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- Abstract
- Introduction
- Theory
- Pulse Design Parameters and Validation
- Temperature Imaging
- Discussion and Conclusions
- References

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Magn Reson Med. Author manuscript; available in PMC 2009 December 17.

Published in final edited form as:

PMCID: PMC2795148

NIHMSID: NIHMS155512

William A Grissom,^{1,}^{2} Adam B Kerr,^{1} Andrew B Holbrook,^{3} John M Pauly,^{1} and Kim Butts-Pauly^{2}

Corresponding author: William Grissom, 203 Packard Building, 350 Serra Mall, Stanford, CA 94305, Email: ude.drofnats@mossirgw

The publisher's final edited version of this article is available at Magn Reson Med

See other articles in PMC that cite the published article.

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 [2–4]. 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 [9–11] 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.

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

$$\widehat{\mathit{b}}=\underset{\mathit{b}}{\text{argmin}}\left\{\frac{1}{2}{\left|\right|\mathit{Ab}-\mathit{d}\left|\right|}_{\mathit{W}}^{2}\right\}=\underset{\mathit{b}}{\text{argmin}}\left\{\frac{1}{2}(\mathit{Ab}-\mathit{d}{)}^{\prime}\mathit{W}(\mathit{Ab}-\mathit{d})\right\},$$

(1)

where ** A** is a non-uniform Discrete Fourier Transform matrix,

$${a}_{ij}=i\gamma \mathrm{\Delta}t{e}^{i({z}_{i}{k}_{z}({t}_{j})+{f}_{i}{k}_{f}({t}_{j}))}\begin{array}{l}i=1,\dots ,{N}_{s}\hfill \\ j=1,\dots ,{N}_{t},\hfill \end{array}$$

(2)

where *γ* is the proton gyromagnetic ratio, Δ*t* is the RF sampling period, and *f* is the frequency offset. The spatial excitation k-space trajectory *k _{z}* (

$${k}_{z}(t)-\gamma {\int}_{t}^{T}{G}_{z}({t}^{\prime})\phantom{\rule{0.16667em}{0ex}}d{t}^{\prime},$$

(3)

where *T* is the pulse duration, and the spectral k-space trajectory *k _{f}* (

$${k}_{f}(t)2\pi (t-T).$$

(4)

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

$$\widehat{\mathit{b}}={({\mathit{A}}^{\prime}\mathit{WA})}^{-1}{\mathit{A}}^{\prime}\mathit{Wd}.$$

(5)

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.

We construct the desired excitation pattern ** d** and the error weighting matrix

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 Δ*z*, desired pass- and stop-band maximum ripples (*δ _{p}* and

$$W={d}_{\infty}({\delta}_{p},{\delta}_{s})\xb7{(TB)}^{-1}.$$

(6)

The function *d*_{∞}(*δ _{p}, δ_{s}*) 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:

$${z}_{p}=\frac{1}{2}\mathrm{\Delta}z(1-W)$$

(7)

$${z}_{s}=\frac{1}{2}\mathrm{\Delta}z(1+W).$$

(8)

The pass- and stop-band error weights (*w _{p}* and

$${w}_{p}=1$$

(9)

$${w}_{s}={\delta}_{p}/{\delta}_{s}.$$

(10)

With these values defined, the spatial components of ** d** and

$${d}_{i}^{z,w}=\{\begin{array}{ll}1\hfill & {z}_{i}\phantom{\rule{0.16667em}{0ex}}\le {z}_{p}0\hfill & o.w.\hfill \hfill \end{array}$$

(11)

$${w}_{ii}^{z,w}=\{\begin{array}{ll}{w}_{p}\hfill & {z}_{i}\phantom{\rule{0.16667em}{0ex}}\le {z}_{p}{w}_{s}\hfill & {z}_{i}\phantom{\rule{0.16667em}{0ex}}\ge {z}_{s}0\hfill & o.w.\hfill \hfill \hfill \end{array}$$

(12)

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 *T _{td}* (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:

$$\mathrm{(f)=-2\pi f\xb7(T-{T}_{td}).}$$

(13)

A conventional ‘linear-phase’ pulse design is characterized by a *T _{td}* that is half the duration of the pulse; a ‘maximum linear-phase’ pulse has a shorter

$${f}_{w}^{+}=\frac{\gamma}{2\pi}{B}_{0}(\alpha \mathrm{\Delta}{T}^{-}+{\delta}_{\mathit{shim}})$$

(14)

$${f}_{w}^{-}=\frac{\gamma}{2\pi}{B}_{0}(\alpha \mathrm{\Delta}{T}^{+}-{\delta}_{\mathit{shim}}),$$

(15)

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:

$${d}_{i}^{f}=\{\begin{array}{l}{e}^{i\mathrm{({f}_{i})}}{f}_{w}^{-}\le {f}_{i}\le {f}_{w}^{+}\hfill \hfill & 0\hfill & o.w.\hfill \end{array}$$

(16)

