Unconventional RF Pulses
and Pulse Pairs
John M. Pauly!
Stanford University
Typical Spin Echo Pulse
Sequence
90
180
RF
TE
A/D
Echo
Pulses designed independently!
Echo time limited!
Large dynamic range, demanding for RF amp
Designing Spin Echoes
0.4
RF Pulses
0.2
90
180
0
−0.2
−0.4
0
Echo
TE
2
4
6
8
10
12
Time, ms
0.3
Design
theEcho
pulses!!
FID the spin echo instead ofSpin
0.2
Matched pulse pairs produce a linear phase echo!
0.1
Non-linear
phase pulses have lower peak power!
0
Much more
selective,
less
demanding
2
4
6
8
10for the
12 RF amp
−0.1
0
Short Echo Times
The pulses can
overlap, allowing
short spin echo
times!
F
Arbitrary flip
angles!
d
e
h
o
e
F
-
-0.5'
-5
-4
-3
-2
Time, ms
-1
I
Phase cycles
0
Lim, et al. Magnetic Resonance in Medicine 32:98︲103 (1994)
Slice Selective Excitation
RF
B1(t)
F{B1(t)}
ω
ω(x) = γGxx
Gz
Choose B1(t) with a
limited range of
frequencies
x
Slice Profile
Small-Tip-Angle Approximation
∆Mxy = γB1(t)dt
RF
k(t) = γ
Z
γB1(t)e
dt
Gz
Mxy(x) =
Z
0
T
ik(t)x
T
t
G(s)ds
Designing Small-TipAngle Pulses
Window
Windowed Sinc
T t
Key Parameters:!
Duration, T!
Number of zeros,
N (here 8)!
Slice Profile
RF Pulse:
Slice Profile:
N zeros
F
T
Width!
N/T
Transition Band!
2/T
N is the Time-Bandwidth Product of the pulse
Large-Flip-Angle Pulses
Fourier-based designs work well to 90°
Mxy
30° Excitation
Mxy
90° Excitation
Mxy
180° Refocusing
Non-linear problem beyond 90°
Spin Domain
Solve for rotations instead of magnetization!
Rotation represented by 2x2 unitary matrix
✓
◆
⇤
α β
Q=
β α⇤
α and β are
α = cos( /2) - inzsin( /2)
β = -i(nx+iny)sin(
Rotation is by angle
/2)!
about axis (nx,ny,nz)T!
Magnitude constraint: αα*+ ββ* = 1
Magnetization from Spin
Domain
Simple to compute slice profiles from spin domain:
*
Mxy = 2α β M0
Mz = (1-2ββ*) M0
Mxy+ = -β2(Mxy-)*
Excitation
Inversion/Saturation
Refocusing
Hard Pulse Approximation
Represent RF as impulses separated by
free precession intervals (discrete time
approximation)!
Spinors are two polynomials AN(z) and
BN(z) in z=eiγGxΔt where Δt is sampling time!
This is invertible, given AN(z) and BN(z) we
can find B1(t)
Shinnar-Le Roux Algorithm
Design BN(z) to approximate the desired
flip angle profile
β(x) = -i(nx+iny) sin( (x)/2)
using a Fourier design (filter design).!
Find a consistent AN(z) using magnitude
constraint, minimum power (min phase)!
Solve for B1(t)
B1(t) = SLR-1(AN(z),BN(z))
Example: Spin Echo Pulse
Desired β profile
1
BN(z) coefficients
FT
sin( (x)/2)
x
BN(z)
SLR
Bloch Eq
RF Pulse
t
1
-1
N
My
Mx
x
Spin Echo Profile
Other Phase Profiles
Linear Phase
Minimum Phase
1
Inversion Profile
0
-1
Minimum phase pulse can have half the
transition width
x
Other Phase Profiles
Minimum Phase
Non-linear Phase
31% peak power
Linear, minimum phase: high peak power!
Optimized non-linear phase reduces peak power!
Same total power (SAR)!
Identical magnitude profile
Designing Spin Echoes
(α90,β90)
(α180,β180)
RF
TE
A/D
Mxy at the echo is
Mxy = (2α90β*90)(-β2180) M0!
Choose
β90 =(1/√2)(iβ2180)
Matches profile and phase
Mxy ≃ |β180|4 M0
Echo
Non-Linear Phase Example
0.4
Phases exactly
cancel!
RF Pulses
0.2
90
180
0
−0.2
−0.4
0
2
4
6
8
10
12
Time, ms
0.3
FID
Spin Echo
90 is twice as
long as 180
0.2
0.1
0
−0.1
0
2
4
6
Time, ms
8
10
Perfect linear
phase spin
echo!
