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conditions of Wiley this is the PRE peerreviewed version of an article that has
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Magnetic Resonance in Medicine. The
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article is expected to be placed here in
September 2015
Title
Designing Hyperbolic Secant Excitation Pulses to Reduce Signal Dropout in GradientEcho Echo-Planar Imaging
Authors
Stephen J. Wastling1,* and Gareth J. Barker1
* Corresponding author
1Institute of Psychiatry, King’s College London, SE5 8AF
Running Title
Reduced Signal Dropout in GE-EPI using HS Pulses
Key Words
Signal dropout, GE-EPI, FMRI, RF excitation, Hyperbolic Secant, Tailored RF
Word Count
4765 words
Figure and Table Count
8 figures and 0 tables
1
Abstract
Purpose
To design Hyperbolic Secant (HS) excitation pulses to reduce signal dropout in the
orbitofrontal (OF) and inferior temporal (IT) regions in gradient-echo echo-planar
imaging (GE-EPI) for FMRI applications.
Methods
An algorithm based on Bloch simulations is used to optimise the HS pulse parameters
needed to give the desired signal response across the range of susceptibility gradients
observed in the human head (approximately ± 250 T m-1). The impact of the HS pulse
on the signal, temporal signal-to-noise ratio, BOLD sensitivity and the ability to detect
resting-state BOLD signal changes was assessed in six healthy male volunteers at 3T.
Results
The optimised HS pulse ( = 4.25,  = 3040 Hz, A0 = 12.3 T, f = 4598 Hz) had a near
uniform signal response for susceptibility gradients in the range ± 250 T m-1. Signal,
TSNR, BOLD sensitivity and the detectability of resting state networks were all partially
recovered in the OF and IT regions, however there were signal losses of up to 50% in
regions of homogeneous field.
Conclusion
The HS pulse reduced signal dropout and could be used to acquire task and resting
state FMRI data without loss of spatial coverage or temporal resolution.
(191 words)
2
Introduction
Functional magnetic resonance imaging (FMRI) data is often acquired using gradientecho echo-planar imaging (GE-EPI) (1,2) because of its speed, and its sensitivity to
blood oxygen level dependent (BOLD) (3) signal changes. However, GE-EPI images
are affected by signal dropout caused by the differences in the magnetic susceptibilities
of materials in the head (4). This dropout hampers the detection of BOLD signal
changes in areas of the brain close to air-bone interfaces such as the orbitofrontal and
inferior temporal regions. These regions have a number of important functions including,
olfaction (5,6), memory (7) and the processing of language (8), rewards (9,10) and
emotional facial expressions (11,12). Therefore the development of a method to reduce
signal dropout – while retaining the underlying advantages of GE-EPI – would be
beneficial in both task-based and resting-state FMRI experiments.
A range of approaches to reduce the problem of signal dropout have previously been
proposed, and have demonstrated differing levels of success. They typically involve a
reduction in the sensitivity to BOLD signal changes and/or temporal resolution.
Straightforward modifications to data acquisition have included shorter echo times (13)
and smaller voxel sizes (4,13-20); although both resulted in reduced sensitivity to BOLD
signal changes (16,21). Methods which aimed to reduce signal dropout by improving
magnetic field homogeneity include localised volume shimming (22,23), passive shims
made from diamagnetic material placed in the mouth or ear canals (24-26), localised
active shimming using resistive coils placed in (27) or near (28) the subject's mouth and
dynamic shim updating (29-31). However each approach had its drawbacks; these
include reduced signal outside the shimmed volume, reduced patient comfort (32), and
the need for dedicated hardware (33). Dual echo EPI, which combines the acquisition of
gradient and spin echo images in a single shot, has also been used, although at a cost
of reduced temporal resolution (34-36). Approaches based on modifying the sliceselection process have included tailored radiofrequency (TRF) pulses (37-39), zshimming (40-49) and their combination (50,51). These can reduce the signal dropout
caused by through-slice susceptibility gradients, but result in reduced temporal
3
resolution and/or signal-to-noise ratio (SNR). Additionally signal dropout caused by inplane susceptibility gradients has been reduced by the addition of compensatory
gradients in the frequency and phase encoding directions (24,52-55).
The current study is based on the TRF pulse approach for reducing signal dropout in
GE-EPI images. As previously demonstrated, the phase dispersion caused by linear
through-slice susceptibility gradients, can be partially cancelled by an RF pulse that
induces a quadratic variation in the phase of the transverse magnetisation (37,38).
Given that the functional form of the TRF pulse was not specified by either Cho et al.
