Direct Signal Control of the Steady-State Response of 3D-FSE Sequences

Abstract

Purpose

Parallel transmission (PTx) offers spatial control of radiofrequency (RF) fields that can be used to mitigate nonuniformity effects in high-field MRI. In practice, the ability to achieve uniform RF fields by static shimming is limited by the typically small number of channels. Thus, tailored RF pulses that mix gradient with RF encoding have been proposed. A complementary approach termed “Direct Signal Control” (DSC) is to dynamically update RF shims throughout a sequence, exploiting interactions between each pulse and the spin system to achieve uniform signal properties from potentially nonuniform fields. This work applied DSC to T2-weighted driven-equilibrium three-dimensional fast spin echo (3D-FSE) brain imaging at 3T.

Theory and Methods

The DSC concept requires an accurate signal model, provided by extending the spatially resolved extended phase graph framework to include the steady-state response of driven-equilibrium sequences. An 8-channel PTx body coil was used for experiments.

Results

Phantom experiments showed the model to be accurate to within 0.3% (root mean square error). In vivo imaging showed over two-fold improvement in signal homogeneity compared with quadrature excitation. Although the nonlinear optimization cannot guarantee a global optimum, significantly improved local solutions were found.

Conclusion

DSC has been demonstrated for 3D-FSE brain imaging at 3T. The concept is generally applicable to higher field strengths and other anatomies. Magn Reson Med 73:951-963, 2015. © 2014 Wiley Periodicals, Inc.

Introduction

Spatial variability of radiofrequency (RF) fields is a prominent issue for MRI at high static magnetic field strengths (≥ 3T) for which parallel transmission (PTx) has emerged as a potential solution by introducing a degree of spatial and temporal control of the RF fields. The simplest way in which this control can be applied is to attempt to create uniform fields by using a suitable linear combination of the available transmit channels. However, this so-called RF shimming is limited both as a result of the electrical properties of the sample, which may cause substantial field modulation, and by the typically small number of available transmit channels whose spatial sensitivity patterns often do not provide sufficient encoding power on their own. Consequently, many researchers have combined RF degrees of freedom with gradient encoding, producing tailored RF pulses with spatial and/or spectral properties that are tailored to an individual subject. For a noncomprehensive set of examples, see Refs. (1–5). These two approaches may be seen as opposite ends of a spectrum spanning the times scale over which the RF modulation is applied. In the case of RF shimming, a field pattern is optimized and then fixed in time, whereas in the case of tailored RF pulses separate waveforms may be applied on each channel—leading to RF field patterns that are modulated over microsecond or millisecond timescales. On the other hand, the process of MR image formation typically takes place over an intermediate timescale; the received signal depends on interactions between multiple RF pulses and the spin system. Such interactions provide an additional mechanism for controlling the evolution of the system. Rather than seeking to control the uniformity of the RF fields or the magnetization created by an individual pulse, the goal of “direct signal control” (DSC) is to influence the properties of the measured signals by changing the RF field pattern on a pulse-by-pulse basis (i.e., by applying dynamic rather than static RF shimming). In order to achieve this, a model of the signal formation process is required.

The extended phase graph (EPG) algorithm (6) provides such a model, which is particularly useful for sequences with a regular repeating time structure; echo amplitudes may be computed by tracking the populations of “dephased” states while they evolve through a sequence. The method has been used extensively to characterize RARE (rapid acquisition with relaxation enhancement) (7), also known as fast spin echo (FSE)- or turbo spin echo (TSE)-pulse sequences (6,8–10). In recent work (11), this framework was extended to include spatial variations of pulse flip angle/phase. The resulting spatially resolved EPG (SR-EPG) framework was used to demonstrate proof of concept DSC experiments for two-dimensional (2D)-FSE imaging, neglecting relaxation effects.

The focus of this work is to achieve uniform signal properties from 3D-FSE with very long echo trains when the RF fields lack spatial uniformity. 3D-FSE sequences have many more pulses than their 2D equivalents (typical echo train lengths are of the order of 100); thus, relaxation must be considered both during echo trains—and crucially, between echo trains. Both the original EPG framework and the SR-EPG extension are generally used to compute echo amplitudes, assuming that magnetization is at thermal equilibrium at the start of each echo train. This is a reasonable model when using single shot acquisitions or long repetition times arising from multi-slice imaging. However, it is not appropriate for 3D-FSE imaging because the repetition time between shots is frequently limited in order to achieve reasonable acquisition durations. Indeed, in order to avoid saturation effects it is often necessary in this case to use a driven equilibrium approach (12) for which an additional refocusing pulse is included, followed by a flip-back pulse designed to return residual transverse magnetization to the longitudinal axis. Echo amplitudes calculated by starting from equilibrium may be considered as the transient response of the system; however, a steady state will arise over multiple repetitions. This steady state will generally be a function of all of the flip angles used in the echo train (including the flip-back pulse), as well as tissue relaxation parameters. In this article, a method for computing the steady-state response is introduced and is used as a model for DSC of 3D-FSE with parallel transmission.