12
Near-Contiguous
Spin Echoes
Sinc Pulses
SLR Pulses
Used for arterial spin labeling,
where the sharp transition is
important
Zun, et al. Magn. Reson. in Med, online (2013)
Pulse Pair
CPMG Condition
Non-Linear
Phase CPMG
90x
180y
180y
180y
RF
Gz
z
x
Imaging!
Volume
x
x
x
y
y
y
y
Goal:
Different absolute
phase at each z
position
CPMG Condition
Non-Linear
Phase CPMG
90x
180y
180y
180y
RF
Gz
z
x
Imaging!
Volume
x
x
x
y
y
y
y
Goal:
Different absolute
phase at each z
position
Echo
Compensation
CPMG
Echoes
90x
180y
180y
180y
RF
Gz
A/D
(2α*90β90)M0
MRSRL
(2α90β*90)(-β2180) M0
Choose
β90 = (1/√2)(iβ180)
=> Mxy,90 ≃ iβ180 M0
Mxy ≃ iβ180 |β180|2 M0
Echo has same phase profile as initial Mxy
Stanford Electrical Engineering
Non-Linear Phase CPMG
Minimum Dynamic Range Pulses
90x
180y
180y
180y
Echoes all have
same profile and
polarity!
RF
Gz
1
Works for any
phase profile,
including frequency
sweeps (SPEN)
B1(t),kHz
B1(t),kHz
1
t
4t
Even/Odd 4Echo
Comparison
1
Echo 3 Slice Profile
1
RSRL
Echo 4 Slice Profile
0.5
Magnetization
Magnetization
Stanford Electrical Engineering
0
- 0.5
-1
-10
0.5
0
- 0.5
-5
0
Position
5
10
-1
-10
-5
0
Position
5
10
he
he
wo
a
diooRF
e.
ed
ee
th
ho
ue
on
RF
n-
Overlapping Pulses
The two pulses don’t need to
be distinct!
Give (α90,β90) and (α180,β180)
βPP = zD β90 α*180 + α90 β180
αPP = z-D α90 α180 - β90 β*180!
-0.5'
-5
-4
-3
-2
Time, ms
-1
2.5 ms Echo Time!
(immediately after end of
pulse)
I
0
D is the echo delay (in
samples)!
Find B1(t) via SLR back
recursion
Lim, et al. Magnetic Resonance in Medicine 32:98︲103 (1994)
Balchandani, et al. Magnetic Resonance in Medicine 67:1077︲1085 (2012)
Pulse Sequence
RF
Gz
Gx
Gy
A
M
:
I
-6
I
-4
I
I
0
-2
I
I
2
4
Time, ms
z-Gradient needs added area!
Same as if it was on constantly until echo time!
Conjugate phase suppresses crushed component from 180
Slice Profiles
:
t
0.6
o.8
0.41
$j0.2
a -0.2
-0.4-
-0.4-
-0.6-
-0.6-
-0.8-
Unchopped
-0.8
-
chopped
- L
a
Frequency, kHz
!2
b
-115
-i
-015
o
0:s
Frequency, kHz
i
_L___j
1.5
Single pulse has spin echo and crushed signal!
Crushed component subtracts out
2
Slice Selective
P Spectra
Pulse pairs with different
echo delays!
PC,
PDE
31
1
Axial 3.3 cm sllice of
brain!
8 rnsec
40
20
0
Frequency(ppm)
1.5T, 250 ms A/D, 2 s TR
-20
Lim, et al. Magnetic Resonance in Medicine 32:98︲103 (1994)
Non-Linear Phase β’s
Many ways to design non-linear phase β
Widely studied for saturation pulses!
Many options:!
Add quadratic phase to a linear phase β!
Use complex Remez algorithm!
Design a minimum phase pulse, factor, and
zero flip!
Use an adiabatic pulse as a prototype
related to an empirically derived performance measure
given by ðn BW FTWÞ [10]. The increase of this error
when applying quadratic phase can be best understood
in connection with Eq. (15). When increasing k, the
envelope in the time domain widens, as shown in Fig. 1.
As the finite pulse length n remains the same, the be-
Quadratic Phase
(Quadratic Phase)
Fig. 1. Effect of different amounts of quadratic phase k on the magnitude of the RF pulse (Remez pulse with n ¼ 512, h ¼ 90!, xp ¼ 0:1p,
and xs ¼ 0:11p). The pulse at k ¼ 0 corresponds to a regular linearphase pulse. For increasing k the RF energy is spread out further and
further away from the original central main lobe.
