), strong (TR = 10 ms, AG = 110 mT*ms/m), and nearly complete spoiling (TR = 15 ms, AG = 280 mT*ms/m). AFI signals were generated using αAFI = 55° for weak (TRAFI1,2 = 10/50 ms, AG1 = 38 mT*ms/m), intermediate (TRAFI1,2 = 10/50 ms, AG1 = 110 mT*ms/m), strong (TRAFI1,2 = 15/75 ms, AG1 = 280 mT*ms/m), and nearly complete spoiling (TRAFI1,2 = 30/150 ms, AG1 = 450 mT*ms/m). The data were then fit using the VFA and VAFI (single SPGR measurement, α = 3°) methods in two ways, one assuming ideal AFI spoiling with non-ideal SPGR data, and one assuming ideal SPGR spoiling using non-ideal AFI data. The accuracy of T1 estimates was then assessed.

Phantom Experiments

To test the effects of the suboptimal AFI regime (TR ≈ T1) on both VFA and VAFI, a set of ten phantoms were created from deionized water doped with Gd-BOPTA to concentrations of C = [0.00 0.10 0.25 0.50 1.00 2.00 2.50 3.00 3.50 4.00] mM. The phantoms were placed in a heated water bath lightly doped with Gd-BOPTA and held at 36.8 ± 0.5 °C for the duration of the experiment. SPGR scans were acquired with TR = 25 ms and α = [4 10 20 30 40 50]° and an AFI scan with TRAFI1 = 30 ms, αAFI = 55° in the nearly complete spoiling regime. Voxel size was 2.2×2.2×2 mm. All SPGR flip angles were used in VFA and VAFI to ensure a precise measurement over the wide range of expected T1 times. The mean and standard deviation of R1 (= 1/T1) was calculated in a single slice ROI encompassing each individual phantom, and used to fit the equation:


where r1 is the relaxivity, and R1,0 is longitudinal relaxation rate for free water. The linear fit was weighted according to the measured variance in the R1 values.

It was shown before that simultaneously increasing TR and acceleration factor R while maintaining a constant acquisition time constant leads to an improvement of the overall SNR efficiency of several steady-state gradient echo techniques (38). In the context of AFI, in addition to signal boost from longer TR, this increase also offers an opportunity to increase the length of the spoiler gradient and improve the accuracy of FA quantification (27). To confirm this effect for the AFI sequence, a uniform bottle phantom doped with Gd-DTPA to a T1 of 1.4 s was scanned using an 8-channel receive-only phased array knee coil. Two AFI scans were acquired (αAFI = 55, nearly complete spoiling with TR = 30 ms, and weak spoiling with TR = 10 ms, 1.5 mm3 voxel). The TR = 30 ms data was subsequently undersampled by a factor of 3 in the phase encode direction and reconstructed with SENSE, such that the experiment time matches AFI with TR = 10 ms. Flip angle maps using weakly spoiled AFI (TR = 10 ms) were also generated for comparison. Noise in the resulting FA maps was estimated by taking the standard deviation in a small ROI over the center of the bottle to avoid regions of artifact in the poorly spoiled AFI image.

In-Vivo Imaging

To demonstrate the application of VAFI to in-vivo brain imaging, one healthy human volunteer was scanned. Informed written consent was obtained in accordance with the local institutional policy. SPGR scans were acquired with TR = 10 ms and αSPGR = [3 18]°, which is optimal for T1 = 1.2 s as described in (33). To demonstrate the benefits of increased TR on spoiling, two AFI scans were acquired with αAFI = 55° in the weak regime (TRAFI1,2 = 10/50 ms AG1 =38 mT*ms/m) and in the complete regime (TRAFI1,2 = 30/150 ms AG1 =450 mT*ms/m). The last AFI scan was repeated with sensitivity encoding (SENSE) (37) acceleration factor R = 3 to demonstrate the feasibility of an overall scan time reduction. All scans were acquired with a 128×96×44 matrix and an isotropic resolution of 2×2×2 mm3 placed near the center of the brain. The scan time was 1:30 minutes for SPGR, 4:13 minutes for AFI with TRAFI1,2 = 10/50 ms, and 12:40 minutes for AFI with TRAFI1,2 = 30/150 ms and R = 1. For AFI with R = 3, the scan time was 4:13 minutes.



