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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 2009 December 17.
Published in final edited form as:
PMCID: PMC2795148

Maximum Linear-Phase Spectral-Spatial RF Pulses for Fat-Suppressed PRF-Shift MR Thermometry


Conventional spectral-spatial pulses used for water-selective excitation in proton-resonance-frequency (PRF-) shift MR thermometry require increased sequence length, compared to shorter wideband pulses. This is because spectral-spatial pulses are longer than wideband pulses, and the TE period starts midway through them. Therefore, for a fixed TE, one must increase sequence length to accommodate conventional spectral-spatial pulses in PRF-shift thermometry. We introduce improved water-selective spectral-spatial pulses for which the TE period starts near the beginning of excitation. Instead of requiring increased sequence length, these pulses extend into the long TE periods common to PRF sequences. The new pulses therefore alleviate the traditional tradeoff between sequence length and fat suppression. We experimentally demonstrate an 11% improvement in frame rate in a PRF imaging sequence, compared to conventional spectral-spatial excitation. We also introduce a novel spectral-spatial pulse design technique that is a hybrid of previous model- and filter-based techniques, and that inherits advantages from both. We experimentally validate the pulses’ performance in suppressing lipid signal, and in reducing sequence length compared to conventional spectral-spatial pulses.


Proton resonance frequency (PRF)-shift MR thermometry is an imaging method that can noninvasively monitor temperature during targeted thermal therapy. It exploits the negative resonance frequency shift of water with increasing temperature to produce maps of temperature change in tissue. Water’s frequency shift creates a phase shift in GRE images acquired during heating, so that temperature maps can be estimated via subtraction of a baseline phase image. However, because fat does not experience a temperature-induced resonance shift, it can corrupt temperature measurements in PRF-shift MR thermometry. Fat modifies the image phase and leads to erroneous temperature estimates in single-echo acquisitions [1].

One solution to this problem is to perform excitation with spectral-spatial (SPSP) RF pulses that are designed to excite water only [24]. Conventional SPSP pulses used in PRF-shift thermometry excite a linear phase ramp across the water spectrum, so that the phase of excited spins is a linear function of their frequency offset. The slope of that ramp is such that the TE period begins midway through the pulse. Long TE periods are desirable in PRF-shift thermometry because temperature sensitivity is proportional to TE. In this work, we demonstrate maximum linear-phase SPSP pulses that excite much larger phase slopes across the water spectrum, such that the TE period starts near the beginning of the pulse. Compared to conventional linear-phase SPSP pulses, for a fixed TE the signal readout can be moved closer to the end of the new pulses and the TR can be shortened, resulting in temperature maps acquired with the same sensitivity but at a higher frame rate. A high frame rate improves a sequence’s robustness to patient motion, which is especially problematic for monitoring thermal ablation in abdominal organs such as the liver. Compared to PRF-shift thermometry without spectrally-selective excitation, the new pulses allow one to exploit the long TE waiting periods common to thermometry sequences to gain spectral selectivity, without extending sequence length. The new pulses make single echo thermometry with SPSP excitation a more compelling choice over multi-echo approaches, since the frame-rate penalty of a multiple echo approach is larger compared to maximum linear-phase fat-suppressed single echo thermometry than it is compared to single echo thermometry with conventional SPSP excitation. For example, a three echo 2DFT imaging sequence [1] with 3.2ms readouts at 3 Tesla would require an (approximately) 18ms TR if the central echo were acquired with a 12ms TE, while a single echo sequence with maximum linear-phase SPSP excitation would require a TR of approximately 14ms, if one assumes that incorporation of the maximum linear-phase pulse does not increase the minimum TR. In this case, the shorter TR with a single echo readout corresponds to a 29% higher frame rate.