Complex Remez algorithm will produce an optimal design
for a specified phase and amplitude profile!
Paper by Schulte et al. tells you what to ask for!
R.F. Schulte et al. Journal of Magnetic Resonance 166 (2004) 111︲122
Zero Flipping
Minimum Phase
Non-linear Phase
zi => 1/zi*
TBW= N has 2N possible phase profiles!!
Max peak amplitude reduced ~1/√N
Adiabatic Pulses
Slice selective inversion!
Insensitive to amplitude
B1(t) = A0sech(βt)e-iμβtanh(βt)t
Mz
0
-1
1
1
1
Mz
0
0
-1
-1
50%
Mz
100%
200%
Adiabatic Pulse Pairs
Self-Refocused Adiabatic Spin Echo Pulses
1081
SLR
e peranner
aukeNova
dded
180◦
pulse
maged
nvenhe SE
pulse
ences
ze =
view
map
ubleed to
ecepr 7T
l was
ained
a 50◦
map.
age is
ceive
nsity
1083
PP
FIG. 5. RF, phase and gradient waveforms for the pulse sequence
which uses the SRA pulse
a SE.
FIG. 8.toInproduce
vivo axial brain
images from a normal volunteer obtained using a quadrature head coil at 7 T. Image obtained using (a) a conventional
Pulse pair with adiabatic 180, and overlapping phase
compensated
matrix
size = 256 × 256, NEX =90!
3, FOV = 22 × 22 cm and
SE sequence and (b) the SRA pulse sequence. Acquisition parameters for both sequences were: TE/TR = 13.5/300 ms, slice thickness =
5 mm and matrix size = 256 × 256, NEX = 2, FOV = 24 × 24 cm2 , and scan time = 2:38 min. A transmit B1 map of the same slice obtained
using a Bloch-Siegert B1 mapping sequence is shown in (e). The calculated receive sensitivity map is shown in (f). c, d: The original images
2
in (a) and (b) after compensation by the receive map. The SLR 180◦ pulse in the conventional SE sequence becomes overdriven at the
center of the brain and underdriven at the edges of the brain where transmit B1 is at its highest and lowest value, respectively. This results in
time = 2 : 38 min.
signal loss and contrast differences. Horizontal cross-sections indicated by the dashed lines on (e) and (f) are plotted in (g). Comparatively,
vivo scans were
also SLR
performed
7 Ttransmit
using
the
the adiabatic
pulse achievesat
greater
B1 uniformity,
particularly at the center of the brain.
scan
In
quadrature head coil. An axial slice through the brain was
SRA pulse
sequence,
respectively.
The images appear
acquired using the SRA
sequence
and
the SE sequence.
For sim- shown in Fig. 8e, reaches its highest and lowest values.
Contrast differences
in the edges of
the brain are apparent
ilar
except
for
a
few
differences
in
contrast
indicated
by the Resonance
Balchandani,
al.pulse
Magnetic
in Medicine
67:1077︲1085
(2012)
the 7 T experiments, we used the
original 4.4et
kHz
Compensates for B1 at 7T!
white arrows. In particular, slowly moving substances such
when evaluating the images. Figure 8g is a plot of hori-
True Self Refocusing Pulses
Another option: start with a minimum phase
excitation, and add phase to α!
Original Mxy is
Mxy = 2α*β M0!
Add phase compensation p to α to delay echo
Mxy = 2(αp)*β M0!
One cycle of linear phase across slice delays echo by
one main lobe width!
Adds one 180 in power (expensive!)
True Self Refocusing Pulses
0.4
Zero echo time!
Self Refocused RF Pulses
0.2
30
60
90
0
−0.2
−0.4
0
1
0.5
1
Time, ms
1.5
2
My Slice Profiles
2.5
90
60
30
0.5
0
−4
−3
−2
−1
0
Position
1
2
3
4
3
4
1
30
60
90
0.5
0
−4
Mz Slice Profiles
−3
−2
−1
0
Position
1
2
No crushed or
unrefocused signal!
Echo delayed by
adding phase, and
RF power
Multi-Dimensional Pulses
Excitation and acquisition are duals!
Any acquisition technique can be used as an
excitation pulse (spiral, EPI)!
EPI RF pulses are most useful for MRS,
Spectral-Spatial pulses!
Do two things with one pulse!
Spectral-Spatial Pulses
Spatial!
Subpulses
Spectral!
Envelope
RF
G
Product Pulse Design:
Sampled spectral pulse (large tip angle)!
Spatial subpulses (small tip angle)!
Also, full 2D design
Spectral-Spatial Profile
Periodic in frequency!
