|B1+|-selective excitation pulse design using the Shinnar–Le Roux algorithm

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jB

þ

₁

j-selective excitation pulse design using the Shinnar–Le Roux

algorithm

William A. Grissom

a,b,c,⇑

_{, Zhipeng Cao}

a,c

_{, Mark D. Does}

a,b,c a

Department of Biomedical Engineering, Vanderbilt University, Nashville, TN, USA b

Department of Radiology and Radiological Sciences, Vanderbilt University, Nashville, TN, USA c

Vanderbilt University Institute of Imaging Science, Nashville, TN, USA

a r t i c l e

i n f o

Article history:

Received 18 December 2013 Revised 9 February 2014 Available online 6 March 2014 Keywords:

RF pulse design Selective excitation Shinnar–Le Roux algorithm Rotating Frame Selective Excitation

a b s t r a c t

A new mathematical treatment and algorithm for the design of jBþ

1j-selective RF excitation pulses is presented and validated. The algorithm is based on a rotated Shinnar–Le Roux pulse design algorithm, wherein the pulse’s frequency modulation waveform is directly designed by the algorithm, and its ampli-tude and sign modulation waveform takes the place of the gradient field. A new pulse configuration is described that enables excitation of large tip-angle slice-selective profiles. Experiments were performed to validate the pulses, and simulations were performed to characterize the pulses’ sensitivity to off-resonance, and to compare them to adiabatic (BIR-4) pulses.

Ó 2014 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction

In 1980 Hoult described ‘Rotating Frame Selective Excitation Pulses’ that selectively excite magnetization based on the strength of the RF transmit ﬁeld (jBþ

1j) they experience[1]. The pulses were intended for use in rotating frame imaging[2–4], but could be used for slice selection in any of several RF gradient-based imaging

methods that have been proposed since[5–8]. The pulses were

based on the assumption of a large and constant B1;xgradient field. When this field was switched on, initially-longitudinal magnetiza-tion would precess around it in the y–z plane. Hoult showed that by modulating the B1;yfield, magnetization could be selectively ex-cited based on the magnitude of the B1;xcomponent (the strength of B1;y was implicitly assumed to be constant across space). The pulses were designed by analogy to B0-selective excitation, where-in B1;xwas treated as the longitudinal gradient field, and B1;ythe perpendicular field responsible for flipping magnetization.

Assum-ing a constant B1;x waveform was played (analogous to the

constant B0 gradient used in conventional slice-selective excita-tion), pulse design then amounted to designing the B1;ymodulation required to obtain the desired slice proﬁle. Somewhat improved design methods and results were described several years later by Karczmar[9]and Hedges and Hoult[10]. To date however, no clear

algorithm has been reported to design the pulses to meet target slice proﬁle characteristics, in analogy to the Shinnar–Le Roux algorithm which is widely used for conventional slice-selective pulse design.

We recast the jBþ

1j-selective pulse design problem as one of designing a frequency modulation waveform rather than a B1;yﬁeld component, and show that the small-tip-angle Shinnar—Le Roux (SLR) algorithm[11–16]can be used to directly design this wave-form for excitations (0–90° tip angles) and inversions. The result is a simple and fast pulse design approach that inherits the ease-of-use of SLR, provides a substantial improvement in the selectivity of the pulses over previous design methods, and enables the exci-tation of larger tip-angles. The following sections formulate the pulse design method, present simulation results that characterize the pulses’ off-resonance sensitivity and compare them to adia-batic pulses, and present experimental results that validate the pulses’ function. Preliminary aspects of this work were presented in Ref.[17].

2. Theory

2.1. RF waveform deﬁnitions

The proposed algorithm directly designs an RF frequency modulation waveform D

x

RFðtÞ that is paired with an amplitude

and sign modulation waveform AðtÞ to comprise a jBþ

1j-selective excitation pulse. Given these waveforms, the pulse can be expressed in terms of its x and y components as:

http://dx.doi.org/10.1016/j.jmr.2014.02.012

1090-7807/Ó 2014 The Authors. Published by Elsevier Inc.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). ⇑ Corresponding author. Address: Institute of Imaging Science, Vanderbilt

Uni-versity, 1161 21st Ave. South, MCN AA-1105, Nashville, TN 37232-2310, USA. E-mail addresses: [email protected](W.A. Grissom), zhipeng.cao@ vanderbilt.edu(Z. Cao),[email protected](M.D. Does).