Figure 1 shows the results of simulations comparing the accuracy and precision of the methods. Over the range investigated, the standard AFI method shows a systematic underestimation of flip angle with decreasing T1 (> 5% for TR/T1 > 0.275). VAFI flip angle errors were less than 0.03%. T1 estimates for VFA were systematically overestimated at small TR/T1, due to propagation of AFI errors into VFA T1 mapping (> 5% for TR/T1 > 0.160). VAFI was able to restore the accuracy of T1 estimates (errors less than 0.07%). T1NR of VAFI exceeded that of AFI for a TR/T1 < 0.085, and αNR was comparable for TR/T1 < 0.150. For shorter T1, the AFI method actually shows better noise properties of FA estimates than VAFI; however this may be artifactual as in this region the FA and T1 estimates are also highly inaccurate and generally not useful. The peak T1NR with these particular settings was observed for a TR/T1 of 0.06, however it will be noted in the next paragraph that T1NR can be optimized for a specific T1 of interest by the choice of an appropriate SPGR flip angle. Remarkably, VAFI with two SPGR measurements shows more uniform noise performance in the range of TR/T1 than VAFI with single SPGR measurements, which is consistent with improved uniformity of noise performance of VFA T1 mapping with extended set of SPGR measurements (more than two) (39).

Figure 1
Monte Carlo simulations of standard VFA (dotted line) and VAFI (dash line for two flip angle fit and solid line for one flip angle fit) over a range of TRAFI1/T1 for flip angle accuracy (a), flip angle-to-noise ratio (b), T1 accuracy (c), and T1-to-noise-ratio ...

Figure 2 shows the results of VAFI T1NR and αNR as a function of TR/T1 and SPGR flip angle. The choice of flip angle has an impact on the T1NR, and just as the case for standard VFA, there exists an optimum angle that maximizes precision for a given T1 time. The optimum angles for T1 = 200, 600, 1000, and 1400 ms were 7.5°, 5.5°, 4°, and 3.5° respectively (TR = 10 ms). As long as the SPGR flip angle is within a reasonable range about the optimum value, there is no significant impact on the precision of the FA map. The choice of AFI flip angle has considerable impact on the precision of both VAFI T1 and FA measurements, although in this case there is no single optimum value for a specific T1 time; the precision of both T1 and FA increase linearly as AFI flip angle increases.

Figure 2
Monte Carlo simulations demonstrating the impact of flip angle selection in the VAFI technique on the T1-to-noise-ratio (a,c) and flip angle-to-noise-ratio (b,d) for a TRAFI1/T1 = 0.150 (dotted line), 0.050 (dot-dash line), 0.030 (dash line), and 0.021 ...

Figure 3 shows the results of simulations assuming ideal AFI spoiling and non-ideal SPGR spoiling for WM and GM tissues. For standard VFA, increasing the gradient spoiling minimizes the dependence of T1 accuracy on the RF phase increment. However, the choice of a proper phase increment (i.e. the choice of [var phi]o which crosses or approaches the zero-axis) and its accuracy and stability remains essential for accurate quantification. In contrast, the use of VAFI greatly reduces spoiling-related errors over a wide range of phase increments, even in the weak SPGR spoiling regime. Specifically, in the weak SPGR spoiling regime, if one chooses a phase increment that does not fall near one of the sharp peaks (ex: [var phi]o = 10° to 35°), VAFI with two SPGR flip angles achieves an error in T1 estimates of < 0.80% (WM), < 1.10% (GM), and < 1.75% (CSF, not shown in plot) and an error in FA < 0.26% (all tissues). Remarkably, these errors and the dependency on the phase increment are further reduced in VAFI with single SPGR flip angle (T1 errors: < 0.54% in WM, < 0.60% in GM, < 0.32% in CSF). The intermediate and strong spoiling regimes produce similar results, with fewer sharp peaks to avoid.