(37), nor Chung et al. (38) we extend the work of Shmueli et al. (39,56), using complex
hyperbolic secant (HS) pulses for signal excitation. HS pulses produce an
approximately quadratic variation in the phase of the transverse magnetisation in the
slice-selection direction (57-59) and hence can be used to reduce signal dropout.
We describe a systematic approach to designing HS pulses for signal recovery in GEEPI images. Bloch simulations are used to determine the HS pulse parameters required
to produce a uniform signal response across the range of susceptibility gradients
typically observed in the head. The limitations imposed on the RF pulse amplitude and
imaging gradient parameters by the MRI scanner hardware are accounted for and an
expression for the bandwidth of a HS pulse (when used for signal excitation) is derived
and used for the first time. The impact of the optimised HS pulse on signal, temporal
signal-to-noise ratio, BOLD sensitivity and detectability of resting-state FMRI networks
is assessed in six healthy male subjects at 3 Tesla.
Theory
It is well known that a linear susceptibility gradient in the direction of slice selection,
Gsus, induces a linear variation in the phase, (z), of the transverse magnetisation, Mxy,
across the slice that is proportional to the echo time (TE) (37):
𝜙(𝑧) = 𝛾𝐺𝑠𝑢𝑠 𝑧𝑇𝐸
4
[1]
This phase dispersion can be compensated for in part of the slice, using a TRF pulse
that induces a quadratic variation in the phase of Mxy (37):
𝜙(𝑧) = 𝑎𝑧 2
[2]
Here a is the design parameter used to tailor degree of quadratic phase variation in the
phase of Mxy. Based on the assumption that the excited slice has a perfectly rectangular
profile, it has been be shown that the signal S, acquired at an echo time TE, in the
presence of a linear susceptibility gradient Gsus, from a voxel with thickness z, using a
TRF pulse is (37):
2
Δ𝑧/2
𝑆 = 𝑀0 √[∫
cos(𝑎𝑧 2
2
Δ𝑧/2
+ 𝛾𝑇𝐸𝐺𝑠𝑢𝑠 𝑧) 𝑑𝑧] + [∫
−Δ𝑧/2
sin(𝑎𝑧 2
[3]
+ 𝛾𝑇𝐸𝐺𝑠𝑢𝑠 𝑧) ⁡𝑑𝑧]
−Δ𝑧/2
Numerical solutions of the signal model given in Equation 3 have previously been used
to design TRF pulses and to highlight the trade-off, controlled by a, between signal
recovery in regions affected by signal dropout and loss of signal in areas unaffected by
susceptibility gradients (37,38). Since the assumption that slices are perfectly
rectangular is not possible to satisfy in practice, and because the functional form of the
TRF pulse was not specified in previous studies (37,38) we describe a method, based
on numerical simulations of the Bloch equations, to determine the HS pulse parameters
required for signal recovery.
A HS pulse with duration TRF has both an amplitude A(t) and phase (t) modulation:
𝐴(𝑡) = 𝐴0 sech(𝛽𝑡)
[4]
𝜙(𝑡) = 𝜇 ln[sech⁡(𝛽𝑡)] + 𝜇 ln 𝐴0
[5]
for -TRF/2 < t < TRF/2. Here A0 is the maximum amplitude of the pulse,  is the
modulation angular frequency and  is a dimensionless parameter that both determines
the sharpness of the slice profile (60) and controls the degree of quadratic phase
induced in Mxy (39,56).
5
As shown in Appendix 1, the maximum amplitude, A0, of an HS pulse used for signal
excitation is a function of the flip angle, , modulation angular frequency,  and :
2
−1 [cos α cosh2 (𝜋𝜇 ) + sinh2 (𝜋𝜇 )]
cos
𝛽
2
2 } + 𝜇2
𝐴0 = √{
𝛾
𝜋
[6]
In the previous work using HS pulses for signal excitation (39,56) it was assumed that
the pulse bandwidth f = / (as for adiabatic inversion (60)). However, the bandwidth
of an excitation pulse is normally defined as the full-width at half-maximum
(FWHM) of the magnitude of the transverse magnetisation, |Mxy|. In this case, as shown
in Appendix 2, the bandwidth is:
1
cosh(𝜋𝜇) [cos⁡𝛼 − 2 √3 + cos 2 𝛼] + cos 𝛼 − 1
β
Δ𝑓 = 2 cosh−1 {
}
1
𝜋
2𝛼−1
+
cos
√3
2
[7]
i.e. it is dependent on the flip angle .