Contents lists available atScienceDirect

Journal of Magnetic Resonance

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~_{B1ðtÞ ¼ jB}þ 1jAðtÞ ^x cos /ðtÞ þ \B þ 1 þ ^y sin /ðtÞ þ \Bþ 1 ; ð1Þ

where /ðtÞ ¼R₀tD

x

RFðt0Þdt0. AðtÞ will be real-valued, with a maxi-mum amplitude of one, and without loss of generality we will

as-sume \Bþ

1¼ 0. In a frame rotating at

x

0þD

x

RFðtÞ, where

x

0 is the Larmor frequency, the pulse comprises two vector components that are illustrated inFig. 1: a transverse component with length jBþ

1jAðtÞ, and a z-directed component with lengthD

x

RFðtÞ=

c

, where

c

is the gyromagnetic ratio. The proposed algorithm operates in this frame.

2.2. Rotated spinor parameters

The SLR algorithm was developed to design a transverse RF ﬁeld waveform that is played simultaneously with a constant gradient waveform for slice selection. In jBþ

1j-selective pulse design by SLR,D

x

RFðtÞ takes the place of the transverse RF ﬁeld waveform, and AðtÞ takes the place of the gradient waveform for slice selec-tion, and is scaled by jBþ

1j rather than by a spatial coordinate. This conﬁguration is achieved by rotating the deﬁnition of the pulse’s spinor parameters

a

and b: whereas conventionally

a

represents rotations about the z-directed gradient ﬁeld (the ‘free precession’ axis [16]) and b represents rotations about the RF ﬁeld with x and/or y components (the ‘nutation’ axis), for jBþ

1j-selective SLR pulse design

a

is redefined to represent rotations about the x-axis, and b is redefined to represent rotations about a field with z and/or y components.

a

will thereby represent rotations about the trans-verse jBþ1jAðtÞ ﬁeld, and b will represent rotations about the z-direc-tedD

x

RFðtÞ=

c

ﬁeld. Deﬁnitions of the magnetization components remain unchanged.

2.3. Target excitation proﬁle

The SLR algorithm is based on relating target magnetization proﬁles (Mx;My, and Mz) to spinor parameter proﬁles (

a

and b) whose discrete Fourier transform (DFT) coefﬁcients can be inverted to obtain the RF pulse that produces them. To apply the algorithm to design anD

x

RFðtÞ waveform that excites a slice along the jBþ1j axis, we must express target excitation proﬁles in terms of the rotated

a

and b parameters. The inverse SLR transform can then compute theD

x

RFðtÞ waveform that corresponds to those parame-ters. Given initial magnetization M

zy,M z þ ıM y, and M x, the magnetization after a pulse with rotated

a

and b parameters will be: Mþ zy Mþzy Mþx 0 B @ 1 C A ¼ ða_Þ2 b2 2a_b ðbÞ2

a

2 _2ab a_b _ab

_aa

_bb 0 B @ 1 C A M zy M zy M x 0 B @ 1 C A: ð2Þ

For initial magnetization at thermal equilibrium (ðM x;M

y;M

zÞ ¼ ð0; 0; 1Þ), the excited transverse magnetization will be:

Mþ x ¼ ab

ab

¼ 2ð

aRb

R

aI

bIÞ ð3Þ Mþ y ¼ I ðaÞ 2 b2 n o ¼ 2ð

aRaI

þ bRbIÞ; ð4Þ

where the R and I subscripts denote the real and imaginary parts of the parameters, respectively. As in conventional linear-phase SLR pulse design and previous jBþ

1j-selective design methods, we will design pulses that produce constant-(speciﬁcally, zero-) phase pro-ﬁles across the excited slice so that Mþ

y ¼ 0. For these pulses bIwill also be zero. If we further restrict our consideration to small-tip-an-gle pulses with AðtÞ waveforms that have zero integrated area, then

a

R 1 and

a

I 0[18]. In this case,

Mþ

x ¼ 2bR; ð5Þ

and Mþ

y ¼ 0. Therefore, bRis the parameter of interest for digital ﬁl-ter design in the jBþ

1j-selective SLR algorithm. Conveniently, because Mþ

x ¼ 2bRalso for a conventional refocused small-tip-angle slice-selective pulse[18], the same ripple relationships provided in Ref.

[16]also apply to jBþ

1j-selective pulse design.