Space
N/2 “Ghost” sidelobes!
Continuous in space
Frequency
Positive Contrast SPIO
Imaging
the simulated profiles of pulse 1 and pulse 2 is plotted in
et al.
Fig. 5a. The profile shows the main spectralBalchandani
passband and
shorter subpulse duration and hence faster gradient refo- sidebands located at ±1.67 kHz. Figure 5b is the simulated
cusing for the same slice thickness. Thus, sideband spacing spatial profile. Simultaneous spatial and spectral selectivwas limited by available gradient amplitude and slew rate. ity as well as the chemical shift immunity of the pulse
A minimum slice thickness of 4 mm was calculated using profile is demonstrated in the simulated 2D spatial-spectral
profile
in Fig. 6. The
main
spectral passband,
the
gradient
amplitude
and slew rate
limits of 40 mT/m
and
FIG.
5. Simulated
subtracted
magnetization
profiles
for the
spinshown
echo produced
by the
SR-SPSP
pulses. (a)sideSpectral profile showing the
150 T/m/s offered by the 1.5T GE CV/i scanner. If insert bands as well as opposing sidebands are visible in this 2D
main passband with a bandwidth of 216 Hz and sidebands spaced ±1.67 kHz from the main passband. (b) Spatial profile for slice selected
gradients with higher amplitude and slew rate limits are profile. In the final implementation, the transmit frequency
withinasthe
passband.
used,
is main
the case
in many animal scanners, more gra- of the pulses is shifted off-center to allow positive contrast
dient area may be accrued during the subpulses enabling imaging. The profile in Fig. 6 has been shifted by −450 Hz.
much thinner slices and making subpulse duration less At this offset frequency, signal from on-resonant water and
of a limitation. The final SR-SPSP pulses shown in Fig. 4 fat is excluded from the main passband as well as the sidesimulator
forofa31.8
range
T2 values.
The results
are shown
in opposing
(Fig. 9a)
and 2Opposing
(Fig. 9b)sidebands
of the exist
SR-SPSP pulse to create
bands and
sidebands.
had
a duration
ms,of
a spectral
bandwidth
of 216 Hz,
the spectralaprofile
SPSPamplitude
pulse does not
exactly
and
spatial
bandwidthsignal
of 4,000loss
Hz. at the center of the in
Fig.a 7a.
Significant
spectral
spinbecause
echo. a RF
waveforms
are identical for
Simulations
in passband
MATLAB byselectivity
explicitly occurs
trace outata rectilinear
trajectory.
passband
aswere
wellperformed
as loss of
the twoexcitation
pulses; k-space
however,
the phase waveforms differ. The
determine the behavior of the SR-SPSP pulse
multiplying the rotation matrices to obtain the spectral
T2 values less than 30 ms. This is expected becauseTo
pulses are played in conjunction with an oscillating gradiof the
exciting
and refocusing spins with short relaxation
and spatial profiles of the pulse. T1 and T2 effects were when
∗
ent waveform.
pulse
duration is long, most of
long
duration
of
the
pulse.
However,
decay
due
to
T
is
refo2
the spectral
profile wasAlthough
generated the
using
a Bloch
neglected. The spectral profile obtained after subtracting times,
Clearly, it would be desirable to have the sidebands as far
186
out as possible, but increased sideband spacing required
Pulse Pair
Envelope
cused by the SR-SPSP pulse as it achieves a spin-echo. Our the energy is concentrated toward the end of the pulse. If tx
◦ and (b) self-refocused
FIG.
3. Real and showed
imaginary components
self-refocused
spectral
pulse 1 with
the echo
phase
φ = −90
simulations
that the Tof2 (a)
value
at which
the RF
profile
is the
time
between
the
peak of the pulse and the end of the
◦
spectral
RF
pulse
2
with
the
echo
phase
φ
=
90
generated
using
the
exact
expressions
for
α
and
β
in
Eq.
[4].
Pulses
very similar
begins to degrade decreases when smaller values are cho- pulse and ∆T is the delayare
between
theto end of the pulse and
those resulting from the approximate solution in Fig. 2 and produce similar spectral profiles. The peak RF amplitude of the pulses is greater
senthefor
thein Fig.
echo
the occurrence of the echo, then the effective TE is given
than
pulses
2. delay ∆T . A cross section along the T2
direction through the center of the spectral band is shown by 2(tx + ∆T ). For a 31.8 ms pulse, 1.7 ms phase encode
Spectral-Spatial
Pulse Pair
in Fig. 7b. Figure 8 is a log plot of the simulated spectral and 4 ms readout gradient, a minimum TE of 12.8 ms can
passband
demonstrating
100-fold
or 40
dB suppresachieved.