Theory

A full introduction to EPG calculations can be found in Refs. (8,13–15); here the discussion will be limited to a brief introduction of notation, which will follow closely from that used in Ref. (11). The magnetization state can be written as a sum over transverse (F_k) and longitudinal (Z_k) states, where subscript k represents the level of dephasing of that state. Echoes are generated by the F₀ state (nondephased transverse magnetization). For FSE with N refocusing pulses, we can write the amplitudes of the N echoes compactly as:

$$\mathbf{I}=\left[{\mathrm{F}}_0\left(1\right),{\mathrm{F}}_0\left(2\right),\dots {\mathrm{F}}_0\left(\mathrm{N}\right)\right]=f\left({\mathrm{\theta }}_0,{\mathrm{\theta }}_1,...{\mathrm{\theta }}_{\mathrm{N}},{\text{T}}_1,{\text{T}}_2\right)=f\left(\mathbf{\theta },{\text{T}}_1,{\text{T}}_2\right)$$ (1)

where F₀(n) is the population of the refocused state at the time of the n^th echo and θ_i is the flip angle of the i^th pulse; θ_i is complex where arg(θ_i) corresponds to the phase of the i^th pulse. The excitation pulse is given index i = 0. T₁ and T₂ are longitudinal and transverse relaxation times, and f signifies application of an EPG algorithm that calculates the F₀(n) for given θ, T₁, and T₂.

Direct Evaluation of Steady-State Echo Amplitudes for Multi-Shot Fast Spin Echo

A driven equilibrium FSE shot with N imaging echoes consists of an excitation pulse with flip angle θ₀, a set of refocusing pulses θ_1...N+1, a flip-back pulse θ_N+2, and finally a recovery period T_REC as shown in Figure 1. Refocusing pulse θ_{N +1} generates an echo that is converted to nondephased longitudinal magnetization (Z₀) by the flip-back pulse. Subsequently, transverse and higher-order longitudinal coherences have small populations that further decay during the recovery period due to relaxation and also due to diffusion (16), which preferentially attenuates higher-order states. However, the Z₀ state grows with longitudinal recovery. Sequence timing dictates that only even-numbered longitudinal states significantly contribute signal to the next echo train. The population of Z₀ at the end of the recovery period is typically two orders of magnitude higher than the next largest contributor (Z₂); therefore, we assume that the only link between each echo train is via Z₀ and consider all other states as zero after the recovery period. Although the system starts from equilibrium magnetization (M = [0 0 M₀]^T) in the first shot, in subsequent shots the starting longitudinal magnetization will approach a dynamic steady-state value M^*₀. Once a steady state is reached, the longitudinal magnetization at the time points indicated in Figure 1 can be written as: (A) M^*₀, (B) M^†₀, (C) M₀(1–E_R) + M^†₀E_R where E_R ≡ exp(−T_REC/T₁). M^†₀ is the longitudinal magnetization directly after the flip-back pulse; in general, this is a function of M^*₀, θ, T₁, T₂, and M₀. Without a simple expression for M^†₀, the only way to evaluate M^*₀ is by brute force, running a transient simulation including Z₀ recovery for multiple shots until a steady state is reached. This so-called “full transient simulation” is used throughout this work as a gold standard, but we now show that a direct approach is possible under certain conditions.

Fig. 1. Schematic of multi-shot, driven-equilibrium three-dimensional fast spin echo sequence.

Because we neglect the effect of higher-order longitudinal states after the recovery period, we may concentrate here just on Z₀, in the EPG terminology just introduced if Z₀ (0) = M^*₀, then Z₀(N + 2) = M^†₀, hence the following equation may be derived from standard EPG relations:

$$Z_0\left(N+2\right)=\frac{i}{2}\text{sin}{\mathrm{\theta }}_{N+2}\mathrm{\{}{F}_{-1}\left(N+1\right)-F_{-1}^{*}\left(N+1\right)\mathrm{\}}{e}^{-\frac{\text{τ}}{2T_2}}+Z_0\left(N+1\right)\text{cos}{\mathrm{\theta }}_{N+2}$$ (2)

where τ is the interecho spacing. Eq. [2] indicates that Z₀(N + 2) is comprised of a contribution from the final echo (F_±1(N+1)) and from preexisting nondephased longitudinal magnetization (Z₀(N+1)) that may have been present from the beginning or could have recovered during the shot. Carr-Purcell-Meiboom-Gill FSE sequences have the general property that the odd and even numbered states are decoupled such that only odd numbered states can contribute to any observed echo, whereas T₁ recovery of Z₀ during the echo train can only contribute to even-numbered states [e.g., Ref. (15)]. As a result, the first term in Eq. [2] depends only on linear operations applied to the magnetization at the start of the shot (M^*₀), whereas the second term contains contributions from recovering magnetization and thus is not necessarily linear in M^*₀. Therefore, we arrive at an expression:

$${\mathrm{M}}_0^{†}=\mathrm{\lambda }\left(\mathbf{\theta },{\text{T}}_1,{\text{T}}_2\right){\mathrm{M}}_0^{*}+\mathrm{\Delta }\left(\mathbf{\theta },{\text{T}}_1,{\text{T}}_2,{\mathrm{M}}_0,{\mathrm{M}}_0^{*}\right).$$ (3)

In order to simplify the problem, we assume that the longitudinal magnetization after the shot is comprised only from transverse magnetization returned to the z-axis by the flip-back pulse, that is, △ = 0. The assumption of “effectively no recovery” is similar to assumptions used by other authors in related circumstances (17–19), and the effect of this simplification is explored in a subsequent section. In this case, M^†₀ = λM^*₀, thus the longitudinal magnetization at point C in Figure 1 is given by M₀(1-E_R) + λ M^*₀ E_R; and given the cyclic nature of the sequence, we equate points A and C to obtain an expression for M^*₀:

$${\mathrm{M}}_0^{*}=\left(\mathbf{\theta },{\text{T}}_1,{\text{T}}_2\right)={\mathrm{M}}_0\frac{1-E_R}{1-\mathrm{\lambda }{E}_R}.$$ (4)