We also introduce a new method for designing slice-selective SPSP pulses that is ideally suited to maximum linear-phase SPSP pulse design. Conventionally, SPSP pulses are constructed in a separable manner, by designing slice-selective and spectrally-selective pulses independently and combining them [2, 5]. One begins with a long pulse that satisfies the desired spectral excitation characteristics, designed using, e.g., filter-based techniques such as the Shinnar-Le-Roux algorithm [6, 7]. A short slice-selective pulse is then produced according to the desired slice profile. That pulse is then replicated to form a pulse train whose total duration equals that of the spectral pulse, and each sub-pulse of the train is weighted by a complex value determined by its temporal location in the spectral pulse. This approach implicitly assumes that movements through spectral excitation k-space occur instantaneously between slice-selective pulses, which constitutes a model error. In reality, spectral k-space traversal is constant and continuous throughout the pulse, resulting in N/2 sidelobes and increasing excitation error away from the center frequency [2, 8] that must be corrected after designing the pulse. The separable approach also does not currently offer the flexibility we desire in our application to set the TE period starting time arbitrarily. In comparison, iterative pulse design techniques [911] do allow the start of the TE period to be flexibly tuned. Another advantage to these techniques is that they are based on an accurate continuous model of spectral k-space traversal, so post-corrections are not required. They also allow the incorporation of ‘don’t care’ regions to improve excitation accuracy in the water and fat bands. However, current iterative pulse design techniques do not allow one to make the intuitive design tradeoffs between pulse duration, pass- and stopband ripple, and transition widths that are afforded by filter-based techniques. Here we introduce a hybrid method for designing small-tip-angle SPSP pulses that combines iterative and filter-based techniques synergistically. Our method will inherit the advantages of iterative pulse design while simultaneously allowing the pulse designer to make intuitive design tradeoffs.


Pulse Design

The iterative multidimensional pulse design method of Ref. [9] designs small-tip-angle pulses by solving the weighted least-squares (WLS) minimization problem


where A is a non-uniform Discrete Fourier Transform matrix, b is a vector of Nt RF pulse samples, and d is a vector containing Ns samples of the desired excitation pattern. We use the diagonal spatial error weighting matrix W to specify ‘don’t care’ regions and to differentially weight pass- and stop-bands to achieve desired ripple levels in the slice (z-) dimension. The elements of A are given by:


where γ is the proton gyromagnetic ratio, Δt is the RF sampling period, and f is the frequency offset. The spatial excitation k-space trajectory kz (t) is defined as the time-reversed integral of the gradient waveform [12]:


where T is the pulse duration, and the spectral k-space trajectory kf (t) is time [2]:


Equation 1 can be solved using, e.g., the Conjugate Gradient (CG) method [9, 13, 14], to obtain the solution:


In the following subsections, we provide details on forming the inputs to the design problem. The steps of our algorithm are summarized in Table 1.

Table 1
The maximum-linear phase spectral-spatial pulse design algorithm.

Desired Excitation Pattern and Error Weighting

We construct the desired excitation pattern d and the error weighting matrix W according to the goals of pulse design. These are: (i) select a slice in the z-dimension, (ii) excite a range of water frequencies corresponding to the expected temperature range, (iii) excite a specified linear phase ramp across the water spectrum, and (iv) suppress the lipid signal. We also consider a conservative shim range. We specify the desired pattern on a discrete, oversampled Nz × Nf z-f grid at locations that are dependent on the spatial and spectral k-space trajectories kz (t) and kf (t).

Slice Dimension

Within the water band, the pulse design formulation of (1) corresponds to a weighted least-squares (WLS) finite impulse response (FIR) filter design problem [6, 7, 15] in the z-dimension. Given a desired slice thickness Δz, desired pass- and stop-band maximum ripples (δp and δs, respectively), and a desired time-bandwidth product (TB), we can tailor the weighting matrix W to achieve these characteristics. The fractional transition width W, which is the ratio of the width of the transition from pass- to stop-band to the total width of the pass- and stop-bands, is given by (cf [7], Eqs. (20, 21)):


The function d(δp, δs) is an empirically-derived polynomial performance measure for min-max FIR filters [16], and is a good approximation for WLS-designed FIR filters [17]. We can then calculate the location of the in- and out-of-slice band edges as:



The pass- and stop-band error weights (wp and ws) are:



With these values defined, the spatial components of d and W in the water band are given by:



Spectral Dimension

Within the water band, the pulse will excite a linear phase ramp that can be tuned according to the desired TE starting point. If we define Ttd (i.e., ‘Time-to-Tip-Down’) to be the TE period starting time measured from the beginning of the pulse, the phase of the desired pattern in the spectral dimension must be:


A conventional ‘linear-phase’ pulse design is characterized by a Ttd that is half the duration of the pulse; a ‘maximum linear-phase’ pulse has a shorter Ttd, so that the TE period starts earlier in the pulse. The boundaries of the water band will be set by the maximum expected positive and negative temperature deviations (ΔT+ and ΔT, respectively), and the expected main field shim range ±δshim. The spectral offsets corresponding to ΔT+ and ΔT are calculated as:



where α = −0.01 ppm/°C is the proton resonance frequency shift with temperature. Combining these parameters with the fact that we want to excite water uniformly and suppress fat, we can define the spectral components of d as:


We can specify the spectral components of W by additionally recognizing that the fat spectrum is between ff±=γ2πB0(3.5×106±δshim):


Dimensions Combined

Combining (11,12) with (16,17) gives the combined desired pattern and weighting matrix elements:



Note that regions outside the water and fat bands are ‘don’t care’ regions in the pulse design, and are given zero error weight.

Parameter Relations for Gradient Design

The gradient waveform design is informed by TB, Δz, and for unaliased pulse designs, the necessary spectral sampling period ΔTf. We first consider ΔTf, which by Nyquist theory should be no larger than the difference between the largest water frequency and the smallest fat frequency:


where FOVf is the spectral FOV of the desired excitation pattern. In this work, we use unipolar flyback gradient trajectories due to their robustness to flow, motion and eddy currents [18], though using bipolar trajectories may improve excitation accuracy for applications with little motion and in which the gradients can be calibrated. The positive and negative flyback gradient lobes are designed so that the duration of one lobe pair is equal to ΔTf, and their area is prescribed by the desired TB and slice thickness in z:


To reduce RF magnitude, the positive gradient lobes are designed to be as long as possible within the ΔTf window. The number of lobes must be at least 2, but may be extended to the start of signal readout without incurring a temporal penalty. The quality of fat suppression will improve with the number of lobes (i.e., the peak fat ripple will decrease), since a longer pulse covers more spectral k-space.

Once the gradient is designed, an appropriate FOV for the design grid in z (FOVz) can be determined as:


where G is the mean value of the positive gradient lobe. We set Nz, the size of the design grid in the z-dimension, equal to the number of time points in the positive gradient lobe, multiplied by an integer oversampling factor. The design grid size in the f-dimension, Nf, is determined by the bandwidth of the k-space trajectory in the spectral dimension, T−1. We set it equal to [left ceiling]FOVf · T[right ceiling], multiplied by the same oversampling factor.

Pulse Design Parameters and Validation

We designed two maximum linear-phase SPSP pulses for 1.5 and 3 Tesla. The slice-selection parameters of the pulses were identical, with TB = 4, slice thickness Δz = 1cm, and ripple levels δp = δs = 0.01. The main field (B0) was assumed to be shimmed within a ±0.4 ppm range. The expected temperature deviations were set to values representative of those expected in high-intensity focused ultrasound therapies, ΔT+ = 70 − 36.8 = 33.2°C and ΔT = 0 [19]. The design grids were oversampled by a factor of 8, and the desired patterns were scaled to unit amplitude. The RF and gradient sampling period was 4 μs. We chose to use 8 gradient lobes with ΔTf = 1.5 ms in the 1.5T pulse, resulting in a total pulse duration of T = 11.86 ms with Nt = 2964, and 7 gradient lobes with ΔTf = 1.02 ms in the 3T pulse, resulting in T = 6.97 ms with Nt = 1742. Note that the total pulse durations are slightly less than the number of lobes times the lobe duration, since half the refocusing area is needed in the final refocusing lobe. The CG algorithm was run for [left ceiling]Nt/5[right ceiling] iterations. We used a non-uniform Fast Fourier Transform algorithm [20] to compute products with the matrix A, which only evaluated RF time points corresponding to the positive gradient lobes. After the design, zeros were inserted into the pulses for time points corresponding to the negative gradient lobes, and the resulting pulses were scaled to excite 30° flip angles. To ensure acceptable gradient heating, we tested the gradient waveforms with vendor-supplied simulation tools for our scanners.

