RF Pulse Design Multi-dimensional Excitation I
M229 Advanced Topics in MRI Kyung Sung, Ph.D.
2018.04.10
Class Business
• Office hours - Instructors: Fri 10-12pm
TAs: Xinran Zhong and Zhaohuan Zhang (time: TBD)
- Emails beforehand would be helpful
• Homework 1 will be out on 4/12 (due on 4/26)
• Papers and Slides
Today’s Topics
• Recap of adiabatic pulses
• Small tip approximation
• Excitation k-space interpretation
• Design of 2D excitation pulses
- Spiral pulse design
Recap of Adiabatic Pulses- A special class of RF pulses that can
achieve uniform flip angle
- Flip angle is independent of the applied B1 field
- Slice profile of an adiabatic pulse is obtained using Bloch simulations
- Does not follow the small tip approximation
✓ 6=Z T
0B1(⌧)d⌧
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▪ Amplitude modulation
▪ Short duration (0.3-1 ms)
▪ Low B1 amplitude
Adiabatic Pulses Non-adiabatic Pulses
▪ Generally multi-purpose(inversion pulses can be used for refocusing, etc.)
▪ Generally NOT multi-purpose (inversion pulses cannot be used for refocusing, etc.)
▪ Amplitude and frequency modulation
▪ Long duration (8-12 ms)
▪ High B1 amplitude (>12 µT)
/
Adiabatic Pulses
• Frequency modulated pulses:
envelop frequencysweep
where
d ~M
dt= ~M ⇥ � ~Beff
Adiabatic InversionA(t)
tω1(t)
t
AFP
ω0
▪ BIR: B1 Insensitive Rotation ▪ Most popular: BIR-4 Pulse
Adiabatic Excitation: BIR Pulses
Adiabatic Excitation: BIR4 Pulse
a b c d
Bloch Simulator
- http://mrsrl.stanford.edu/~brian/blochsim/
Small Tip Approximation
Bloch Equation (at on-resonance)
d ~Mrot
dt= ~Mrot ⇥ � ~Beff
where
d ~M
dt=
0
@0 !(z) 0
�!(z) 0 !1(t)0 �!1(t) 0
1
A
When we simplify the cross product,
Small Tip Approximation
d ~M
dt=
0
@0 !(z) 0
�!(z) 0 !1(t)0 �!1(t) 0
1
A
Mz ⇡ M0 small tip-angle approximation
sin θ ≈ θ cos θ ≈ 1 Mz ≈ M0 → constant
First order linear differential equation. Easily solved.
Mr(⌧, z) = iM0e�i!(z)⌧/2 · FT 1D{!1(t+
⌧
2)} |f=�(�/2⇡)Gzz
Solving a first order linear differential equation:
(See the note for complete derivation)
To the board ...
Small Tip Approximation
- For small tip angles, “the slice or frequency profile is well approximated by the Fourier transform of B1(t)”
- The approximation works surprisingly well even for flip angles up to 90°
Fourier transform
Shaped Pulses30� 90�
Pauly, J. J. Magn. Reson. 81 43-56 (1989)
small-angle approximation still works reasonablywell for flip angles that aren’t necessarily “small”
Multi-Dimensional Excitation RF Pulses
▪ 1D vs. N-D RF pulses ▪ Small tip angle approximation revisited ▪ Excitation k-space interpretation ▪ 1D examples in excitation k-space ▪ Excitation k-space integrals ▪ 2D excitation pulse design steps ▪ 2D spiral pulse design example
▪ EPI pulse design, spectral-spatial pulses(next lecture)
What is Multi-Dimensional Excitation?