We can specify the spectral components of ** W** by additionally recognizing that the fat spectrum is between
${f}_{f}^{\pm}={\scriptstyle \frac{\gamma}{2\pi}}{B}_{0}\phantom{\rule{0.16667em}{0ex}}(3.5\times {10}^{-6}\pm {\delta}_{\mathit{shim}})$:

$${w}_{ii}^{f}=\{\begin{array}{ll}1\hfill & {f}_{w}^{-}\le {f}_{i}\le {f}_{w}^{+}\phantom{\rule{0.16667em}{0ex}}\text{or}\phantom{\rule{0.16667em}{0ex}}{f}_{f}^{-}\le {f}_{i}\le {f}_{f}^{+}\hfill \\ 0\hfill & o.w.\hfill \end{array}$$

(17)

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

$${d}_{i}={d}_{i}^{z,w}{d}_{i}^{f}$$

(18)

$${w}_{ii}=\{\begin{array}{ll}{w}_{ii}^{z,w}\hfill & {f}_{w}^{-}\le {f}_{i}\le {f}_{w}^{+}\hfill \\ 1\hfill & {f}_{f}^{-}\le {f}_{i}\le {f}_{f}^{+}\hfill \\ 0\hfill & o.w.\hfill \end{array}$$

(19)

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

The gradient waveform design is informed by *TB*, Δ*z*, and for unaliased pulse designs, the necessary spectral sampling period Δ*T _{f}*. We first consider Δ

$$\mathrm{\Delta}{T}_{f}\le {\text{FOV}}_{f}^{-1}={\left({f}_{w}^{+}-{f}_{f}^{-}\right)}^{-1},$$

(20)

where FOV* _{f}* 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 Δ

$$\frac{2\pi \xb7TB}{\gamma \mathrm{\Delta}z}.$$

(21)

To reduce RF magnitude, the positive gradient lobes are designed to be as long as possible within the Δ*T _{f}* 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* (FOV* _{z}*) can be determined as:

$${\text{FOV}}_{z}={\left(\frac{\gamma}{2\pi}\overline{G}\mathrm{\Delta}t\right)}^{-1},$$

(22)

where is the mean value of the positive gradient lobe. We set *N _{z}*, the size of the design grid in the

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 Δ*z* = 1cm, and ripple levels *δ _{p}* =

To inform our choices of *T _{td}*, we repeated 1.5T pulse designs over a range of

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 *T*_{td}. The metrics generally increase when *T*_{td} is shorter or longer than half the pulse’s duration. **...**

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.

$$\text{NRMSE}=\frac{{\left|\right|\mathit{Ab}-\mathit{d}\left|\right|}_{\mathit{W}}}{{\left|\right|\mathit{d}\left|\right|}_{\mathit{W}}},$$

(23)

also increases with shorter *T _{td}*. However, there is a large range of

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 Δ*T _{f}* at 3T, which results in larger gradient magnitude and a faster traversal of

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 *G _{x}*, and a constant

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 *T _{td}* of 1.04 ms in the experimental data, which is in good agreement with the desired

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 *T*_{td} = 1.04 ms via a linear **...**

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.

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).

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 *T _{td}* 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.

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 **...**

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 *B*_{1}, 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 *B*_{1} 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 *k _{z}* 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 *T _{td}* is, the larger the peak

**Support**

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

*α*- proton resonance frequency shift with temperature
*z*_{i}- z-location
*k*(_{z}*t*)- z-dimension excitation k-space trajectory waveform
*k*(_{f}*t*)- f-dimension excitation k-space trajectory waveform
*T*- duration of RF pulse
*t*- time
*N*_{t}- number of time points in the RF pulses
- Δ
*t* - sampling period of pulses
*γ*- gyromagnetic ratio
*N*_{z}- Size of z-dimension of design grid
*N*_{f}- Size of f-dimension of design grid
*i*- imaginary number, also spatial index
*b*- vector of RF pulse samples
*A*- Fourier system matrix
*a*_{ij}- elements of the system matrix
*A* *d*- vector of samples of desired pattern
*W*- diagonal matrix specifying Region of Interest (ROI) for pulse design
- (
*δ*_{p}δ ), peak pass and stop-band ripple amplitudes_{s}- Δ
*z* - slice thickness
*TB*- time-bandwidth product
*W*- fractional transition width in z dimension
*d*_{∞}- optimal FIR filter performance measure
- (
*z*_{p}z ), in- and out-of-slice edges_{s}- (
*w*_{p}w ), in- and out-of-slice error weights_{s}*T*_{td}- time to tip-down
- (Δ
*T*^{−} - Δ
*T*^{+}), lower/upper temperature change bounds *δ*_{shim}- shim bound
- ( ${f}_{w}^{-},{f}_{w}^{+}$)
- lower/upper boundaries of water band
- $\left({f}_{f}^{-},{f}_{f}^{+}\right)$
- lower/upper boundaries of fat band
- Δ
*T*_{f} - spectral sampling period
- FOV
_{f} - spectral FOV of the spectral-spatial trajectory
- FOV
_{z} - 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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