A similar expression was derived in Ref. (19) for the simpler scenario of 90° excitation and 180° refocusing pulses; in that case λ simply represents T₂ decay throughout the echo train. In this work, λ can be defined more generally as the fraction of the starting longitudinal magnetization that is returned to the z-axis after the shot, and it may be directly determined from a standard transient EPG calculation by evaluating:

$$\mathrm{\lambda }\left(\mathbf{\theta },T_1,T_2\right)=\frac{Z_0\left(N+2\right)}{Z_0\left(0\right)}.$$ (5)

The steady-state echo amplitudes, I_ss(n) may thus be computed as:

$${\mathbf{I}}_{\mathbf{\text{SS}}}=f_{SS}\left(\mathbf{\theta },{\text{T}}_1,{\text{T}}_2\right)=M_0^{*}\left(\mathbf{\theta },{\text{T}}_1,{\text{T}}_2\right)f\left(\mathbf{\theta },{\text{T}}_1,{\text{T}}_2\right),$$ (6)

where both components can be obtained from the results of a single standard EPG calculation.

Spatially Resolved Extended Phase Graphs and Direct Signal Control Optimization

All of the preceding theory can be generalized to include spatial variability of the RF fields (11). The steady-state echo amplitudes may thus be written as:

$${\mathbf{\text{I}}}_{\mathbf{\text{SS}}}\left(n,r\right)=f_{\text{SS}}\left(\mathbf{\theta }\left(\mathbf{\text{r}}\right),{\text{T}}_1,{\text{T}}_2\right)$$ (7)

where now for clarity, the echo amplitudes are written as a function of both echo number n and spatial location r. For parallel transmission, the spatially variable flip angles may be written as linear combinations of contributions from multiple channels (20,21) leading to the expression:

$$\mathbf{\theta }\left(\mathbf{r}\right)=\mathbf{\text{A}}\mathbf{\sigma }\left(\mathbf{r}\right).$$ (8)

In Eq. [8], the flip angles are decomposed into a product between A_ij, the nominal flip angle applied on channel j for pulse i, and σ_j(r), the dimensionless transmit sensitivity of channel j. Both A and σ(r) can be complex, and σ(r) is written as a column vector so that θ(r)is a single set of pulse amplitudes/phases relevant to spatial location r. The goal of DSC is to find an optimized set of flip angles A that will lead to echo amplitudes conforming to a target T(n,r) by performing the following minimization:

$${\text{min}}_{\text{A}}{\left\{{\displaystyle \sum _{n\in \mathbf{\text{W}},\mathbf{\text{r}}\in \mathbf{\Omega }}(}\right|{\mathbf{\text{I}}}_{\mathbf{\text{SS}}}\left(n,\mathbf{\text{r}}\right)|-\mathbf{\text{T}}(n,\mathbf{\text{r}}))}^2+\eta {{\displaystyle \sum _{n\in \mathbf{\text{W}},\mathbf{\text{r}}\in \mathbf{\Omega }}\left(\frac{\partial }{\partial n}\text{arg}({\mathbf{I}}_{\mathbf{\text{SS}}}(n,\mathbf{\text{r}}))\right)}}^2+\mathbf{\beta }\mathrm{\Vert }\mathbf{\text{A}}{\mathrm{\Vert }}_F^2\}$$ (9)

where Ω and W represent subsets of spatial locations and echoes, respectively, to be included in the optimization. The first term aims to minimize the difference between the magnitude of the steady-state echo amplitudes and the (real valued) target echo amplitudes. This is similar in spirit to magnitude least-squares RF shimming (22) and allows more degrees of freedom in the optimization because the spatial distribution of echo phase is not important (spatial smoothness constraints could be included, however, if necessary). The second term in the cost function aims to penalize temporal phase oscillations because these would affect the resulting images. The third term aims to control the RF power (and global specific absorption rate (SAR)); the weighting parameters η and β are empirically determined.

Calculation of appropriate refocusing flip-angle trains in the absence of RF inhomogeneity is a well-studied area with many interesting strategies, including pseudosteady-state (PSS) (10), transitions between pseudo steady states (TRAPS) (23), hyperechoes (14), and tissuespecific signal modulation (24,25). In this article, we seek solutions with spatially uniform signals and assume that an “ideal” set of flip angles exists as a starting point that would yield desired ideal signal properties in the absence of RF inhomogeneity. This will be referred to as the “base sequence” denoted by θ_base, which is not spatially variable [i.e., θ_base(r) is the same for all r]. The actual echo amplitudes I_SS(n,r) will vary spatially in the presence of RF inhomogeneity, and the goal is to use dynamic RF shimming to restore the ideal behavior. Therefore, the natural target for optimization with Eq. [9] is to define T(n,r) as the echo amplitudes that would result if there was no inhomogeneity, that is,

$$\mathbf{T}\left(\mathrm{n},\mathbf{r}\right)={\mathbf{I}}_{\mathbf{\text{SS}}}\left({\mathbf{\theta }}_{\mathbf{base}},{\text{T}}_1,{\text{T}}_2\right)$$ (10)

where T₁ and T₂ are those of an appropriately chosen reference tissue.