Methods
Hyperbolic Secant (HS) Pulse Design Algorithm
Using conventional GE-EPI, the ability to detect brain activations of equal magnitude
varies across the brain because susceptibility gradients cause signal loss and
reductions in the BOLD sensitivity. To reduce this variability we propose an algorithm to
optimise the parameters of an HS pulse (, , A0, and f) to ensure that the signal
response across a range of through-plane susceptibility gradients is as uniform as
possible. We quantify the signal uniformity is using the ratio of the mean signal, 𝑠̅, over
the range Gsus,min < Gsus < Gsus,max to the standard deviation of the signal, s, over the
same range (i.e. 𝑠̅ /𝜎𝑠 ).
The algorithm requires, as inputs, the following properties about the object being
imaged: the longitudinal relaxation time, T1, the transverse relaxation time, T2, and the
range of susceptibility gradients in the direction of slice selection, Gsus,min < Gsus <
6
Gsus,max, over which the signal dropout is to be reduced. Additionally it requires the
following properties of the scanner hardware: the maximum obtainable gradient
amplitude in the direction of slice selection, Gz,max, and the maximum radiofrequency
amplitude that can be produced, B1,max.It also requires a number of parameters of the
pulse sequence that will be used to acquire the MRI data, specifically: the repetition
time, TR, the echo time, TE, the slice thickness, z, as well as the RF pulse duration,
TRF.
The flip angle, , of the RF pulse is set to the Ernst angle (61) to maximise the steadystate signal (although this is not a requirement of the algorithm, and other values can be
used, if appropriate). The signal, s, at each value of⁡ in the range 2 ≤ ≤ 8 (The search
space of 𝜇 was set from preliminary simulations, and previous studies (39,56)) and Gsus
is then determined, for this particular value of  using the following six steps:
1. The value of  is set to its maximum value to minimise stop-band ripple in the
slice profile (56). The value of  is limited by both B1,max and Gz,max, so it is set to
the smallest of values given by expressions 8 and 9:
𝛾𝐵1,𝑚𝑎𝑥
[8]
𝜋𝜇
𝜋𝜇 2
cos −1 [cos 𝛼 cosh2 ( 2 ) + sinh2 ( 2 )]
√{
} + 𝜇2
𝜋
𝜋𝛾Δ𝑧𝐺𝑧,𝑚𝑎𝑥
1
cosh(𝜋𝜇) (cos⁡𝛼 − 2 √3 + cos 2 𝛼) + cos 𝛼 − 1
−1
2 cosh (
)
1
2
+
cos
𝛼
−
1
2 √3
[9]
2. The peak amplitude of the RF pulse, A0, is set using Eq. 6.
3. The frequency bandwidth of the RF pulse, f, is calculated using Eq. 7.
4. The amplitude of the trapezoidal slice selection gradient and the area of the sliceselection refocussing gradient are calculated using:
7
𝐺𝑧 =
2𝜋Δ𝑓
𝛾Δ𝑧
[10]
𝑇𝑅𝐹 + 𝑇𝑟𝑎𝑚𝑝
𝐴𝑟𝑒𝑓 = −𝐺𝑧 (
) − 𝐺𝑠𝑢𝑠,𝑚𝑒𝑎𝑛 𝑇𝐸
2
[11]
Here Tramp is the duration of the slice selection gradient ramps and
Gsus,mean = (Gsus,max + Gsus,min) / 2. The dependence of the slice refocussing
gradient area on the range of susceptibility gradients is essentially the same as
for z-shimming. It allows signal recovery from regions with a range of
susceptibility gradients not centred on zero.
5. Given the slice-selection gradient, slice-refocusing gradient, susceptibility
gradient, Gsus, along with T1 and T2, and the now-known parameters determining
the shape of the HS pulse, the steady-state values of the x and y components of
the transverse magnetisation, mx and my, are determined at the echo time at
Nz(=101 in the examples shown) spatial positions equally spaced in the range –
z ≤ z ≤ z, using Bloch simulation (62). This range of z deliberately
encompasses twice the slice width such that the impact of the imperfect slice
profile on the total voxel signal can be accounted for in the following step.
6. The voxel signal, 𝑠, is calculated by numerical integration of mx and my using:
Δ𝑧
𝑠 = √[∫
2
Δ𝑧
𝑚𝑥 (𝑧)𝑑𝑧⁡⁡] + [∫
−Δ𝑧
2
[12]
𝑚𝑦 (𝑧)𝑑𝑧⁡⁡⁡] ⁡⁡
−Δ𝑧
The limits of integration in Eq. 12 deliberately encompass twice the slice width
such that the impact of the imperfect slice profile is accounted for.
The mean 𝑠, and standard deviation of the signal s, are calculated over the range
Gsus,min < Gsus < Gsus,max. Given optimal (the value of  at which the uniformity (𝑠̅/𝜎𝑠 ) is
maximised) the value of  is determined using step 1 above, the value of A0 is using Eq.