Figure 3
Simulation results showing the dependence of T1 accuracy on SPGR phase increment for weak (a, b), intermediate (c, d), and strong (e, f) spoiling regimes in prototypical white matter (T1/T2 = 1000/70 ms, D = 0.70 × 10−3 mm2/s) (a, c, e) ...

Figure 4 shows the results of simulations assuming non-ideal AFI spoiling and ideal SPGR spoiling for WM and GM. The behavior of spoiling-related errors is similar for the VFA and VAFI. Unlike the case of SPGR spoiling, the choice of a proper AFI phase increment is essential for accurate FA and T1 quantification for all techniques. The proper phase increment (i.e. the choice of [var phi]o which crosses zero-axis) depends on the spoiling regime, tissue type, and estimation technique used. For weak spoiled AFI, the optimal values of [var phi]o are 29° (WM), 20° (GM), and 50° (CSF), whereas in the nearly complete spoiling regime, these values are 39° (WM), 41° (GM), and any [var phi]o for CSF (not shown in plot). For the VAFI technique in the weak regime, these values are 30° (WM), 25° (GM), and 55° (CSF), while in the nearly complete regime these values are 43° (WM), 44° (GM), and any [var phi]o for CSF. However, small deviations in [var phi]o about the optimal value produce minimal errors in FA and T1 accuracy for all tissues when the nearly complete AFI regime is used.

Figure 4
Simulation results showing the dependence of T1 and flip angle accuracy on AFI RF phase increment for weak (a–d), intermediate (e–h), and nearly complete (i–l) gradient spoiling (see text for details) using standard (dotted line) ...

Phantom Experiments

Figure 5 compares FA maps in Gd phantoms estimated using standard AFI and VAFI. A significant discontinuity between the short T1 vials (phantoms 6–10) and the background media (longer T1) can be seen in AFI FA map (Fig. 5a). These errors, which are due to a violation of the AFI assumption (Eq. [1]), are minimized in the VAFI FA map (Fig. 5b), which agrees well with the expected slowly varying value of FA in the imaging plane. The growing differences between local FA values of AFI and VAFI with Gd concentration may be appreciated in Table 1. A similar trend may be observed in the calculated T1 values, with T1 errors reaching 19% for the highest Gd concentration. Corresponding R1 measurements versus Gd concentration show expected linear relaxivity curves (Eq. [6]) for VAFI (Fig. 6). The VAFI-derived relaxivity for the Gd-BOPTA solution was 3.9361 s−1mM−1 (R2 = 0.9996), which compares well to the reference value of 4.0 s−1mM−1 at 37° C (personal communication with Dr. Sophie Laurent, NMR and Molecular Imaging Laboratory, University of Mons, Belgium). This relaxivity measurement was significantly underestimated by standard AFI VFA (r1 = 3.5192 s−1mM−1, R2 = 0.9935).

Figure 5
A map of flip angle, in degrees, of the T1 phantom generated using the standard AFI (a) and VAFI (b) techniques. The nominal FA was 55° (see table 1 vial labeling). A significant error not related to the slowly varying B1 spatial distribution ...
Figure 6
Plot of R1 vs. gadolinium concentration for the vial phantoms, using standard VFA and VAFI. The standard method shows a significant underestimation of R1 at short T1 times (due to errors in the AFI FA map propagating into the T1 estimate) and the relaxivity ...

Figure 7 illustrates the effect of increasing the SENSE acceleration factor on the accuracy and precision of AFI maps for the strong spoiling regime and a comparison with an AFI map in the short TR (weak spoiling) regime. Simulated AFI steady-state signals grow with increasing TR faster than the SNR loss due to R-factor, giving rise to an overall increase in SNR efficiency for realistic reduction factors (R < 8) and constant scan time (Fig. 7a). Indeed, long TR (strong spoiling) AFI (Fig. 7b) demonstrates improved precision over a short TR (weak spoiling) AFI (Fig. 7c) obtained in an equivalent scan time (R=3). The weak spoiling regime in the short TR AFI also leads to significant artifacts in the estimated FA map (Fig. 7c).