Fig. 2illustrates the target b proﬁle conﬁguration. Unlike con-ventional slice-selective excitation, a jBþ

1j-selective slice proﬁle can-not be centered at jBþ

1j ¼ 0, since excitation cannot occur with zero RF field. Thus, the slice profile must be shifted away from this point. A slice-selective excitation is conventionally shifted using fre-quency modulation of the RF pulse; however, this would result in complex b DFT coefficients, and subsequently a complex-valued D

x

RFðtÞ waveform. TheD

x

RFðtÞ waveform must be real-valued to be physically realizable, which dictates that the b DFT coefﬁcients must be purely imaginary, since a small-tip RF pulse designed by SLR is

p

=2 out of phase with its b DFT coefficients[16]. The required purely imaginary b DFT coefficients can be obtained by specifying an odd and dual-band (anti-symmetric) b profile[19]. Thus, the tar-get b profile must be real-valued, dual-band, odd, and zero at jBþ1j ¼ 0. The correspondingD

x

RFðtÞ will be real-valued and odd. 2.4. b ﬁlter design

A real-valued, odd, and dual-band b profile and its correspond-ing DFT coefficients can be designed in several ways. For well-separated passbands (i.e., centered at sufficiently high jBþ

1j), the process can start with a conventional single-band linear-phase fi-nite impulse response filter designed using a weighted-least squares method. That filter is then duplicated, and the duplicates are frequency modulated to opposite center frequencies and sub-tracted from each other. This is equivalent to modulation of the single-band filter by a sine function at the center frequency. For very close passbands (i.e., passbands close to jBþ

1j ¼ 0) however, ripples from one band can distort the other. In these cases, an odd, dual-band b filter can be designed directly using weighted-least-squares. The distortions could also be mitigated using a phase-correction method[20]. Once the b filter is designed, assum-ing small excitation angles the inverse SLR transform reduces to a simple scaling of the filter coefficients to obtain the D

x

RFðtÞ waveform.

2.5. AðtÞ waveform construction

The SLR algorithm conventionally designs an RF pulse that accompanies a constant gradient waveform. In jBþ

1j-selective pulse Fig. 1. Illustration of ﬁeld and axis conﬁguration for jBþ

1j-selective pulse design by SLR. The pulse is expressed in terms of its amplitude and sign modulation waveform AðtÞ and its frequency modulation waveform DxRFðtÞ. Here these components are depicted in the frame rotating at frequencyx0þ DxRFðtÞ. Whereas conventionally the SLR algorithm designs the x and y components of an RF waveform, in jBþ

1 j-selective pulse design it is modiﬁed to design the z-directed frequency modulation waveform DxRFðtÞ, by rotating the deﬁnitions of the pulse’s spinor parametersa and b.

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design, AðtÞ replaces the gradient waveform. In the small-excita-tion angle regime, the

a

proﬁle at the end of a pulse with duration T is[18]:

aðjB

þ 1jÞ ¼ e ı2cjBþ1j RT 0AðtÞdt_; ð6Þ

and the b proﬁle is: bðjBþ1jÞ ¼ ı 2e ıc 2jB þ 1j RT 0AðsÞds Z T 0

DxRFðtÞe

ıcjBþ 1j RT t AðsÞdsdt: ð7Þ ¼ ı 2e ı 2jB þ 1jkð0Þ ZT 0

DxRFðtÞe

ıjBþ 1jkðtÞdt; ð8Þ where kðtÞ ,

c

RT

t AðsÞds is the pulse’s jB þ

1j-frequency trajectory. From Eq.(6), it is evident that if AðtÞ is constant and comprises no pre- or rewinder lobes before or after the D

x

RFðtÞ waveform to achieve zero total area, then

a

I–0, which is unacceptable. Zero to-tal area could be achieved by adding a negative rewinder lobe to AðtÞ with the same area as the main lobe, but according to Eq.(8)

this would create a nonzero bIsinceD

x

RFðtÞ would deposit energy at negative frequencies only, as depicted in the middle column of

Fig. 3. A real and odd b proﬁle can only be produced ifD

x

RFðtÞ deposits energy anti-symmetrically as a function of frequency, and therefore cannot be produced with this trajectory. Placing the rewinder lobe at the beginning of the pulse would also lead to non-zero bI. The desired symmetric kðtÞ can be restored by splitting the rewinder lobe, so that half is played at the beginning and half at the end, as shown in the right column ofFig. 3. With this conﬁguration,

a

¼ 1 and bI¼ 0 as required. This AðtÞ waveform conﬁguration is analogous to a balanced gradient waveform conﬁguration for con-ventional slice-selective excitation, which is commonly used for refocusing pulses in spin echo sequences and for excitation pulses in balanced steady-state free precession sequences[21].