“No Effective Recovery” Assumption

Although the assumption Δ(θ,T₁,T₂,M₀,M^*₀) = 0 does not hold for any general set of refocusing pulses, the existence of the flip-back pulse in the driven equilibrium FSE sequence helps, as can be seen from Eq. [2], where we note that the second term Z₀(N+1) cos θ_N+2 corresponds to Δ. If the flip-back pulse θ_N+2 = – 90° (as is usually the case) then Δ = 0 irrespective of Z₀(N +1). In general, RF inhomogeneity means that cos θ_N+2 is likely to be small but finite. Similarly for FSE sequences commonly used for imaging, we can expect Z₀(N + 1) to be small because the excitation results in purely transverse magnetization, and recovery is then impeded by the refocusing pulses. Therefore, Δ is the product of two small numbers. To illustrate this, simulations were performed for two different sets of refocusing pulses (N = 100): (i) 180° pulses; (ii) pseudo-steady-state refocusing with echo amplitude 0.5M₀ (15), for a range of θ₀ and θ_N+2. Figure 2 displays the error caused by using Eq. [6] to compute I_SS compared with the “true” steady-state echo amplitudes estimated by full transient simulation over 10 shots. Simulations used echo spacing 4.4 ms, repetition time (TR) = 2500 ms, T₁ = 3651ms, T₂ = 1429 ms [relevant to cerebrospinal fluid at 3T, (26)]. As expected, very low error can be achieved if θ₀ = 90° with 180° refocusing pulses irrespective of θ_N+2. Similarly, as predicted by Eq. [2], θ_{N + 2} = −90° leads to zero error for either set of refocusing angles. For the more general case, as Figure 2b shows for the PSS example, a large range of flip angles lead to below 3% simulation error.

Fig. 2.

Percentage error in predicted echo amplitudes when calculated using direct estimation method (Eq. [6]) over a range of excitation (θ₀) and flip-back (θ_N+2) pulse angles. (a) 180° refocusing pulses. (b) Variable refocusing angles designed to yield a pseudosteady state with echo amplitudes of 0.5M₀. Contours are drawn for errors of 3%, 10%, and 20%.

Methods

All imaging was performed on a Philips 3T Achieva MRI system (Best, Netherlands) fitted with an eight-channel PTx body coil (2kW available peak power per channel) (27) with an eight-channel receive-only head coil used for signal reception. The body coil is configured to have a default “quadrature” mode that produces a quadrature excitation at the center of the bore when not loaded. Amplitudes and phases of each channel are thus defined relative to the quadrature mode (i.e., a relative drive of amplitude 1.0 and phase 0.0 on each channel represents quadrature mode). For brain imaging, the quadrature mode excitation is qualitatively similar in both inhomogeneity pattern and range of achieved B₁⁺ to a more standard birdcage device. More detailed data on typical B₁⁺ variation in the brain from this coil can be found in Ref. (4).

Base Sequences for T2 Weighted Imaging

The methodology outlined in the theory section is entirely general and could be used to obtain a uniform response for any base sequence. In this work, we consider T₂ weighted brain imaging examples at 3T, with two base sequences given in Table 1. Both generate 100 imaging echoes, and in both cases the angles were fixed after the sixth refocusing pulse as a pseudo-steady-state was assumed to have been reached. The first is a standard PSS sequence designed to yield echo amplitudes of 0.5M₀. Flip angles were computed with the one-ahead algorithm (15,28); the final refocusing pulse θ_N+1 was computed using Eq. [20] in Ref. (15). The second base sequence uses higher flip angles at the start in order to move toward the ideal static pseudo-steady-state [SPSS, Ref. (10)], which yields the highest possible signal for a given flip angle at the cost of producing large oscillations in the first echoes. To compensate, five dummy echoes are included to delay data readout until the signal stabilizes.

Table 1. Base Sequence Flip Angles.

	θ0	θ1	θ2	θ3	θ4	θ5	θ6...N	θN+1	θN+2
Base sequence 1	90°	90°	53.1°	43.6°	39.8°	38.3°	37.8°	71.1°	–90°
Base sequence 2	90°	137.1°	81.2°	56.4°	45.6°	40.9°	37.8°	109.4°	–90°

Both sequences generate 100 imaging echoes: N = 100 for sequence 1, but N = 105 for sequence 2 because this includes five dummy echoes. Pulses 1 – N +1 have a phase offset of 90° to fulfill the CPMG condition, which has not been explicitly written here.

Direct Signal Control Optimization

The minimization outlined in Eq. [9] was formulated with cerebrospinal fluid (CSF) as the reference tissue [T₁ = 3651ms, T₂ = 1429 ms; Ref. (26)]. CSF was chosen because it often dominates the visual appearance of T₂ brain images, and this type of imaging is frequently used to visualize fluid pathology with long T₂. Minimization used the self-organizing migrating algorithm [SOMA, (29,30)] implemented in MATLAB (MathWorks, Natick, MA). SOMA is a stochastic “leader following” method for nonlinear optimization; a population of 40 with 40 migrations was used. Although the optimization could be allowed to vary every flip angle independently, for long echo trains this leads to a substantial number of variables and long computation times. Both base sequences in this study use a constant flip angle for the majority of the scan. In keeping with this, the rows in A corresponding to pulses 6 to N were represented by a single row in the optimization (i.e., the same “RF shim” was used for pulses 6 to N). In optimizing base sequence 1, W = {1,...,N}, that is, echoes 1 to N, were included in the cost function. However, for base sequence 2, W = {6,...,N}, reflecting the presence of five dummy (discarded) echoes. Note that regardless of which echoes were included in the cost function, all pulses were included in the optimization. The cost-function weighting parameters were set empirically (using L-curve methods) as η = 0.2 and β = 10⁻³, and optimization was constrained so that no refocusing pulses exceeded a nominal flip angle of 180° on any channel (this constrains peak power because pulse durations are fixed).