6; the value f using Eq. 7, and the values of Gz and Aref are determined using Eqs. 10
8
and 11. Together these constitute the optimal pulse parameters for this range of
susceptibility gradients, scanner hardware limits and pulse sequence parameters.
Simulation of Slice Profile of an Exemplar HS Pulse
To demonstrate the effectiveness of the pulse design algorithm, an HS pulse was
designed specifically for FMRI data acquisition in the human head using a GE-EPI
sequence on a 3 T GE Discovery MR750 system (General Electric, Waukesha, WI,
USA). In this case, the inputs to the algorithm were: T1 = 1.6 s, T2 = 66 ms (for cortical
grey matter at 3T (63,64)), -250 Tm-1 < Gsus < 250 Tm-1 (53,65), Gz,max = 36 mTm-1
(lower than the hardware limit of 50 mTm-1 in order to reduce acoustic noise and
vibration), B1,max = 20 T, TR = 2 s, TE = 30 ms, z = 3 mm and TRF = 5 ms. The pulse
duration was chosen to match the excitation pulse used as standard on the GE
Discovery MR750 system to ensure that approximately the same number of slices could
be collected during each TR period. The flip angle was set to the Ernst angle  = 73°
Phantom Validation of an Exemplar HS Pulse
To validate the algorithm described above and to demonstrate the ability of HS pulses
to reduce the signal loss resulting from through-plane susceptibility gradients a series of
images were obtained of uniform spherical phantom (T1 = 170 ms and T2 = 25 ms part
number: 2359877, General Electric, Waukesha, WI, USA). All data were acquired using
the same 3T system as above. A quadrature head coil was used for signal transmission
and reception. Initially the scanner was shimmed using the in-built automatic procedure.
To model the effects of different through-plane linear susceptibility gradients, the shim
gradient in the slice-selection direction was then deliberately mis-set to values in the
range -500 Tm-1 < Gsus < 500 Tm-1. At each setting of the shim gradient a single 3
mm axial slice with a field-of-view of 32 cm and a 64×64 acquisition matrix was acquired
with a TR = 5 s and TE = 30 ms using an HS pulse with TRF = 5 ms,  = 90°,  = 4.25, 
= 3040 Hz. The TR and flip angle were chosen to avoid steady-state effects. The
quadrature coil and large field-of-view were selected to enable straightforward
measurements of the signal and background noise. The signal from a circular region-ofinterest (with a radius half of that of the phantom) in the centre of the phantom as a
9
function of the `susceptibility' gradient (induced by mis-setting the shim) was calculated
using FSL (FMRIB's Software Library - www.fmrib.ox.ac.uk/fsl). This was compared to
the voxel signal determined by Bloch simulation (scaled to match the phantom data).
In Vivo Validation
Data Acquisition
A series of scans were performed on six healthy male volunteers (five right handed, one
left handed) to determine the impact of the HS pulse on the signal, temporal signal-tonoise ratio (TSNR) (66,67), BOLD sensitivity and the ability to detect resting-state BOLD
signal changes. The scanning of healthy volunteers was carried out under an approval
from the London Camberwell St Giles National Research Ethics System Committee
(“Development of Magnetic Resonance Imaging and Spectroscopy Methods” study
reference: 04/Q0706/72). All data were acquired using the same 3T system as above
with an eight-channel phased array head coil for signal reception and the body coil for
RF transmission. Thirty-six 3 mm slices with 0.3 mm slice gaps were prescribed parallel
to the line intersecting the anterior and posterior commissure for all scans. The field-ofview was 21.2 cm and the acquisition matrix was 64 × 64. The subjects' breathing
pattern was tracked using a respiratory bellows and a pulse oximeter was used to
monitor cardiac activity throughout. FMRI paradigms were presented using a projector
and screen at the rear of the scanner bore viewed via a mirror attached to the head coil.
A pair of resting-state functional MRI scans was acquired using a conventional GE-EPI
sequence and GE-EPI with the HS pulse. The conventional GE-EPI acquisition used a
Shinnar-Le Roux (SLR) RF excitation pulse. In both cases a CHESS pulse (68) was
used for fat suppression. For both scans TR = 2 s, TE = 30 ms, the flip angle was 73°
and the ASSET acceleration factor was 2. Slices were collected top-down sequentially.
The order of the two scans was counterbalanced across the six subjects. For each
acquisition four hundred and fifty volumes of data (15 minutes) were acquired, preceded
by four dummy acquisitions, whilst the subject was at rest. Subjects were instructed to
keep their eyes open and to look at a cross hair.