2.6. Maintaining selectivity at large tip-angles

Fig. 4a shows that as a jBþ

1j-selective pulse is scaled to excite a large tip-angle,

a

Igrows and degrades the excited proﬁle by creat-ing a large unwanted Mycomponent (Eq.(4)), particularly in the stopband. This is a consequence of the Hilbert transform

relation-ship between the amplitude and phase of

a

for a minimum-power

pulse[16], and it is impossible to design an

a

ﬁlter that has the magnitude response required by a large-tip SLR pulse and a zero phase response, for any nonzero ﬂip angle. To reduce the unwanted stopband excitation and achieve more accurate large-tip-angle excitations and inversions, the two halves of theD

x

RFðtÞ waveform can be reﬂected and played out on the pre- and rewinding AðtÞ lobes. The amplitude of the whole pulse is also divided by two to achieve the target tip angle. With this modiﬁcation, the size of the

a

I component is reduced dramatically, and selectivity is re-stored at large tip-angles. This works because, for each half of the pulse, time-reversal ﬂips the phase of

a

, and the division by two increases its amplitude, which reduces its phase by the Hilbert transform relationship. This leads to a combined

a

parameter for each half that is dominated by j

a

j2 when the pulse halves are

R profiles DFT Coefficients RF Waveforms

Centered Single-Band Shifted Single-Band Even Dual-Band Odd Dual-Band Real Imaginary

Fig. 2. Target b proﬁle conﬁguration for jBþ

1j-selective pulse design. (Top row) A conventional single-band slice proﬁle would be centered at the zero position on the selection axis, which is impossible for a jBþ

1j-selective pulse since excitation cannot occur with zero RF ﬁeld. (Second row) Frequency modulation can be applied to shift the slice proﬁle away from jBþ

1j ¼ 0, but this results in a complex pulse that is not physically realizable since DxRFðtÞ must be real-valued. (Third and fourth rows) A dual-band pulse with symmetric slice shifts corresponds to a purely imaginary (even dual-band) or purely real (odd dual-band) RF pulse; to obtain a purely real DxRFðtÞ, an odd dual-band b proﬁle is used (box). The bIproﬁle is zero in all cases.

Fig. 3. Amplitude and sign modulation waveform (AðtÞ) construction. (Left Column) With no pre- or rewinding, the jBþ

1j-frequency trajectory kðtÞ visits both positive and negative frequencies when DxRFðtÞ is on, enabling DxRFðtÞ to deposit energy anti-symmetrically as required to obtain a purely real b profile (Eq.(8)). However, the total area under AðtÞ is non-zero, leading to a complex-valuedaprofile which is unacceptable (Eq.(6)). (Middle Column) By adding a rewinder to the waveform, the area under AðtÞ is zero soa¼ 1 as required, but DxRFðtÞ only deposits energy when kðtÞ is negative, leading to a complex b profile. (Right Column) Splitting the rewinder between the beginning and end of the pulse restores the desired trajectory, anda¼ 1 as required.

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played back-to-back with their time-reversed copies.Fig. 4b shows that with this modiﬁcation, selective inversions can be designed.

2.7. Overall algorithm

Given the RF digital-to-analog conversion dwell time Dt

(seconds), the time-bandwidth product TB, the pulse type (small-tip, excitation, inversion, or saturation), the tip angle h (radians), the passband width PBW (Gauss), the passband center PBC (Gauss), and the passband (d1;e) and stopband (d2;e) ripple levels (units of M1

0 ), the steps of the proposed jB þ

1j-selective pulse design algorithm are:

1. Calculate the half-pulse duration T and the number of samples in the half-pulse n: T ¼ c TB 2pPBW ð9Þ n ¼ 2 T 2Dt ð10Þ where

c

is the gyromagnetic ratio in radians per second per Gauss.

2. Calculate the inputs for finite impulse response (FIR) b filter design. The required inputs to the commonly-used MATLAB (Mathworks, Natick, MA, USA) firls function for weighted least-squares FIR filter design are:

n : The number of samples in the filter: ð11Þ

f ¼ 0 ð1 wÞTB n ð1 þ wÞ

TB

n 1

:Normalized band edges: ð12Þ

m ¼ 1 1 0 0½ : Amplitude of frequency response at band edges:ð13Þ wts ¼ 1 d1

d2

:Band error weights: ð14Þ

Here, w is the fractional transition width, which is:

w ¼d1ðd1;d2Þ

TB ð15Þ

where the b ripples ðd1;d2Þ are calculated from ðd1;e;d2;eÞ and the pulse type using the analytic relationships and d1function deﬁned in Ref.[16]. We note that there are other options for linear-phase FIR ﬁlter design that could be used in place of firls, such as the Parks–McClellan algorithm[22,23]. −0.4 −0.2 0 0.2 0.4 −0.4 −0.2 0 0.2 0.4 −0.4 −0.2 0 0.2 0.4 −2 −1 0 1 2 −200 −100 0 100 200 −2 −1 0 1 2 −1000 −500 0 500 1000 −2 −1 0 1 2 −1000 −500 0 500 1000 −1 0 1 Time (ms) RF (t )/2 (Hz) Time (ms) Time (ms) A (t) 0 2 4 6 8 10 −0.1 0 0.1 0.2 0 2 4 6 8 10 −0.5 0 0.5 1 0 2 4 6 8 10 −0.5 0 0.5 1 αI /M0 |B1 +_{| (Gauss)} _|_B 1 +_{| (Gauss)} _|_B 1 +_{| (Gauss)}