Figure 7
(a): Effect of SENSE on the SNR of the AFI sequence when simultaneously increasing reduction factor R and TR. Losses due to coil geometry (g-factor) were not considered. (b): Flip angle map acquired with AFI in strong spoiling regime (long TR) for SENSE ...

In-Vivo Imaging

Figure 8 shows a representative axial T1 map from the in-vivo experiment using fully sampled well-spoiled (Fig. 8a,b) and poorly-spoiled (Fig. 8d,e) AFI sequences. Table 2 shows the values taken from ROIs positioned in WM, GM, and CSF. For the weak AFI spoiling regime (TR1 = 10 ms), incomplete spoiling leads to underestimated FA measurements and overestimated T1 for all tissues, which is in accordance with theoretical predictions (see Fig. 4 and Ref. (27)). In the complete spoiling regime, both VFA and VAFI compare similarly to previously published values in the selected WM/GM ROIs (40), while VAFI reaches comparable or even better precision in less time (using one less SPGR measurement). Remarkably, in CSF, VAFI provided superior accuracy of T1 estimates based on the previously published reference CSF T1 values at 3T (41). The errors observed for the standard VFA method may be due to the fact that the spoiling regime used in this experiment resulted in incomplete spoiling for the higher SPGR flip angle and nearly complete spoiling for the smaller SPGR flip angle. Because VAFI uses only the smaller SPGR flip angle, the spoiling errors in SPGR are less likely to affect T1 estimates, as also follows from the theoretical analysis (see Fig. 3). Finally, we demonstrate a SENSE-accelerated, well-spoiled T1 map using VAFI, which shows high image quality (Fig. 8c) and T1 values (Table 2) comparable to the non-accelerated VAFI (Fig. 8b).

Figure 8
In-vivo T1 maps in units of seconds in a healthy human volunteer comparing nearly complete AFI spoiling (a–c) versus weak spoiling (d, e) using standard VFA versus single flip angle VAFI techniques. For a well-spoiled sequence, both VFA and VAFI ...


The variable flip angle SPGR T1 mapping continues to be an accurate and efficient method for mapping T1 over a large three dimensional volume; however a large limitation to this technique is the need for an accurate estimate of an actual flip angle, especially at higher field strengths (3T and above). Among many methods for measuring flip angle, AFI (25) has emerged as a very attractive choice for correcting VFA measurements because of its high time-efficiency and similar pulse sequence design. In spite of these benefits, AFI often presents its own limitations due to the need for a short TR to maintain the validity of the TRAFI1,2 << T1 assumption. The available tradeoffs may lead to inaccurate FA and T1 mapping and elongated acquisition times in some practical applications. We presented a novel technique, VAFI, which provides a T1 – independent flip angle map and simultaneously yields accurate T1 values. Briefly described in (32), VAFI maintains high accuracy in regimes that are problematic for the standard AFI technique (TRT1), allowing its use in applications involving a long TR (for proper spoiling) or with T1-shortening contrast agents (such as dynamic contrast-enhanced imaging). It was previously noted that the VFA technique alone provides no unique solution for M0, T1, and flip angle (13). With the addition of dual-TR AFI data, a unique solution can be found even with only a single SPGR data point. An interesting approach to yield both flip angle and T1 values was proposed in subsequent work, which generalizes AFI into a pulse sequence of N arbitrary TR periods (42). The practical advantage of the VAFI approach compared to the multiple TR method is much shorter data acquisition times, which are minimized when using a single SPGR measurement to augment the standard dual-TR AFI scan.

