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 ksung@mednet.ucla.edu 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.