In order to minimize the computation times, efficient methods for evaluating EPG calculations were devised, which are discussed in the Appendix, available online as supporting information. Because the computational load increases linearly with the number of voxels, the spatial domain Ω was decimated to consist of an array of sparse control points distributed through the 3D region of the calculation (in this case, the brain of the subject), similar to the approach used in Ref. (11). In pilot work (data not shown), it was found that tetrahedral lattices with spacing 25 mm allow the whole brain to be characterized by approximately 75 control points, with little loss in performance. Further acceleration was achieved by using the MATLAB parallel computing toolbox on a Dell Precision T5600 Workstation (Round Rock, TX) with 12 cores used simultaneously. Using all of these methods together, the DSC optimization could be performed in approximately 6 minutes.

Data Acquisition

The proposed methodology has been tested on phantoms and in vivo. In this article, we present results using both base sequences on a healthy volunteer, who gave written informed consent prior to enrolment, as well as phantom data used for validation. 3D-RF field maps were acquired using a two-stage method. First, a quantitative B₁⁺ map was obtained for the coil’s quadrature mode using the AFI sequence (31) with modified spoiling (32) nominal flip-angle 80°, TR = 30 ms/150 ms, and acquired voxel size 5 × 5 × 5 mm³ giving scan duration 4m40s. Next, a series of eight low flip-angle spoiled gradient echo (SPGR) scans was acquired using the same resolution and field of view, but with TR = 10 ms and nominal flip angle 1° (total duration 2m04s). Relative transmit sensitivity was obtained by dividing each SPGR scan by the sum of all SPGRs, and overall B₁⁺ was obtained by scaling with the quantitative B₁⁺ map. A similar method was used in Refs. (33,34). Dimensionless transmit sensitivity [σ(r)] was then obtained by dividing measured B₁⁺by the nominal value. Sensitivity maps were actually obtained for linear combinations of the transmit channels (35), which were inverted to obtain per channel maps. The brain was automatically segmented by applying the FSL brain-extraction tool to the SPGR image data; this was used as a mask for the optimizations.

3D-FSE images were obtained using both base sequences with the quadrature mode of the coil as well as DSC dynamic shimming. All sequences used TR = 2500 ms between shots, echo spacing 4.4 ms, acquired resolution 1 × 1 × 1mm³ with frequency encoding in the head-foot direction, and sensitivity encoding (SENSE) reduction factor of 1.7 applied in both phase-encoded directions to give total imaging duration of 5m12s for each scan. All RF pulses were nonselective. A hybrid radial-linear phase encoding scheme was used so that each shot filled an approximately radial profile in k_y–k_z, with the echo closest to the center of k-space occurring for the 50th imaging echo. Images were produced directly on the scanner using the standard reconstruction software. An alternative to DSC is to perform static RF shimming in which a single set of channel weights is determined by optimizing for a uniform RF field (as opposed to optimizing directly for uniform echo amplitudes) and applied to all pulses. This method was simulated using the acquired RF sensitivity maps; shimming was performed using magnitude least-squares optimization (22) with varying regularization (equivalent to the third term in Eq. [9]).

Echo amplitudes resulting from the optimized and quadrature-mode flip angles were predicted using a forward model for the entire 3D domain rather than just the decimated grid used for optimization. Although optimization was performed using the direct steady-state method (Eq. [6]), the solutions were subsequently evaluated using full transient simulations.

A validation experiment was performed with the same 3D FSE sequences using a 10 ml sample tube filled with approximately 0.01 mM concentration of MnCl₂ solution. Frequency encoding was aligned with the long axis of the tube; and phase encoding was switched off, allowing direct measurement of echo amplitudes. T₁ was measured with multiple inversion recovery experiments as 2292 ms. T₂ = 613ms and transmit sensitivity σ = 0.95 were determined by fitting calibration data using fixed 180° and 90° refocusing pulses to an EPG model. Note that σ ≠ 1 because of a lack of precision in the scanner’s in-built power scaling step.

Specific Absorption Rate Control and RF Safety

The scanner is treated as a normal system when in quadrature mode, and reliable SAR estimates can be obtained.

The 3D FSE sequences used in this work have intrinsically low SAR because of their long TRs and low refocusing flip angles. The quadrature mode generic SAR model for an adult male predicted the average head SAR to be 0.16 W/ kg and 0.19 W/kg, respectively, for the two base sequences. SAR for the DSC dynamic shimming sequences was estimated by scaling these values with the relative RF power, calculated from ||A||²_F (c.f., Eq. [9]). This estimate does not account for interference between electric fields that may alter the true local SAR. In line with locally adopted rules, a factor of 10 safety margin was included; the relevant regulatory limit is 3.2 W/kg for head scanning (36), and scans were thus limited to a predicted SAR of 0.32 W/kg. Preliminary modeling (unpublished data) confirms that this approach respects all relevant regulatory limits and is conservative from a safety point of view.

Results

Figure 3 presents the results of the validation experiment with data showing recorded signals for 10 echo trains, the first starting from thermal equilibrium. The “full transient simulation” of 10 cycles of the FSE sequence converges to the same steady state predicted by the direct simulation method. There also is very good agreement between these simulations and the experimental data for both base sequences; the RMS error is 0.18% for sequence 1 and 0.26% for sequence 2. This excellent agreement suggests that assumptions made in modeling the sequence were reasonable. Note that the experimental data were not fitted to the predictions; rather, the global scaling constant was estimated from a calibration scan. Very small differences can most likely be attributed to diffusion effects, which are not included in the simulations (16).

Fig. 3. Results of phantom experiments comparing full transient simulation, direct computation of steady state, and experiment.