Similar to VAFI, DESPOT1-HIFI (24) is a variation on the SPGR sequence, which also utilizes simultaneous fitting to map FA and T1. To generate unique information to fit for an additional FA parameter, DESPOT1-HIFI utilizes an inversion pulse prior to SPGR readout. Both VAFI and DESPOT1-HIFI require some assumptions about RF pulse properties. In the case of DESPOT1-HIFI, it is assumed that the adiabatic inversion pulse is completely insensitive to any parameters affecting the spin system (B0, B1, T1), which requires careful pulse sequence design. In VAFI, it is assumed that the pulse flip angle scales linearly with transmit field. Previous work (35) has shown that flip angle linearity is a valid assumption over the range of values herein investigated. DESPOT1-HIFI may be more time-efficient than VAFI as it avoids a longer TR. However, DESPOT1-HIFI FA maps may suffer from decreased accuracy in areas with long T1 (e.g., CSF, edema, or T1 "black hole" lesions in multiple sclerosis). As noted in (24), these problems can be resolved through the acquisition of additional SPGR flip angles and IR-SPGR inversion times tuned specifically for the longer T1, which would increase overall scan time of DESPOT1-HIFI accordingly. Another potential concern is the reduction of signal in DESPOT1-HIFI due to magnetization inversion. A detailed comparison of both techniques, including their SNR efficiency, is a subject of a future work.

We demonstrated that, with properly tuned parameters (proper choice of AFI and SPGR flip angle, phase increment of RF spoiling, and spoiler gradient area for the tissue of interest, the accuracy and precision of VAFI fit can surpass that of the standard VFA fit (when used with AFI for flip angle calibration) (Fig. 1). This is true even when one SPGR measurement is eliminated, further increasing the overall time efficiency of the technique. This improvement comes from the explicit utilization of least squares estimation for solving Eq. [9], a benefit which is not available with linearized calculation of the flip angle and T1 in the standard AFI and VFA methods. However, this improvement is at the cost of longer processing times than those required for original AFI and VFA approaches. An alternative and potentially faster minimization approach is to solve VAFI Eq. [5] through interleaved estimations of flip angle by AFI and T1 maps by VFA (43). However, this approach obviously requires a full set of SPGR measurements to allow a separate VFA fit, and its convergence and noise performance have yet to be studied. Image reconstruction can be easily performed after data acquisition, similar to the majority of quantitative parameter mapping methods. The current implementation uses a brute-force optimization method, and as a result the algorithm requires lengthy processing times (~42 ms/voxel, or about 6 hours for a high resolution whole-brain dataset). An important next step will be to substantially reduce this time through the use of more efficient optimization methods utilizing a C-code implementation or commodity graphics processing unit hardware (44). An additional speed up is expected from initializing the algorithm using T1/FA values from the regular VFA/AFI fitting procedures.

A major advantage of the traditional VFA technique is the ease with which acquisition settings can be optimized, either for a single T1 time using a pair of “ideal” flip angles (33) or using sets of combined ideal angles to image over a wide range of T1 (10). Our simulation results show that VAFI can be similarly optimized through the proper choice of flip angle for the proton-density weighted SPGR measurement. Because of the non-linear nature of VAFI, we found this optimal angle through a numerical solution instead of an empirical formula. It is important to note that the optimal choice of this flip angle for VAFI is neither the Ernst angle, nor the same as the smallest ideal angle for a VFA experiment. However, we observed that the optimal SPGR flip angle is always close (within one degree) to the smallest of the "ideal" angles optimizing VFA experiment design (33). Therefore, the “ideal” angle formula may still be utilized as a close approximation to the optimal VAFI SPGR flip angle when designing experiments. Simulations also showed that VAFI with multiple SPGR flip angles will result in improved efficiency in the wider T1 range, similar to VFA T1 mapping with optimized experiment designs (39,45).

A major limitation of both AFI and VFA is the need for proper spoiling parameters to ensure accurate quantification (27). While strong AFI spoiling is still a necessity for VAFI, simulations and in vivo studies have shown that the need for strong SPGR spoiling for the case of VAFI may be greatly reduced if a single flip angle SPGR measurement is used. This may allow reduction of spoiling gradients for SPGR part of VAFI and, as a result, more time-efficient overall mapping of the flip angle and T1 than that is available with AFI and VFA alone. Notably, the dependence of T1 accuracy on the choice of RF phase increment becomes much smaller. This is similar to a previous modification of the VFA method, which proposed an SPGR RF phase increment near the plateau of the curve rather than near a zero-crossing in order to reduce the sensitivity of T1 to this value (46). However, the previous method requires an additional correction to the subsequent T1 maps, which is not necessary in VAFI.