10° 90° 90°, Split and Reflected

(a)

|M_xy| Mx My 0 2 4 6 8 10 −1 −0.5 0 0.5 1 0 2 4 6 8 10

(b)

Mz |B1 +_{| (Gauss)} _|_B 1 +_{| (Gauss)} Split and Reflected ,

180° 180°

Fig. 4. Maintaining selectivity at large tip-angles. (a) At small tip-angles such as 10°,aI 0 as assumed in the design. As the tip angle is increased to 90°,aIincreases dramatically, and degrades selectivity of the pulse outside the passband. By splitting the 90° pulse into its two halves, reﬂecting those halves and playing them out during the pre- and rewinding AðtÞ lobes, the selectivity of the pulse is restored. (b) The split-and-reﬂect strategy also enables selective inversions.

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3. Run the FIR filter design tool to design the b filter, producing length-n coefficient vector h. The frequency response of h should have an average amplitude of 1 in its passband; with firlsthis is determined by the entries of the vector m (Eq.

(13)).

4. Sine-modulate the b ﬁlter coefﬁcients to the desired passband center: hj 2hjsin

cPBC jDt

T 2 ; j ¼ 0; . . . ; n 1: ð16Þ 5. Split and reﬂect the modulated ﬁlter, producing length-2n

coef-ﬁcient vector ~h: ~ hj¼ hn=21j=2; j ¼ 0; . . . ;n2 1; hjn=2=2; j ¼n2; . . . ; 3n 2 1; hn1ðj3n=2Þ=2; j ¼3n2; . . . ;2n 1: 8 > < > : ð17Þ

6. Scale to the desired tip angle and divide by the dwell time to get the sampledD

x

RFðtÞ waveform

x

RF(in units of radians/s):

DxRF;j

¼ h Dt ~

hj; j ¼ 0; . . . ; 2n 1: ð18Þ

7. Build the sampled, normalized AðtÞ waveform a:

aj¼ 1; j ¼ 0; . . . ;n 2 1; 1; j ¼n 2; . . . ; 3n 2 1; 1; j ¼3n 2; . . . ;2n 1: 8 > < > : ð19Þ 3. Methods 3.1. Experiments

Phantom experiments were performed to validate control of ﬂip angle, time-bandwidth product, and centering of the pulses. jBþ

1j-Selective pulses were designed in MATLAB1and deployed on

a 31 cm 4.7 T Varian spectrometer (Agilent, Santa Clara, CA, USA) with a 38 mm Litz volume coil (Doty Scientiﬁc, Columbia, SC, USA) for transmit and receive and a 50 mL, 3 cm diameter/10 cm long vial phantom containing a CuSO4solution with T1 200 ms. The pulses were used for excitation in a 3D gradient-recalled echo sequence with FOV 30 30 100 mm, 32 32 32 matrix size, 50 or 100 ms TR and 5 ms TE as measured from the center of the pulses. The pulses were sampled with a 4

l

s dwell time, and frequency modulation was converted to phase modulation. To account for ﬁnite RF ampliﬁer rise times, 40-sample ramps were placed on either end of the AðtÞ waveforms, which were paired with 20-sample rewinders with opposite sign to cancel the area of the ramps. These ramps and rewinders are visible on the waveforms inFig. 5(a and c). Because a homogeneous volume coil was used for transmission and reception, the jBþ

1j-selective proﬁle of each pulse was measured by repeating the 3D scans over a range of RF amplitudes corresponding to a de-sired jBþ

1j measurement range. Two additional scans were collected to calculate an off-resonance ﬁeld map. Those scans had the same volume coverage and matrix size, and a 50 ms TR, but used a 1 ms, 30° Gaussian excitation pulse and TEs of 5 and 6 ms, so that a ﬁeld map could be calculated from their phase difference[24]. Then, from each jBþ

1j-selective excitation pulse’s set of 3D acquisitions, the signal for each jBþ

1j was calculated from the corresponding images as the magnitude of the complex average of signal from voxels with off-res-onance within ±5 Hz (so as to obtain on-resoff-res-onance proﬁle measure-ments), and inside an object mask derived by thresholding one of the off-resonance map acquisition images at 15% of the peak image magnitude.