10
Following previous work, in which alternative methods were presented to reduce signal
dropout (22,65), FMRI with a breath-hold paradigm was used to assess changes in the
BOLD sensitivity [28] caused by the HS pulse. The breath-hold task causes a
hypercapnic stress, similar to carbon dioxide inhalation (69). This reliably increases
cerebral blood flow (CBF), and hence causes global increases in the BOLD signal
across grey matter. As before, a pair of functional MRI scans was acquired during which
the subjects performed a breath-hold task. The same scan parameters were used as for
the resting-state task, and the order was again counterbalanced across subjects. For
each breath-hold experiment one hundred and fifty two volumes of data (5 minutes and
4 seconds) were acquired. The subject was visually cued to perform interleaved blocks
of paced breathing (48 s) and breath holding on expiration (16 s) finishing with a block
of paced breathing (48 s).
Data Analysis
All imaging data analysis was carried out using FSL. TSNR maps were calculated from
the resting-state data sets. For each subject, to remove the effect of subject motion, all
of the volumes from both acquisitions were registered to the first volume of the
corresponding conventional GE-EPI data using MCFLIRT (70). The brain was extracted
using BET (71). The data were high pass filtered using a Gaussian weighted leastsquares line fit with a cut-off = 50 s (0.01Hz) to remove signal drifts (72). TSNR was
calculated voxel-wise as the ratio of the temporal mean to the temporal standard
deviation of the resting-state FMRI data sets. Subject specific maps of the percentage
change in TSNR between the data acquired with conventional GE-EPI and GE-EPI
were then calculated.
BOLD sensitivity was assessed using the FMRI data sets acquired whilst the subject
performed the breath-hold task. Motion correction was performed using MCFLIRT. The
brain was extracted using BET and the resulting data were spatially smoothed using a
Gaussian kernel with a 5 mm FWHM. The data were then scaled, by a single
11
multiplicative factor such that the overall mean signal was 10000. The time series from
each voxel was temporally high pass filtered using a Gaussian weighted least-squares
line fit, with a cut-off = 50 s. The regions of the brain showing significant changes in
BOLD signal in response to the breath-hold stimulus were found using FILM (73).
Specifically the box-car design, with a delay of 8 s was convolved with a Gaussian
function with a standard deviation of 7.48 s and peak lag of 5 s (22,65). This was fitted
to the pre-processed time series signals using the general linear model (GLM) with local
autocorrelation correction. To reduce the impact of head motion six covariates from the
motion correction procedure were added to the model. This resulted in an unthresholded t-statistic map for each subject and each acquisition method. Subject
specific maps showing the difference in the t-statistic between the two acquisition
methods were then calculated. These were masked to show only regions where the GEEPI signal increased when the HS pulse was used in place of the SLR pulse. The signal
from the respiratory bellows (not shown) was inspected for both acquisition types to
ensure each subject performed the task as instructed.
The effect of the HS pulse on the ability to detect resting-state BOLD signal changes
was assessed using the resting-state FMRI data. Motion correction and brain extraction
were performed as above, and the resulting data were spatially smoothed using a
Gaussian kernel with a 6 mm FWHM as suggested by Van Dijk et al. (74). The data
were band pass filtered (0.01 to 0.08 Hz) to remove the effect of signal drifts and to
reduce the impact of cardiac and respiratory noise (75). The data were scaled, by a
single multiplicative factor such that the overall mean signal was 10000. The spatial
transformations needed to register each functional data set into MNI space were
calculated using FLIRT (70,76). Seed-based regression analyses were performed in
each subject’s native space to determine whether the fluctuations in the resting-state
signals in the regions of recovered signal in the orbitofrontal and inferior temporal
regions were correlated with fluctuations from a seed region placed in the default mode
network. Three 4mm spherical regions of interest were defined in MNI space. The first
ROI was placed in the posterior cingulate (x = 0 mm, y = – 53 mm, z = 26 mm) (74), a
node of the default mode network (77). The second and third ROIs were placed in
12
“control” regions, not expected to give a BOLD signal: in the lateral ventricle (x = 27
mm, y = – 8 mm, z = 32 mm) and an area of white matter (x = –19 mm, y= – 36 mm, z =
17 mm), respectively. The three ROIs were transformed from MNI standard-space into
each subject's native space using the inverse transformations calculated in the preprocessing stage. The mean time courses from each of these regions were then
extracted. The regions of the brain where the resting-state BOLD signal was correlated
with signal changes in the posterior cingulate ROI were found using a GLM with local
autocorrelation correction, as implemented in FILM (73). To reduce the impact of head
motion six covariates from the motion correction procedure were included in the model.
In addition, covariates from the ROI in the ventricle, white mater and global brain signal
were also included, in order to reduce the effect of physiological noise (74,78). The
resulting z-statistic maps were thresholded using clusters determined by z > 2.3 and a
corrected cluster significance threshold of P = 0.05.