Figure 4 shows in vivo images as reconstructed by the scanner in the native sagittal plane, all windowed in the same way. All images are free from ghosting artifacts that might indicate unstable echo signals. Both sequences in quadrature mode suffer from B₁ inhomogeneity with low signal at the periphery of the brain; this is particularly true for sequence 1. Both images acquired with DSC appear visibly more uniform. Figure 5 shows ratios between images depicted in Figure 4; the images are masked to show brain only and were registered together using the image registration toolkit (37) prior to analysis. Ratios are examined in order to eliminate the effect of the unknown absolute receiver sensitivity profile. Figure 6 is a combination of Figures 4 and 5 for a coronal section through the same reconstructed images. The coronal images show that, in addition to low signal in the periphery, in quadrature mode the shading effect is left–right asymmetric for both base sequences. The ratio images are all clearly tissue-dependent, with CSF and soft brain tissue changing in different ways. Figures 5a and 6e show the ratios of the two quadrature mode images: Sequence 1 produces much lower signals in the periphery of the brain, and a left–right asymmetry is visible on the coronal section (sequence 1 is more asymmetric than sequence 2, hence the ratio itself is also asymmetric). Although large signal differences exist in the CSF (ratios > 1.5), the soft brain tissue also shows spatial signal differences, with the ratio sequence 2/ sequence 1 approximately 1.3 in peripheral areas. After dynamic shimming, the two sequences produce visibly more uniform signal levels through the brain. Examining the ratios of the dynamic shimmed and quadrature images for each base sequence, we see that the effect is stronger for sequence 1 (Fig. 5c; Fig. 6g)—in which soft tissue signals are boosted by 30% to 40% in peripheral regions after optimization and signals are reduced in the center by up to 20%. The effect is similar but less pronounced for sequence 2 (Fig. 5d; Fig. 6h). Here CSF and soft tissues behave slightly differently—CSF signals are boosted in the periphery and remain the same in the center of the brain. However, soft tissue signals are around the same in the periphery and are reduced in the center of the brain. Finally, the ratios of the two shimmed images (Fig. 5b; Fig. 6f) show little spatial variation, which is consistent with each having substantially achieved the target uniform signal properties.

Fig. 4. In vivo images acquired using both base sequences (see Table 1) with quadrature excitation and Direct Signal Control dynamic shimming.

Fig. 5. Ratios of acquired images (postregistration and brain extraction).

(a) Base sequence 2 (Seq.2) quadrature divided by base sequence 1 (Seq.1) quadrature. (b) Seq.2 dynamic shim divided by Seq.1 dynamic shim. (c) Seq.1 ratio of dynamic shimmed image to quadrature image. (d) Seq.2 ratio of dynamic shimmed image to quadrature image.

Fig. 6. Combined Figures 4 and 5 for coronal reformat through the same three-dimensional data.

Figure 7 shows predicted inhomogeneity, measured as normalized root mean square error (NRMSE) with respect to the ideal (σ= 1) echo amplitudes from full transient simulations for DSC, quadrature, and static RF shimming. The static shimming optimizations were performed with multiple regularization settings penalizing RF power. Inhomogeneity is plotted against predicted head SAR and also maximum flip angle; shaded areas on the graphs show solutions forbidden either because SAR exceeds 0.32W/kg or the maximum nominal flip angle exceeds 180° (peak power constraint). The DSC solutions contain significantly reduced error with little SAR increase, which cannot be accomplished with static RF shimming.

Fig. 7. Simulated static shim experiments alongside quadrature and direct signal control.

Normalized root mean square error (NRMSE) values are computed from the difference between a full transient simulation of the indicated solution and the ideal response of the base sequence. Solutions exceeding the SAR/peak power constraints are shaded. (a,c) Predicted SAR versus NRMSE; SAR is not allowed to exceed 0.32 W/kg (10% of the regulatory limit) (b,d) Maximum nominal flip angle versus NRMSE; regions with flip angle exceeding 180° violate the peak power constraint. Base sequence 2 contains higher flip angles at the start; thus, it is more restricted by both the maximum flip angle and SAR constraints.

Figure 8 shows the optimized flip angles and phases for each channel for the two base sequences, along with transmit sensitivities in an axial plane given below (axial planes best reflect the sensitivity variation of the transmit coil used). It is also possible to represent transmit sensitivities in a Fourier basis [e.g., Ref. (38)] where the modes are labeled by integer j such that a phase increment of j × c × 45° is added to channel c. Mode 0 corresponds to the quadrature mode of the coil. The flip angles on each channel can also be represented in this domain, such that the overall flip angle is the sum of the Fourier mode sensitivities weighted by the Fourier domain “flip angles.” In this representation, the base sequences in quadrature mode would have nonzero flip angles only for mode 0. Contributions from non-zero modes are an indicator of the contribution of PTx to the solution because these are not accessible without parallel transmission.

Fig. 8.

(a)-(d) Direct signal control optimized pulse amplitudes and phases (A_ij) for both base sequences. The pulses are numbered from 0 to N + 2 where 0 corresponds to the excitation and N + 2 corresponds to the flip-back pulse. Note the approximately 90° phase difference between excitation and refocusing pulses on (c)-(d) required to maintain the CPMG condition. This is allowed to vary because echo instability is modeled by the spatially resolved extended phase graphs and penalized in the optimization. (e)-(f) Flip angles corresponding to a Fourier domain representation; flip angles for modes other than quadrature mode (k = 0) are multiplied by 2 to aid visibility. (g) Transmit sensitivity of each channel [σ(r) is a dimensionless quantity]. (h) Transmit sensitivity of each Fourier mode (see results section for more detail). The total flip angle for each pulse is the sum over the channel sensitivities weighted by that channel’s (complex) nominal flip angle, or equivalently, the sum over the Fourier modes weighted by that mode’s (complex) nominal flip angle.