3.2. Simulations

Simulations were performed to characterize the sensitivity of jBþ

1j-selective pulses to off-resonance, and to compare them to BIR-4 adiabatic pulses [25] in terms of off-resonance sensitivity and threshold jBþ

1j. A hard pulse approximation-based Bloch simu-lator was used[16], with a 2

l

s dwell time for the off-resonance simulation, and a 4

l

s dwell time for the BIR-4 comparison. The simulations assumed excitation of1_{H, so that} c

2p¼ 4257 Hz=Gauss. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 5 10 0 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 |_B1+_{| (Gauss)} a.u. a.u. a.u. Measured |Mxy| Profiles −1 −0.5 0 0.5 1 −100 0 100 −1 0 1 Pulses −3 −2 −1 0 1 2 3 −100 0 100 −2 −1 0 1 2 −50 0 50 −1 0 1 Time (ms) RF (t )/2 (Hz) A (t) A (t)

(a)

(b)

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30° 15° TB 6 TB 2 0.2 Gauss 0.4 Gauss 0 0.2 0.4 0.6 0.8 1 5 RF (t )/2 (Hz) RF (t) /2 (Hz)

Fig. 5. Experimental results. (a) 15° (Solid black) and 30° (dashed) TB = 2 excitations. The signal intensity is higher for the 30° excitation. (b) 15° TB = 2 (solid) and 6 (dashed) excitations. The TB = 6 pulse has narrower transitions between the passband and stopband. (c) 15° TB = 2 excitations, centered at 0.2 Gauss/850 Hz (solid black) and 0.4 Gauss/1.7 kHz (dashed). The pulses’ proﬁles are centered in the intended locations on the jBþ

1j-axis.

1

MATLAB code to design jBþ

1j-selective pulses, and to generate most of the ﬁgures in this article is available athttp://www.vuiis.vanderbilt.edu/grissowa/.

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4. Results 4.1. Experiments

Fig. 5shows the pulses played out in the experiments and the resulting measured jBþ

1j-selective profiles.Fig. 5a shows a compar-ison of signal profiles for nominal 15° and 30° excitations, with duration 2.83 ms, 0.4 Gauss/1.7 kHz slice width, TB = 2 (Gaussian-like profile), and centered at 0.4 Gauss/1.7 kHz. The signal intensity from the 30° excitation is larger and consistent with increased

excitation and T1-weighting. Fig. 5b shows a comparison of

TB = 2 (2.37 ms) and TB = 6 (6.13 ms) pulses and signal profiles, with a nominal 15° flip angle, 0.5 Gauss/2.1 kHz slice width, and centered at 0.5 Gauss/2.1 kHz. The TB = 6 pulse has narrower tran-sition regions from stop to pass, reflecting the higher selectivity it was designed to have.Fig. 5c shows a comparison of the 15° TB = 2 (5.74 ms) excitations, centered at 0.2 Gauss/850 Hz and 0.4 Gauss/ 1.7 kHz. The two pulses’ profiles are centered in the intended loca-tions, but otherwise appear very similar.

4.2. Simulations

Fig. 6shows the off-resonance simulation results. Four jBþ 1 j-selective pulses were simulated: two 3.1 ms TB = 2 pulses at 30° and 90°, centered at 2 and 4 Gauss/8.5 and 17 kHz, with 0.3 Gauss passband width, and two 12.5 ms TB = 8 pulses, for the same flip angles, profile centering and passband widths. All four designs used d1;e¼ d2;e¼ 0:01. The two-dimensional patterns of unwanted excitation due to off-resonance appear the same for a given dura-tion. This suggests that off-resonance sensitivity primarily depends on pulse duration and the shape of the AðtÞ waveform, rather than on the flip angle and profile centering, which are characteristics that determine the shape and amplitude of theD

x

RFðtÞ waveform. As might be expected, near jBþ

1j ¼ 0, the shorter 3.1 ms pulse ap-pears to have a wider frequency bandwidth over which unwanted excitation is insigniﬁcant. Further, in all cases the unwanted exci-tation decays rapidly as jBþ

1j increases. Note that the jMxyj patterns

shown inFig. 6are Hermitian symmetric about the jBþ

1j axis, and are therefore displayed only for positive off-resonance frequencies.

Fig. 7 shows the BIR-4 comparison results. A 4.7 ms, TB = 4 jBþ1j-selective pulse was designed to excite a 45° tip angle, with a passband width of 0.4 Gauss/1.7 kHz, and ripples d1;e¼ 0:01 and d2;e¼ 0:4. The high d2;ewas used to reﬂect the fact that the stop-band above the passstop-band was a ‘don’t-care’ region. The passstop-band was placed as close to jBþ

1j ¼ 0 as possible, so direct weighted-least squares dual-band FIR ﬁlter design was used to design the b ﬁlter. Two BIR-4 pulses were then designed: one with the same 4.7 ms duration as the jBþ

1j-selective pulse, and one longer 5.9 ms pulse.