Results
Simulation of Slice Profile of an Exemplar HS Pulse
The result of the optimisation procedure was an HS pulse with parameters:
 = 4.25,  = 3040 Hz, A0 = 12.3 T, f = 4598 Hz, a trapezoidal slice-selection gradient
with amplitude Gz = 36 mTm-1 and a trapezoidal slice-refocusing gradient of area Aref = 0.093 s mTm-1.
The amplitude and phase modulation of the optimised HS pulse and the accompanying
slice-selection gradient are shown in Figure 1. Bloch simulations of the steady-state
slice profile and phase variation for grey matter (with TE, TR, T1 and T2 as per the
design parameters using the optimised HS pulse are shown in Figure 1 a-c. Bloch
simulations of the normalised steady-state voxel signal as a function of the throughplane susceptibility gradient are shown in Figure 1 d-e. As before, the signal for each
value of Gsus was calculated numerically using Eq. 12 from the transverse
13
magnetisation. This was normalised relative to the steady-state signal from a perfectly
rectangular slice with thickness z.
The simulation results in Figure 2 show that (as expected) the HS pulse produces a
lower signal than a conventional RF pulse without quadratic phase variation in regions
of relatively low susceptibility gradient, but recovers signal at higher offsets. For
susceptibility gradients less than ±154 Tm-1, the signal from the HS pulse is reduced to
between 48.2 and 51.8% of the signal from a conventional RF pulse. Signal is
recovered for through-plane susceptibility gradients more extreme than ±154 Tm-1,
with the HS pulse providing a highly uniform normalised voxel signal for susceptibility
gradients in the design range (i.e. -250 Tm-1 < Gsus < 250 Tm-1 ), and appreciable
signal recovery even well outside this range.
Phantom Validation of an Exemplar HS Pulse
Figure 3 confirms that in a phantom, the pulse produces a near uniform signal for
susceptibility gradients in the range -250 Tm-1 < Gsus < 250 Tm-1. Additionally it is
clear that the variations in signal from the phantom experiments and simulations
(including the asymmetry in the pulse response) closely match, validating the use of
Bloch simulation in pulse design algorithm.
In Vivo Validation
Figure 4 shows representative slices through the orbitofrontal and inferior temporal
regions of each subject from data acquired using GE-EPI with the conventional SLR
excitation pulse and the HS pulse. Comparing the two sets of images acquired using the
HS pulse to their SLR equivalents, it can be seen that signal is partially recovered in the
orbitofrontal (OF) and inferior temporal (IT) regions in all six subjects. However, as
expected from the Bloch simulations described above, there is a visible reduction in the
global signal-to-noise ratio in the images collected using the HS pulse.
14
Figure 5 shows maps of the TSNR for each subject for data acquired with the SLR and
HS excitation pulses, again for slices through the OF and IT regions. Figure 6 shows
maps of the percentage change in TSNR. The TSNR in regions affected by
susceptibility gradients increases to a level comparable to unaffected voxels when the
HS pulse is used. However, the HS pulse results in decreases in TSNR of up to 60% in
large areas of the brain.
Figure 7 shows the result of the breath-hold experiment used to determine the changes
in BOLD sensitivity as a result of the HS pulse. It shows subject specific maps of the
difference in the un-thresholded t-statistic between the two acquisition methods. For
clarity, and to highlight changes BOLD sensitivity in the regions of recovered signal, the
difference map is masked to show only regions where the GE-EPI signal increased
when the HS pulse was used in place of the SLR pulse. The HS pulse results in
increases in the t-statistic in the majority of voxels in the areas of signal recovery in the
OF and IT regions for all six subjects. Inspection of the signal from the respiratory
bellows (not shown) demonstrated each subject performed the breath-hold task as
instructed.
Figure 8 shows the impact of the HS pulse on the result of a seed based analysis of the
resting-state FMRI data. When the HS pulse is used there is a significant correlation in
all six subjects of the resting-state BOLD signal in the region of signal recovery in the
orbitofrontal cortex with a seed in the posterior cingulate. As highlighted in the figure this
region of correlated signal is not observed in the data acquired with the conventional
SLR pulse.
Discussion and Conclusion
Building on the previous experiments using tailored RF pulses (37-39) we have
developed an algorithm to systematically optimise the parameters of a HS pulse to
recover signal in regions of the brain affected by susceptibility induced dropout. In
contrast to previous approaches to TRF design, which used either trial-and-error or an
15
analytic model based on unrealistic assumptions of the slice profile, Bloch simulations
were used to determine the HS pulse parameters. The effectiveness of these
simulations was confirmed in phantom experiments. In addition an expression for the
bandwidth of a HS excitation pulse was derived for the first time, enabling the amplitude
of the slice selection gradient to be calculated correctly.