To inform our choices of Ttd, we repeated 1.5T pulse designs over a range of Ttd values to investigate this parameter’s influence on pulse characteristics. We found that as one moves from a linear-phase SPSP pulse design to a maximum linear-phase design (i.e., Ttd is set less than half the pulse width), the pulse’s peak RF magnitude increases, as well as its SAR relative to a linear-phase design. This is shown in Fig. 1(a,b) for the 1.5T pulse. In a linear-phase design, the flip angles induced by each sub-pulse generally add coherently in the passband, but incoherently in the stopband, while in a maximum linear-phase design, both pass- and stopband spins are excited early in the pulse (note the high RF magnitude early in the pulse in Fig. 2) so that phase can accrue in the passband, and subsequent sub-pulses act to flip spins in the stopband back to equilibrium but leave the passband largely unperturbed. Therefore, in general the sub-pulses of a maximum linear-phase pulse add less coherently than in a linear-phase pulse, requiring more RF power to reach the same final flip angle. Figure 1c shows that the Normalized RMS Excitation Error (NRMSE), defined as

Figure 1
Dependence of (a) peak B1, (b) SAR relative to a linear-phase pulse, and (c) normalized RMS excitation error (NRMSE) on the TE period starting time Ttd. The metrics generally increase when Ttd is shorter or longer than half the pulse’s duration. ...
Figure 2
Maximum linear-phase SPSP pulse for 1.5T. The peak RF magnitude of 0.075 G is reached near the beginning of the pulse. This reflects the early point in time at which the TE period starts, which is indicated by the dashed vertical line at 1.13 ms.


also increases with shorter Ttd. However, there is a large range of Ttd over which these metrics remain within acceptable limits. Furthermore, as shown in Fig. 1(d,e), no significant deviation from the desired phase profile in the water band is seen at even the extreme values of Ttd. Instead, the higher NRMSE for extreme values of Ttd is due to larger flip angle ripples in the water and fat bands. While a modest ripple in the water slice is not a significant issue for PRF-shift thermometry, the quality of fat suppression is degraded somewhat for extreme Ttd values, and Fig. 1f shows that for Ttd = 0 ms, the peak fat flip angle was 1.04° across all z for a 30° water excitation, while at Ttd = 6ms, the peak fat angle was a lower 0.17°. To summarize, these results indicate that the choice of Ttd should be guided by peak RF and SAR constraints, as well as the desired quality of fat suppression, but that in the context of PRF-shift thermometry, the quality of excitation in the water band is not significantly influenced by Ttd. In both our designs, we chose to start the TE period 3/4 of the way through the first positive/negative gradient pair. This resulted in Ttd = 1.13 ms for the 1.5T pulse and Ttd = 0.77 ms for the 3T pulse.

Figure 2 plots the 1.5T pulse and gradient waveforms. Note that the peak RF magnitude is reached near the beginning of the pulse. The peak magnitude of the pulse is 0.075 G. The SAR of this pulse (measured as the integrated squared magnitude) is only two times that of a 30° sinc pulse with TB = 4 and the same peak magnitude. The 3T pulse had a higher peak magnitude of 0.17 G due to the lower bound on ΔTf at 3T, which results in larger gradient magnitude and a faster traversal of kz-space. Its SAR was 1.17 times that of a 30° sinc pulse with the same peak magnitude.