Finally, Figure 9a shows simulated echo amplitudes for the center of k-space echo of each sequence for a midsagittal plane. Features such as the peripheral image shading and higher signal from base sequence 2 agree qualitatively with Figure 4 and Figure 5a. As illustrated by Figure 2, the direct steady-state simulation method is not exact when θ_N+2≠–90°, which can be the case for the optimized sequences or in the quadrature mode if RF inhomogeneity is present. Figure 9b shows the error compared with the full transient simulation, assumed to be the gold standard. Figure 9c illustrates the proportion of voxels in the 3D domain (the whole brain, not just the slice shown) for which this error is less than a given percentage. The fraction of voxels with error less than 3% in quadrature mode is 76% for base sequence 1 and 94% for base sequence 2. This rises to 97% and 98% respectively for the dynamic RF shimmed solutions.

Fig. 9.

(a) Simulated echo amplitudes using direct method. These echo amplitudes are relevant to cerebrospinal fluid. They are not direct predictions of image signal values obtained from a wide range of brain tissues. (b) Errors caused by using direct method with respect to full transient simulation. (c) Percentage of voxels in full three-dimensional domain with simulation error below the given value. For example, for base sequence 1 in quadrature mode, only 76% of voxels have lower than 3% error, whereas 94% of voxels meet this criterion for base sequence 2 in quadrature, as do 97% and 98% for the respective base sequences after dynamic shimming.

Discussion

This article explores the use of DSC—the direct optimization of image properties via a model, rather than RF field or individual pulse properties—as a means for correcting spatially variable signal and contrast caused by RF inhomogeneity in 3D-FSE imaging, with T₂ weighted brain imaging at 3T used as an example. For the requisite signal model, SR-EPGs were extended to allow for prediction of a dynamic steady-state formed over multiple TR periods. In vivo scanning resulted in excellent image-quality matching predictions from theory.

In vivo images were acquired with two base sequences in quadrature mode and then DSC. Although both quadrature mode images display shading artifacts due to RF inhomogeneity, these are more prominent for base sequence 1. This is consistent with model predictions in Figure 9a and the error values plotted on Figure 7. The dynamically shimmed images display an obviously more uniform appearance than do their quadrature mode equivalents. Noticeable left–right and superior–inferior shading artifacts have been removed. Whereas the latter are more severe, the former are perhaps more intrusive because they break anatomical symmetry. Although ratios between shimmed and quadrature images (Fig. 5c–5d; Fig. 6g–6h) indicate that signals have changed in different ways for the two base sequences, the ratio between the two shimmed images (Fig. 5b; Fig. 6f) is much more uniform in appearance. This is consistent with both shimmed images being more spatially uniform or at least having the same degree of uniformity; and it is in keeping with Figure 7, which suggests that although the quadrature mode NRMSE values differ, the NRMSE after dynamic shimming should be approximately the same for both sequences.

However, as Figure 8 indicates, the optimized pulse amplitudes and phases have not converged to a common solution; instead the separate solutions are more in line with the intended behavior of the respective base sequences. This can be seen from the differing tissue-specific responses visible on the ratios (Fig. 5b; Fig. 6f). For example, sequence 2 produces more signal in CSF than does sequence. 1. The Fourier representation of the optimized flip angles (Fig. 8e–8f) can help to visualize this by separating the quadrature mode (mode 0), which dominates for this coil configuration as it produces the most B₁⁺ field per-unit input power, from higher order modes that add some spatial control and directly reveal the contribution from parallel transmission. The optimized solution for base sequence 1 retains the property of having the same flip angle for excitation and first refocusing pulses (mode 0 remains the same for θ₀ and θ₁), whereas for base sequence 2, θ₁ is much larger than θ₀— as with the unoptimized versions. In both sequence variants, significant contributions are made, especially from modes +1 and −1 for the first two pulses. These modes allow higher flip angles to be produced at the periphery of the brain. In contrast, static RF shimming uses only these higher-order mode contributions to attempt to correct for nonuniformity (i.e., no dynamic modulation). Figure 7 summarizes simulations of the expected performance of static shimming when compared with DSC and quadrature excitation. The DSC solutions achieve much lower NRMSE than what is achievable by static shimming within SAR and peak power limits.

However, it should be noted that the SAR values used in this work—derived from the computed values for quadrature excitation, scaled by the RF power relative to quadrature mode—are not necessarily accurate, the charts in Figures 7a and 7c are more indicative of average power. For brain imaging at 3T, we have found a generally good correspondence between average power and the head average SAR; and it is this value rather than local 10g SAR that is generally the limiting factor. Nevertheless, a factor 10 margin of error was introduced for safety evaluation purposes. For situations with larger fields of view and/or higher B₀ field strengths, this would no longer be appropriate. In those circumstances, predictions from SAR models could be used both to assess safety and also to penalize/constrain solutions by suitably modifying the cost function; such methods are actively under development in the wider field [e.g., (39–41)] and could readily be adopted into the proposed methodology when necessary.