The 4.7 ms BIR-4 pulse design used D

x

0

RF¼ 100

p

=T radians/s, b¼ 10, and

j

¼ tan1₂₀_[25]_{. These parameters were empirically} selected to match the threshold jBþ

1j and passband ripple of the jBþ

1j-selective pulse. The 5.9 ms BIR-4 pulse design used the same D

x

0

RFand b, but its longer duration enabled use of a less-aggressive

j

¼ tan1_{15. All pulses are plotted in}_{Fig. 7}_{a. Note that there is a}

p

þ

p

=8 phase shift (not shown) between the central and outer lobes of the BIR-4 pulses’ AðtÞ waveforms, to affect the 45° tip an-gle.Fig. 7b plots the jMxyj proﬁle of each pulse at 0 Hz. All three pulses have approximately the same threshold jBþ

1j, and approxi-mately the same ripple across the passband. The longer 5.9 ms BIR-4 pulse achieved the same threshold jBþ

1j as the 4.7 ms BIR-4 pulse, without requiring a large

j

.Fig. 7c compares the off-reso-nance sensitivity of the three pulses. The pulses all have similar off-resonance sensitivity near jBþ1j ¼ 0, in the transition up to their passbands. In the passband, the jBþ

1j-selective pulse appears to have similar off-resonance sensitivity to the 4.7 ms BIR-4 pulse, but the 5.9 ms BIR-4 pulse is signiﬁcantly more robust to off-reso-nance than either 4.7 ms pulse.

5. Discussion

The proposed algorithm extends the attractive properties of the Shinnar–Le Roux algorithm to the design of jBþ

1j-selective pulses. These include speed and the ability to predict slice proﬁle charac-teristics analytically, and to thereby make tradeoffs between pulse

0 200 400 600 800 1000 0 1 2 3 4 5 0 200 400 600 800 1000 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 30° 3.1 ms, TB 2 90° 12.5 ms, TB 8 |_B₁+_{| (Gauss)} _|B 1 +_{| (Gauss)} Δ f (Hz) |M xy | (/ M0 ) Δ f (Hz)

Fig. 6. Off-resonance simulation. The top row shows jMxyj patterns excited by 3.1 ms TB = 2 pulses at 30° and 90°, centered at 2 and 4 Gauss/8.5 and 17 kHz, with 0.3 Gauss/ 1.3 kHz passband width, across a 1 kHz range of off-resonance frequencies (Df0). The bottom row shows 12.5 ms TB = 8 pulses, for the same ﬂip angles, proﬁle centering and passband widths. The pattern of unintended excitation appears to depend primarily on the duration of the pulse, rather than the amplitude or centering of the pulses. There is larger unintended excitation for lower jBþ

(7)

parameters before ever designing a pulse and evaluating it. This eliminates the need for a guess-and-check approach to pulse de-sign and makes the dede-sign process more accessible to non-experts. Further, previous methods for jBþ

1j-selective pulse design focused on the design of the y-component of the RF field, and assumed that the amplitude of the overall field was independent of that compo-nent[9,10]. By directly designing the frequency modulation wave-form rather than the y-component of the RF field, the proposed algorithm eliminates a source of approximation error in the pulse design that may have hampered previous methods.

The development of the algorithm was based on an assumption of small excitation angles, and it was shown that without the de-scribed split-and-reﬂect conﬁguration, the pulses’ selectivity se-verely degrades when they are scaled to excite large tip-angles. This degradation was attributed to an increasing nonlinear phase variation in the

a

proﬁle that grows with ﬂip angle. Unfortunately, the degradation cannot be mitigated by explicit design of an

a

ﬁlter with a zero phase response combined with use of the full inverse SLR transform rather than the small-excitation version used in the described algorithm, since as noted earlier it is impossible to design an FIR ﬁlter with the required

a

magnitude response and zero phase response. Nor can the degradation be mitigated by adjusting the areas of the pre- and rewinding AðtÞ lobes: this ap-proach could eliminate ﬁrst-order phase variation in the

a

proﬁle in the slice, but would leave a phase roll across the

a

and b proﬁles, and consequently nonzero

a

I and bI that would degrade the Mxy profile further. While the described split-and-reflect modification to the pulses enables pulses designed by the algorithm to excite selective large-tip-angle profiles up to 180°, there will still be some loss in selectivity due to the bandwidth narrowing effect [26]. Attaining the most accurate large-tip excitations will require the development of a novel approach to inverting the b profile along a bipolar trajectory, subject to a zero-phase

a

. Recent advances in multidimensional SLR pulse design may lead to the development of such a method in the future[27,28]. The design of jBþ

1j-selective refocusing pulses remains an open problem and will require a dif-ferent problem formulation than that developed here. Previous re-ports of jBþ1j-selective pulse design approaches [9,10] did not address the design of pulses with tip angles greater than 90°.