In our first scanner experiment we imaged the 1.5T pulse’s excitation profile to validate its excitation pattern. Imaging was performed on a 1.5T GE MR Scanner (GE Healthcare, Waukesha, WI) in a 1% copper sulfate phantom. The profile was acquired using a spin-echo sequence in which the SPSP pulse’s z-gradient is played out on Gx, and a constant Gy pulse is used to simulate the spectral dimension [2]. The spin-echo was timed to image the profile at the end of the pulse. Prior to acquisition with our pulse, an image was acquired using a hard pulse of the same flip angle and TE starting time, so that transmit/receive and other sources of image non-uniformity could be divided out of the SPSP pulse’s profile image. We also performed Bloch equation simulations to compare to our imaging results.

The magnitude and phase of the 1.5T pulse’s excitation profile are shown in Fig. 3. The simulated magnitude patterns were normalized by sin (π/6), i.e., the nominal transverse magnetization magnitude assuming unit equilibrium magnetization. The magnitude images show no excitation in the fat band, a smooth transition in the ‘don’t care’ band between fat and water, and a 1 cm slice in the water band. They show a uniform FWHM in the water band, and no unwanted sidelobe excitation in the fat band. The FWHM of both the experimental and simulated average water magnitude profiles was 0.97 cm. Via a linear fit of Eq. (13) to the excited phase in the water band, we measured a Ttd of 1.04 ms in the experimental data, which is in good agreement with the desired Ttd of 1.13 ms. The excited magnitudes in the water band are also near 1, indicating that the desired 30° flip angle was achieved. Figure 4 plots the excitation profiles at f = 0 Hz and z = 0 cm. The slice and spectral profiles are in good agreement between simulation and experiment. The desired in- and out-of-slice ripple bounds are indicated on the simulated slice profile. At f = 0, the out-of-slice ripple was below the ripple bound for all z. The in-slice ripple almost entirely stayed within ripple bounds, but fell slightly below the lower bound at the very edge of the profile. In units of normalized flip angle (the same units as the desired pattern d), the peak simulated out-of-slice ripple across the water band was 0.02, while the peak simulated fat excitation was 0.015. Across water frequencies, the in-slice excitation was between 1.04 and 0.95. The spectral phase profiles do not significantly deviate from the desired profile (Eq. (13)).

Figure 3
Simulated (a,c) and experimental (b,d) magnitude and phase of the 1.5T pulse’s excitation pattern. The pulse produces no excitation in the fat band, and a 1 cm slice in water band. From the experimental data, we measured Ttd = 1.04 ms via a linear ...
Figure 4
(a–c) Simulated and (d–f) experimental excitation profiles. (a,d) Slice profiles at f = 0 Hz, with pass- and stop-band ripple bounds indicated in the simulated profile (dashed lines). Spectral magnitude (b,e) and phase (c,f) profiles at ...

Temperature Imaging

Fat-suppressed thermometry

Our second experiment was designed to verify the accuracy of fat-suppressed PRF-shift thermometry at 1.5T using the pulse described in the previous Section for excitation. We chose to perform thermometry in whipping cream due to its high fat content. A vial (2.5-cm diameter) of whipping cream (Lucerne Foods, Pleasanton, CA, USA) was heated to 50°C in a microwave and was transferred to a bath of water to cool during imaging. Imaging was then performed every minute using a 2DFT SPGR sequence with parameters: TR = 50 ms, FOV = 28 cm, resolution = 256 × 256. Data was acquired in two separate dual-echo imaging runs, one using sinc excitation and one using our SPSP pulse. The TE’s were 21 and 32.2 ms, corresponding to a π fat/water phase shift between the two echoes. An additional room-temperature phantom was placed next to the water bath and was used as a phase reference. Temperature mapping was again performed using baseline subtraction, but with the phase of the last image used as the reference. Absolute temperature was simultaneously recorded using a fiber-optic temperature probe (Luxtron Model 790; Luxtron Corporation, Santa Clara, CA). Figure 5 shows an image acquired before heating using the maximum linear-phase pulse for excitation. It shows the locations of the warm vial, probe, reference phantom, cool water bottle, and the guide tube that was used to ensure consistent placement of the probe between imaging runs.