One must consider the time efficiency of VAFI compared to the traditional VFA and AFI approaches when used separately. In the context of T1-insensitive flip angle mapping, VAFI requires a short TR SPGR acquisition in addition to AFI, which may increase the overall scan time insignificantly (by 6 % in our un-accelerated version, compared to standard AFI scan time). In the context of T1 mapping, VAFI requires that both the AFI and SPGR scans be matched in spatial resolution, at the resolution of interest for T1 mapping. Although this eliminates the need for one SPGR measurement without a loss of precision, an AFI scan with the optimal n = TRAFI2/TRAFI1 = 5 is six times longer than a normal SPGR acquisition, which may make straightforward high resolution (~1 mm3) T1 mapping with VAFI less efficient than with alternative VFA-based T1 mapping techniques (24). While such full resolution VAFI may potentially improve accuracy of FA/T1 mapping if there is a partial volume effect between very short (problematic AFI regime) and long T1 components, several options may help minimize imaging times and simultaneously maintain accurate FA/T1 mapping with VAFI. One approach is to acquire an AFI dataset at a lower spatial resolution along with a standard VFA acquisition of two SPGR images at the desired resolution, and to use low-resolution VAFI to fit for FA. Such method would still benefit from high accuracy due to proper spoiling and T1-independence of FA maps, but would not benefit from a reduced number of VFA SPGR measurements and a gain in T1 precision from simultaneous least squares fitting. As we confirmed in the paper, parallel imaging is also a very suitable option for acceleration of VAFI, as the signal boost from longer AFI TR efficiently compensates SNR loses from parallel MRI (38), an effect mostly unavailable for inherently short TR techniques (24). Parallel MRI was used to accelerate AFI (R = 3) resulting in the acquisition of a VAFI T1 mapping in under 5 min with a minimal effect on the map quality and accuracy (< 3.6% error in white matter/gray matter T1). The method can be further accelerated for increased coverage or higher resolution imaging using systems with larger numbers of receive channels or lower resolution VAFI mapping as discussed above. The other recommended parameters for AFI part of in vivo protocol are TRAFI1 = 30 ms, TRAFI2 = 150 ms, αAFI = 55°, AG1 = 450 mT*ms/m, [var phi]0 = 44°, and a SENSE acceleration factor of 3. Recommended SPGR parameters are TR = 10ms, αSPGR = 3° AG = 38 mT*ms/m, and [var phi]0 = 30°.


The variable flip angle method is a rapid and accurate method for measuring T1, as long as it can be paired with an equally rapid and accurate method for measuring and correcting flip angle. In this study, we presented a method that combines VFA and AFI into a single procedure, named VAFI. This allows the TR of the AFI sequence to be extended to accommodate a large spoiler gradient for proper quantification, while removing the main AFI requirement of TRAFI1,2 << T1. We showed in both Monte Carlo simulations and phantom scans that this method is highly accurate at estimating both FA and T1 in the presence of very short T1 times, with greater precision than the traditional VFA approach. We showed in simulations and in-vivo scans that proper spoiling of the AFI sequence is essential to accurate quantification, while the spoiling requirements of the SPGR portion of the experiment are considerably reduced, resulting in a much shorter SPGR acquisition time. The proposed VAFI method has a high potential to become a widely used T1/flip angle mapping approach in a variety of high-field imaging applications, particularly those with shortened T1 values such as dynamic contrast enhanced perfusion MRI.


This work was supported by NIH NINDS R01NS065034 and in part by NIBIB R21EB009908. We would like to thank to Dr. Alex Frydrychowicz and Dr. Ian Rowland for assistance with phantom experiments.

Contributor Information

Samuel A. Hurley, Medical Physics, University of Wisconsin, Madison, WI.

Vasily L. Yarnykh, Radiology, University of Washington, Seattle, WA.

Kevin M. Johnson, Medical Physics, University of Wisconsin, Madison, WI.

Aaron S. Field, Radiology, University of Wisconsin, Madison, WI.

Andrew L. Alexander, Medical Physics and Psychiatry, University of Wisconsin, Madison, WI.

Alexey A. Samsonov, Radiology, University of Wisconsin, Madison, WI.


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