In addition to the pulse construction described here, Ref.[9] de-scribes a ‘transposed sinc pulse’ conﬁguration (Fig. 6in Ref.[9]), which is equivalent to playing the ﬁrst half the waveforms pre-sented here with twice theD

x

RFðtÞ amplitude. While the shorter duration of these pulses is attractive, compared to the full pulses their excitation proﬁles are degraded since the jBþ

1j-frequency tra-jectory visits only positive frequencies, leading to increased b amplitude in the stopband and corresponding undesired excita-tion. Inversion pulses constructed this way also exhibit substan-tially degraded and narrowed proﬁles. It is possible that future work will reveal an approach to design these pulses that can accu-rately account for or mitigate these effects.

There remain multiple questions to be answered regarding the use and performance of jBþ

1j-selective pulses, which have not been previously addressed and are beyond the scope of the present work. One question is the pulses’ sensitivity to T1and T2 relaxa-tion: compared to conventional slice-selective excitations, jBþ

1 j-selective pulses can cause magnetization to rotate through many cycles of phase in the y–z plane, and it is possible that relaxation during these rotations will degrade localization. Requirements on RF amplifier performance should also be investigated: because jBþ1j-selective pulses used for slice selection would have higher power demands than conventional gradient-based slice-selective pulses, amplifier nonlinearities such as compression, droop, and phase distortions may be an issue. Preemphasizing the waveforms or leveraging recent developments in amplifier technology such as Cartesian feedback may mitigate these distortions[29,30].

Our results showed that jBþ

1j-selective pulses achieve similar jBþ1j thresholds as BIR-4 pulses of the same duration. The proposed algorithm will enable the user to directly specify the jBþ

1j-threshold and the jBþ

1j range over which a given tip angle is desired, within a given tip angle error tolerance, and the algorithm will produce the shortest possible pulse that meets those requirements. Given the results presented inFig. 7though, replacing adiabatic pulses with jBþ1j-selective pulses may only be feasible when broad robustness to off-resonance is not a primary design objective. Future work may also reveal a connection between jBþ

1j-selective and adiabatic pulses, that could enable the straightforward design of adiabatic pulses using the SLR algorithm.

−1 0 1 −200 −100 0 100 200 0 0.5 1 −3 −2 −1 0 1 2 3 −10 0 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 −3 −2 −1 0 1 2 3 0 0.5 10 0 200 400 600 800 1000 1 0 1 0.5 0.5 |B1

+_{|-Selective 4.7 ms} _{BIR-4 4.7 ms} _{BIR-4 5.9 ms}

|_B₁+_{| (Gauss)} _|_B 1 +_{| (Gauss)} _|B 1 +_{| (Gauss)} 0 1 |M xy | (/ M0 ) Time (ms) Time (ms) |M xy| (/ M0 ) |B1 +_{| (Gauss)} RF (t )/2 (kHz) RF (t )/2 (Hz) A (t) _{A (t)} |B1 +_|-Selective BIR-4

(a)

(b)

(c)

|B1

+_{|-Selective 4.7 ms} _{BIR-4 4.7 ms} _{BIR-4 5.9 ms}

Fig. 7. BIR-4 comparison. (a) Comparison of 45° RF waveforms, showing the 4.7 ms TB = 4 jBþ

1j-selective excitation pulse, and two BIR-4 pulses: a duration-matched 4.7 ms pulse with parameters tuned for minimum threshold jBþ

1j, and a longer 5.9 ms pulse with the same transition width but better robustness to off-resonance. (b) Simulated jMxyj proﬁles at 0 Hz, illustrating that the three pulses have approximately the same threshold jBþ

1j at which a 45° excitation is achieved. (c) Simulated jMxyj patterns over a 1 kHz off-resonance (Df0) range. The three pulses have similar off-resonance sensitivity for low jBþ1j. As jB

þ

1j increases, the 4.7 ms jB þ

1j-selective and BIR-4 pulses have similar distortion patterns, but the 5.9 ms BIR-4 pulse has a more uniform excitation pattern across jBþ

(8)

6. Conclusion

A new mathematical treatment and algorithm for jBþ

1j-selective RF pulse design was introduced and characterized in simulations and experiments. It is based on the direct design of a frequency modulation waveform using a rotated Shinnar–Le Roux slice-selec-tive pulse design algorithm, and it inherits the desirable properties of the Shinnar–Le Roux algorithm. It can design accurate small-tip, 90° excitation/saturation, and inversion pulses. The pulses may be useful in RF gradient-encoded imaging, or as an alternative to adiabatic pulses.

Acknowledgment

The authors would like to thank Dr. Adam Kerr for insightful discussions.

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