Figure 5
Image of the experimental setup prior to heating, using the maximum linear-phase pulse for excitation.

Figure 6 plots the measured temperature change curves during the cooling experiment, in a voxel adjacent to the temperature probe tip. Performing excitation with a (wideband) sinc pulse resulted in highly inaccurate PRF temperature measurements. The measurements were also strongly dependent on TE, particularly at higher temperatures. In comparison, the curves measured using maximum linear-phase excitation track the probe’s curve very closely. The measurement Normalized RMS errors (NRMSE) for sinc excitation were 0.211 (TE = 21ms) and 0.441 (TE = 32.2ms), while the errors for SPSP excitation were 0.066 (TE = 21ms) and 0.053 (TE = 32.2ms).

Figure 6
Temperature during cooling using (a) sinc and (b) maximum linear-phase excitation, measured at the tip of the fiber-optic probe (indicated by the ‘Probe’ arrow in Fig. 5). Fat and water nominally accrue π phase difference between ...

Dynamic heating

In this experiment we validated the frame-rate enhancement afforded by the 3T pulse described in the ‘Pulse Design Parameters and Validation’ section. Using that pulse and a second, linear-phase SPSP pulse that was also designed with our method, we imaged a phantom during ultrasound heating using a 2DFT SPGR sequence. Imaging was performed on a 3.0T GE Signa Excite MR Scanner (GE Healthcare, Waukesha, WI), and heating was performed using an InSightec ExAblate 2000 HIFU system (InSighted Ltd., Tirat Carmel, Israel). Images were acquired during two separate sonications, one for each pulse. Sonication was performed on a gel-based QA phantom that was immersed in a bath of distilled water. A vial of peanut oil was positioned adjacent to the phantom to validate fat suppression. Sonication parameters were: duration = 50 seconds, frequency = 0.5 MHz, acoustic power = 40 W. The difference in Ttd was 2.5 ms between the two pulses. Imaging parameters were: FOV = 20 cm, resolution = 256 (frequency) × 128 (phase), TE = 12 ms, TR = 23 ms (linear-phase), TR = 20.5 ms (maximum linear-phase). Temperature mapping was performed using baseline subtraction [21], with the phase of the first image before heating as a reference.

Figure 7 shows a magnitude image acquired prior to heating, as well as a phase image acquired during heating, using the maximum linear-phase 3T pulse for excitation. The location of the hot spot is indicated on both images, and is visible in the phase image. The signal void corresponding to the vial of oil is also indicated. Figure 8 plots the phase change in the central voxel of the hot spot for both pulses. The curves match very well, indicating equivalent temperature sensitivity despite the shorter duration of the maximum linear-phase sequence. The maximum temperature change recorded was 18.8°C for the maximum linear-phase pulse, and 19°C for the linear-phase pulse. In this imaging scenario, the frame rate using the linear-phase pulse was 2.94 seconds per image, while the frame rate using the maximum linear-phase pulse was 2.62 seconds per image. Thus, the maximum linear-phase pulse improved the frame rate by 11%, with the same TE.

Figure 7
Magnitude image acquired prior to heating and a phase image acquired during heating, using the maximum linear-phase pulse for excitation. The black arrow indicates the location of the hot spot, which is visible in the phase image. The white arrow indicates ...
Figure 8
Phase change relative to baseline in the central voxel of the hot spot during heating. The phase differences match very well between the two pulses, indicating equivalent temperature sensitivity despite the shorter duration of the maximum linear-phase ...