The algorithm was used to optimise the parameters of an HS for typical FMRI
experiments at 3T. A series of experiments were then performed in six healthy
volunteers to assess whether these improvements in signal translated to improvements
in the TSNR, BOLD sensitivity and detectability of resting-state BOLD signal changes.
The results demonstrate the potential benefits of the using HS pulses for signal
excitation in FMRI experiments.
Signal was recovered in parts of the OF and IT regions. The areas of unrecovered
signal may be caused by susceptibility gradients in the frequency and phase encoding
directions. As predicted by Bloch simulations of the pulse, the localised signal recovery
comes at the cost of approximately 50% loss of signal in regions of the brain unaffected
by through-slice susceptibility gradients. These changes in signal translate to increases
in TSNR in the OF and IT regions, although these are accompanied by losses of up to
60% in TSNR elsewhere. Additionally in the majority of voxels where signal was
recovered in the OF and IT regions the sensitivity to BOLD signal changes also
increased.
Seed-based analysis of the resting state FMRI data suggests that regions of the
orbitofrontal cortex, normally obscured by signal dropout, may be functionally connected
to the default mode network. This result, of clear importance to the modelling of such
networks across the whole brain, is in agreement with the findings of Dalwani et al. (79).
The optimised HS pulse results in signal and TSNR losses in regions of homogeneous
field, but allows detection of apparently meaningful BOLD signal changes in regions
inaccessible to conventional approaches. In addition, unlike many of the previous
16
approaches to reducing signal dropout, our optimised HS pulse does not compromise
temporal resolution or spatial coverage, is non-invasive and does not require additional
specialist hardware, although larger group sizes and/or longer acquisitions may be
needed to overcome the resulting global reduction in signal. As such, while we expect
our approach to have applications for task based fMRI studies hypothesising effects in
“problematic” OF and IT regions (e.g. studies of olfaction, memory and the processing
of language, rewards and emotional facial expressions.). We believe it is particularly
suitable for event related and resting-state FMRI as it does not cause a loss of temporal
resolution.
Appendix 1: Derivation of the Peak RF Amplitude a Complex Hyperbolic Secant
Pulse Used for Signal Excitation
The maximum amplitude, A0, of a complex hyperbolic secant pulse used for signal
excitation, with a flip angle, , can be derived from the expression for the longitudinal
magnetisation given in Eq. 17 of Silver et al. (60):
𝑀𝑧 (Δ𝜔)
𝜋𝛥𝜔 𝜋𝜇
𝜋𝛥𝜔 𝜋𝜇
= tanh (
+ ) tanh (
− )
𝑀0
2𝛽
2
2𝛽
2
[13]
𝛾𝐴0 2
𝜋𝛥𝜔 𝜋𝜇
𝜋𝛥𝜔 𝜋𝜇
√
+ cos [𝜋 (
) − 𝜇 2 ⁡] sech (
+ ) sech (
− )
𝛽
2𝛽
2
2𝛽
2
Here  is the offset frequency, which in the presence of a slice-selection gradient
is equal to Gzz. At the centre of the slice  = 0, and the z-magnetisation is:
𝑀𝑧 (Δ𝜔)
𝛾𝐴0 2
𝜋𝜇
𝜋𝜇
√
= cos [𝜋 (
) − 𝜇 2 ⁡] sech2 ( ) − tanh2 ( )
𝑀0
𝛽
2
2
[14]
Recognising that the z-magnetisation at the slice centre can also be written in terms of
the flip angle, :
𝑀𝑧 (Δ𝜔)
= cos 𝛼
𝑀0
17
[15]
Then:
𝛾𝐴0 2
𝜋𝜇
𝜋𝜇
√
cos⁡(𝛼) = cos [𝜋 (
) − 𝜇 2 ⁡] sech2 ( ) − tanh2 ( )
𝛽
2
2
[16]
Therefore the RF amplitude as a function of ,  and the flip angle, , is:
2
−1 [cos(𝛼) cosh2 (𝜋𝜇 ) + sinh2 (𝜋𝜇 )]
cos
𝛽
2
2 } + 𝜇2
𝐴0 = √{
𝛾
𝜋
[17]
Appendix 2: Derivation of the Bandwidth of a Complex Hyperbolic Secant Pulse
Used for Signal Excitation