tx 1and
for
this
pulse
the simulatedbe
profiles
of pulse
and∆T
pulse
2 is
plotted
in are 2.7 and 3.7 ms,
Clearly,
it would
be desirable tothat
havea the
sidebands
as far
sion
efficiency
on-resonant
water
may required
be achieved
respectively.
transmit
frequency
of the pulses is offset
Fig.using
5a. The profile
shows theThe
main
spectral passband
and
out
as possible,
butfor
increased
sideband
spacing
at ±1.67
Figure 5b is acquire
the simulated
shorter
thesesubpulse
pulses. duration and hence faster gradient refo- sidebands located
by ±450
Hz kHz.
to selectively
signal from off-resonant
Simultaneous
spatial and
spectral
cusing for the same slice thickness. Thus, sideband spacing spatial profile.
spins
in the vicinity
of the
SPIOselectivnanoparticles.
was limited by available gradient amplitude and slew rate. ity as well as the chemical shift immunity of the pulse
Pulse
Sequence
AFinal
minimum
slice
thickness of 4 mm was calculated using profile is demonstrated in the simulated 2D spatial-spectral
and main
In Vivo
Experiments
in Fig. 6. The
spectral
passband, sidethe gradient amplitude and slew rate limits of 40 mT/m and profile shownPhantom
Figure
showsbythe
RF phase,
gradi150
T/m/s9offered
the RF
1.5Tamplitude,
GE CV/i scanner.
If insertand
bands
as wellLabeled
as opposing
sidebands
are
visible
in this 2D
Cell Preparation
ent waveforms
foramplitude
the sequences
utilize
1 the final implementation,
profile. In
the transmit frequency
gradients
with higher
and slewwhich
rate limits
are version
is shifted
to allow positive
used, as is the case in many animal scanners, more gra- of the pulses We
usedoff-center
ferumoxides
(Feridixcontrast
IV; Berlex Laboratories,
profile in NJ),
Fig. 6which
has beenhad
shifted
by
−450
Hz. content of 11.2 mg
dient area may be accrued during the subpulses enabling imaging. TheWayne,
a total iron
FIG. 4.thinner
Real and
imaginary
of (a) SR-SPSP
RF less
pulse 1At
andthis
(b) offset
SR-SPSP
RF pulsesignal
2. Thefrom
SPSPon-resonant
pulses are obtained
by
frequency,
water and
much
slices
and components
making subpulse
duration
Fe/mL, and clinical-grade protamine sulfate (PS; American
the spectral
Figs. 2a,b
with small-tip
spatial
subpulses.
is excluded
from the main passband as well as the sideofsampling
a limitation.
Thepulse
finalenvelopes
SR-SPSPinpulses
shown
in Fig.angle
4 fat
Pharmaceuticals
Partner, sidebands
Schaumburg,
sidebands. Opposing
exist IL), which was prehad a duration of 31.8 ms, a spectral bandwidth of 216 Hz, bands and opposing
pared
as
a
stock
solution
of
1
mg/mL
in distilled water, for
in the spectral profile because a SPSP pulse does not exactly
and a spatial bandwidth of 4,000 Hz.
Simulations were performed in MATLAB by explicitly trace out a rectilinear
excitation
k-space
trajectory.
labeling human bone marrow stromal cells (hBMSC; courTo determine
behavior BioServices,
of the SR-SPSP
pulse
multiplying the rotation matrices to obtain the spectral
tesy the
of Cognate
Sunnyvale,
CA) (21,22). The
and
refocusing
spins
with
short
relaxation
and spatial profiles of the pulse. T1 and T2 effects were when exciting
original ferumoxide solution was added into a tube conneglected. The spectral profile obtained after subtracting times, the spectral profile was generated using a Bloch
Spin echo image of frequency
shifted signals!
2D Profile
taining hBMSC complete media (Cognate), then PS was
applied into the ferumoxides mixed media, and shaken
well for 5 min. Finally, the labeling media was combined
with equal amounts of hBMSC complete media and the
hBMSC were incubated in the labeling media for 12–
15 h. The final concentration of Fe and PS was 50 and
6 µg/mL, respectively. In our experiment, we estimated that
a maximum
of 1,500 µg
Fe Magnetic
was used. Resonance in
Balchandani,
etofal.
Positive contrast
Medicine 62:183︲192 (2009)
Conclusions
Much to be gained by focusing on the MR signal
instead of the RF pulses!
Benefits include!
Sharper profiles!
Shorter echo time!
Better control of the signal