A key aspect of the presented work is that the optimization seeks to control steady state and not transient echo amplitudes. This is particularly important for optimization of the last two pulses of the pulse train because these affect signals in subsequent shots (and hence the final steady state), but they do not contribute to the transient response of the shot in which they are played out. The proposed approach enables the effect of all pulses on the observed steady-state echo amplitudes to be evaluated, allowing optimization of all on the same terms. Figure 8 shows that the last two pulses are indeed changed from their starting values by the optimization. The proposed “direct” estimation of the steady state is an important means for performing this optimization because it allows for fast calculation. The method is based on the assumption that Z₀ at the end of the shot can be expressed as a fraction of the value present just prior to the excitation. The related assumption that Z₀ does not recover during the FSE shot used elsewhere in the literature [see Refs. (17–19)] is only truly the case for a 90° excitation pulse and 180° refocusing pulses, making it unsuitable for low refocusing angle PSS sequences of the type used. It is the addition of the flip-back pulse in the driven equilibrium variant of the sequence that makes use of the assumption more generally reasonable. Of course, RF inhomogeneity means that this condition is violated; Figure 9b shows the errors in using the direct estimation method. The largest errors occur for base sequence 1 with quadrature excitation, and in this case only 76% of the voxels in the 3D calculation have an error lower than 3%; whereas this figure is above 97% when using dynamic shimming. Although the optimization is not constrained to ensure that θ_N+2 = − 90°, the optimization has a natural tendency to keep this close to −90° so that the remaining transverse magnetization is efficiently stored longitudinally. This tends to keep simulation error low, as seen.

DSC relies on a large nonlinear optimization (Eq. [9]). The SOMA method was chosen because of prior experience of good performance; however, many optimization methods exist. Although convergence to a global optimum for a given target T(n,r) cannot be guaranteed, it should be noted that the starting point—the base sequence—is the operating point that would be used for imaging in normal practice. The present method improves on this by iteratively updating the inputs to the system, so it will always drive to a better operating point. Exploring how to ensure that improved performance will approach optimal performance remains a matter for future research. In order to make the method tractable for online optimization, many steps were taken to minimize computation time: In addition to the direct steady-state calculation just discussed, an approximate method that involves dropping large numbers of the states from the calculation was also implemented (see Appendix, which is available online as supporting information). Finally, because the SR-EPG simulation for each spatial location is effectively independent, approximately linear improvements in speed can be obtained by reducing the number of control points and by using parallel computation. Using these methods, acceptable computation times for this proof-of-concept study (approximately 6 minutes for 9 flip angles and 8 channels using 75 control points with 12-fold parallel computation) were achieved using MATLAB. Online clinical implementation would require acquisition of B₁⁺ maps before optimization. Much progress has been made by others in accelerating these mapping approaches [e.g., Ref. (42)], and computation times of a few minutes could feasibly be tolerated if run concurrently with other scanning. Further reductions could be achieved by implementing in a more efficient language and by taking advantage of highly parallel graphics processors.

Although the method proposed is general, some of the methodology used in the present study is specific to the example implementation. For example, the base sequences had only nine independent flip angles (Table 1), which meant that the optimization could naturally be performed with only nine unknowns per channel. This makes for a calculation that is tractable in a limited amount of time, and is not unreasonable for PSS sequences in which the PSS is established by the first few pulses (10). In principle though, however every pulse could be included independently in the optimization or these could be subdivided in other ways: for example, in a TRAPS sequence (23) the ramp of increasing flip angles could be scaled by a common RF shim setting—or in a “tissue-specific” modulation sequence (24,25), the different “phases” of the signal evolution curve could use different RF settings. Additionally, signals have been optimized for specific reference tissue which in this case was actually CSF. In practice, the contrast between tissues is also an important consideration. In choosing a single reference tissue with particular properties, we expect that similar tissues will behave in a similar way—CSF was selected as the reference tissue in this work so that it could be representative of signals from fluids and tissues with very long T₂. Alternatively, it would be possible to use multiple tissue types in the optimization, or to even specifically optimize the contrast between two tissues (i.e., the difference in predicted echo amplitudes), and this will be left as the subject of future work.

Conclusion

DSC via dynamic RF shimming employs interactions between multiple RF pulses to control the evolution of signals throughout a sequence. This is useful because in many circumstances a given transmit coil may not provide sufficient spatial degrees of freedom with which to produce uniform B₁⁺ fields by using static RF shimming alone. Another way of obtaining additional control is to use “tailored” RF pulses that contain some degree of gradient encoding (4,43–45). A recent study (46) used a similar approach to reduce inhomogeneity artifacts in T₂ weighted 3D-FSE brain images at 7T. This approach may be viewed as an extension of static RF shimming because it uses a fixed spatial modulation over time. In the present article, short nonselective RF pulses were used throughout. Spatially tailored pulses have the drawback of longer durations, leading to larger interecho spacings as well as complicated frequency responses. For the 3T examples in this article, dynamic RF shimming alone gives good results. However, it is possible that in other situations with more extreme inhomogeneity (e.g., 7T imaging) some degree of spatial encoding with tailored RF pulses may be desirable. The two approaches are not mutually exclusive. The concept of DSC is agnostic to the subelements of the sequences used, and the SR-EPG framework provides an overarching model to combine these effects. By directly evaluating expected echo amplitudes, we can sidestep issues such as explicit phase matching of RF pulses to maintain the CPMG condition and to constrain amplitudes of larger angle pulses while being less restrictive with those with smaller angles. This same approach could be taken for designing sequences of tailored RF pulses acting together. Interactions could then be used to give additional degrees of freedom that would allow the required spatial encoding (hence duration) of the individual pulses to be reduced.

The concept of DSC is to achieve the desired image properties by considering the pulse sequence as a whole rather than by designing the properties of individual pulses. This article has demonstrated that this approach can be used for 3D-FSE imaging with long pulse trains to produce high quality images with excellent signal uniformity despite operating with nonuniform RF fields. It does so by considering not just the action of one pulse, or even one train of pulses, but the dynamic equilibrium reached by applying a train multiple times.