The bandwidth of a complex hyperbolic secant pulse used for signal excitation is best
defined as the full-width at half-maximum (FWHM) of the magnitude of the transverse
magnetisation, |Mxy|. This can be derived from Eq. 13. The cosine term including the
pulse amplitude, A0, can then be written in terms involving the flip angle, , and  using
Eq. 16:
𝑀𝑧 (Δ𝜔)
𝜋𝛥𝜔 𝜋𝜇
𝜋𝛥𝜔 𝜋𝜇
= tanh (
+ ) tanh (
− )
𝑀0
2𝛽
2
2𝛽
2
[18]
2 𝜋𝜇
𝜋𝛥𝜔 𝜋𝜇
𝜋𝛥𝜔 𝜋𝜇 cos 𝛼 + tanh ( 2 )
+ sech (
+ ) sech (
− )[
]
𝜋𝜇
2𝛽
2
2𝛽
2
sech2 ( 2 )
Simplifying the tanh and sech terms using standard trigonometric identities:
𝜋Δ𝜔
𝑀𝑧 (Δ𝜔) cosh ( 𝛽 ) + cosh(𝜋𝜇) cos(𝛼) + cos(𝛼) − 1
=
𝜋Δ𝜔
𝑀0
cosh (
) + cosh(𝜋𝜇)
𝛽
2
Rearranging to make  the subject (and using 𝑀02 = |𝑀𝑥𝑦 | + 𝑀𝑧2 ):
18
[19]
[20]
2
Δ𝜔 =
|𝑀𝑥𝑦 |
cosh(𝜋𝜇) [cos(𝛼) − √1 − ( 𝑀 ) ] + cos(𝛼) − 1
0
𝛽
cosh−1
𝜋
2
|𝑀 |
√1 − ( 𝑥𝑦 ) − 1
𝑀0
{
}
At the centre of the excited slice, for a flip angle , the magnitude of the transverse
magnetisation is |𝑀𝑥𝑦 |/𝑀0 = sin 𝛼.The half-width at half-maximum (HWHM) is the value
of  at which |𝑀𝑥𝑦 |/𝑀0 =
Δ𝜔𝐻𝑊𝐻𝑀
sin 𝛼
2
i.e.:
1
cosh(𝜋𝜇) [cos(𝛼) − 2 √3 + cos 2 (𝛼)] + cos(𝛼) − 1
β
−1
= cosh {
}
1
𝜋
2
√3 + cos (𝛼) − 1
2
[21]
Therefore the bandwidth, defined as the FWHM, in Hz is:
1
cosh(𝜋𝜇) [cos(𝛼) − √3 + cos 2 (𝛼)] + cos(𝛼) − 1
β
2
Δ𝑓 = 2 cosh−1 {
}
1
𝜋
√3 + cos 2 (𝛼) − 1
2
[22]
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24
Figures
Figure 1. (a) Amplitude and (b) phase modulation of the optimised HS pulse with (c) the
accompanying slice selection and slice refocussing gradients. Steady-state (d) slice
profile and (e) phase variation in grey matter of the optimised HS pulse.
Figure 2. Simulated normalised steady-state grey matter voxel signal as a function of
through-slice susceptibility gradient for the optimised HS pulse (blue) compared to a
conventional RF pulse without quadratic phase variation (green)
25
Figure 3. Variation in the mean signal from an ROI in the centre of the phantom as a
function of susceptibility gradient for a HS pulse. The error bars represent the standard
deviation of the signal in the ROI. The (scaled) voxel signal calculated by Bloch
simulation is shown for comparison.
26
Figure 4. Representative slices through the OF regions of six subjects from images
acquired with GE-EPI with the SLR (row 1) and HS (row 2) RF pulses and through the
IT with the SLR (row 3) and HS (row 4) RF pulses. Regions of signal recovery in the
OF, right and left IT regions are highlighted with connected red, yellow and green circles
respectively.
27
Figure 5. Representative slices from TSNR maps through the OF regions of six subjects
from images acquired with GE-EPI with the SLR (row 1) and HS (row 2) RF pulses and
through the IT with the SLR (row 3) and HS (row 4) RF pulses. Regions of TSNR
recovery in the OF, right and left IT regions are highlighted with connected red, yellow
and green circles respectively.
28
Figure 6. Percentage change in TSNR in six subjects in representative slices through
the OF (row 1) and IT (row 2) regions.
Figure 7. Subject specific changes in BOLD sensitivity quantified by the difference in the
un-thresholded t-statistic maps from the breath-hold experiment. Representative slices
through the OF and IT regions are shown for all six subjects
29
Figure 8. Thresholded z-statistic maps, showing voxels in which the resting-state BOLD
signal changes were significantly correlated with the signal variation from a seed in the
posterior cingulate. Maps are shown for each subject acquired with the SLR and HS
pulse for representative slices though the OF region. Changes in correlation in the OFC
are highlighted by green circles.
30