Discussion and Conclusions

In this work, we have introduced maximum linear-phase spectral-spatial excitation pulses for fat suppression in PRF-shift thermometry. We used the term maximum linear-phase to loosely denote a pulse that excites the largest possible phase slope across the spectral dimension given a spectral sampling scheme and constraints on peak B1, SAR, and the quality of fat suppression. Our pulses are ideally suited to PRF-shift thermometry, in that they exploit the long, unused TE waiting periods common to PRF sequences to perform fat suppression. They therefore provide fat suppression without increasing sequence length, compared to wideband excitation. We validated experimentally that the pulses suppress fat signal. We also showed experimentally that they provide temperature sensitivity equivalent to linear-phase spectral-spatial excitation, while improving the PRF-shift imaging frame rate. There may be other applications for maximum linear-phase SPSP pulses, for example in fast spin echo imaging [22] and spectroscopy [23].

We also introduced a spectral-spatial pulse design method that is a hybrid of model-and filter-based techniques [7, 9], and draws advantages from both. The method allows the pulse designer to tune the TE starting time arbitrarily, an ability afforded by the method’s model-based characteristics. It also inherently compensates for excitation k-space trajectory nonuniformity, which improves the slice profile across the spectral dimension without post-correction [5, 8]. Though not explored here, the method could also incorporate peak B1 constraints via Lagrange multipliers, or an integrated pulse power penalty via Tikhonov regularization [9]. The construction of the desired excitation pattern and error weights, and the design of the kz trajectory, are informed by filter-based pulse design techniques. This allows the user to intuitively specify pulse characteristics in the slice-select dimension, such as time-bandwidth and ripple levels, and ensures that an acceptably accurate pulse will be produced. This obviates the ‘guesswork’ of manually iterating on the desired pattern construction and trajectory design until an accurate pulse is reached, and effectively jointly optimizes the two design inputs.

We have found that the earlier the TE period starts in the pulse, i.e. the smaller Ttd is, the larger the peak B1 magnitude and SAR become. Excitation error, which we showed primarily manifests as degraded fat suppression, may also increase to an unacceptable level if the TE period starts too early in the pulse. These factors may limit the benefit of using maximum linear-phase pulses, particularly at high field strengths or when a large flip angle is desired. However, it is likely that a smaller Ttd than that corresponding to a linear-phase SPSP pulse can always be used, since those pulses do not typically operate at RF power and SAR limits. In the scenarios we investigated, we found that peak B1 and excitation error can be kept to reasonable levels when the TE period is set to start at about the same time it would for sinc excitation (e.g., 0.8–1.6 ms into the pulse). Furthermore, even if a user wishes to maintain the peak B1 of a linear-phase pulse in their sequence, our design technique could still be useful since it would allow them to improve fat suppression by extending a pulse’s duration to the beginning of signal readout, with the same Ttd as the conventional pulse.



This work was supported by NIH grants R01 CA111981, RO1 CA121163, P01 CA067165 and U41 RR019703.

List of Symbols

proton resonance frequency shift with temperature
kz (t)
z-dimension excitation k-space trajectory waveform
kf (t)
f-dimension excitation k-space trajectory waveform
duration of RF pulse
number of time points in the RF pulses
sampling period of pulses
gyromagnetic ratio
Size of z-dimension of design grid
Size of f-dimension of design grid
imaginary number, also spatial index
vector of RF pulse samples
Fourier system matrix
elements of the system matrix A
vector of samples of desired pattern
diagonal matrix specifying Region of Interest (ROI) for pulse design
s), peak pass and stop-band ripple amplitudes
slice thickness
time-bandwidth product
fractional transition width in z dimension
optimal FIR filter performance measure
s), in- and out-of-slice edges
s), in- and out-of-slice error weights
time to tip-down
ΔT+), lower/upper temperature change bounds
shim bound
( fw,fw+)
lower/upper boundaries of water band
lower/upper boundaries of fat band
spectral sampling period
spectral FOV of the spectral-spatial trajectory
FOV of the design grid in the z-dimension
mean value of the positive gradient lobe


Second submission to Magnetic Resonance in Medicine for consideration of publication as a full paper


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