Multi-dimensional excitation occurs when using multi-dimensional RF pulses in MRI/NMR, i.e. 2D or 3D RF pulses
1D vs. N-D RF Pulses
▪ 1D pulses are selective along 1 dimension, typically the slice select dimension
▪ 2D pulses are selective along 2 dimensions • So, a 2D pulse would select a long cylinder instead of a slice • The shape of the cross section depends on the 2D RF pulse
z
Selective along z only
x
y
z
x
y
Selective along z
Selective along y
2D/N-D Pulse Design Requires: - Specific B1 waveform - Specific gradient waveforms
Excitation k-space Interpretation
Small Tip Approximation
Small Tip Approximation
Let us define:
Gz(t)
B1(t)
k(t,t0)
t7
T
t1 t2 t3 t4 t5 t6
One-Dimensional Example
Consider the value of k at s = t1, t2, … t7
One-Dimensional Example
• This gives magnetization at t = t0, the end of the pulse
• Looks like you scan across k-space, then return to origin
Evolution of Magnetization During Pulse
• RF pulse goes in at DC (kz = 0)
• Gradients move previously applied weighting around
• Think of the RF as “writing” an analog waveform in k-space
• Same idea applies to reception
Other 1D Examples
Gz(t)
B1(t)
k(t,t0)t0
Other 1D Examples
Gz(t)
B1(t)
k(t,t0)t0
Other 1D Examples
Gz(t)
B1(t)
k(t,t0)t0
Multiple Excitations
• Most acquisition methods require several repetitions to make an image - e.g., 128 phase encodes
• Data is combined to reconstruct an image
• Same idea works for excitation!
Simple 1D Example
Sum the data from two acquisitions
Same profile as slice selective pulse, but zero echo time
Small Tip Approximation Solution as k-space Integral
=
Substituting and changing the order of integration:
• The magnetization is the inverse transform of p(k)
• We want this to be a unit delta, multiplied by a weighting function
where
Small Tip Approximation Solution as k-space Integral
Multiply and divide by |k’(s,t)|:
W(k(s,t))Unit Delta
If we assume W(k) is single-valued
Small Tip Approximation Solution as k-space Integral
k-space weighting
k-space sampling
So, the inverse Fourier transform of the k-space weighting will give us the excitation profile!
Small Tip Approximation Solution as k-space Integral
k-space weighting
Design of 2D Excitation Pulses
2D Pulse Design
1. Choose a k-space trajectory
2. Choose a weighting function
3. Design the RF pulse
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B1(s) =1
�W (~k) · |k0(s, t)|
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1. Choose a k-space trajectory
• Select a k-space trajectory that uniformly covers k-space
- k-space extent (−kmax, kmax) ⇒ spatial
resolution
- sampling density (∆k) ⇒ spatial FOV
1. Choose a k-space trajectory
• Two most common choices:
• Spiral is common for pencil beams
• EPI is common for spectral-spatial pulses
2. Choose a weighting function
• An excitation profile is the inverse Fourier transform of the weighting function
• If you know what excitation profile you want, its Fourier transform will be the weighting function
• Localized excitation ⇒ low-pass k-space weighting
k-space weighting Excitation profile
2. Choose a weighting function
3. Design the RF pulse
B1 needs to be scaled for flip angle
2D Spiral Pulse Design
• Two major choices: - Resolution ∆r
• Smallest volume / minimum transition width
- Field-of-View (FOV)
• Distance to center of first sidelobe
k-Space
Trajectory
kx
ky
kmax
�kmax
Δk =2kmax
2N
N Turn
Spiral
FOV = 2NΔr
Δr =1
2kmax
Impulse
Response
2D Spiral Pulse Design
• Spiral Gradient Design
- Constant angular rate spiral
- Constant slew rate spiral
2D Spiral Pulse Design
• Truncated Jinc Weighting
Minimum transition width, but residual ripples
2D Spiral Pulse Design
• Windowed Jinc Weighting
Window Function
Doubled transition width, but smoother response
2D Spiral Pulse Design
• Calculation of the RF pulse - given W(k) and k(s,t)
- needs to be scaled for flip angle
- |k’(s,t)| is an estimate of the density compensation function d(t)
Conclusions
- N-D RF pulses are selective in N-dimensions
- The small tip approximation can be extended to describe the excitation k-space
- The small tip approximation solution can be used to show that the excitation profile of an N-D pulse is given by the inverse Fourier transform of the excitation k-space weighting
Conclusions
- An N-D RF pulse can be designed by:
• Choosing a k-space trajectory
• Choosing a k-space weighting function
• Then calculating the B1(t) and G(t) functions
Next Class
• N-D pulses with EPI trajectory
• Spatial-spectral pulses
• Matlab demo of N-D pulses
Thank You!
Kyung Sung, PhD [email protected] http://kyungs.bol.ucla.edu
- Further reading • Read “Spatial-Spectral Pulses” p.153-163
- Acknowledgments • John Pauly’s EE469b (RF Pulse Design for MRI) • Shams Rashid, Ph.D.