1. Radiofrequency Pulse Shapes and Functions
G Practical MRI 1
Radiofrequency Pulse Shapes and Functions

2. Small Tip-Angle Approximation
It is easier to solve the Bloch equation after making the following assumptions:
At equilibrium Mrot = [0 0 M0] (initial condition)
RF pulse is weak leading to a small tip angle θ < 30°
The Bloch equations become:
= Why ?
No Off-Resonance Effects  ωrf = ω0
Mz = M0cos(theta); Cos(theta) = 1 – theta^2/2 + …  Mz = M0
We also turn off the RF field before observing the evolution of the magnetization

3. Solution of the Bloch Equation
The transverse and longitudinal components are decoupled:
We are usually interested in the transverse component, as it determines the time signal detected:

4. A k-Space Analysis of Small-Tip-Angle Excitation

5. Useful Quantities to Describe RF Pulses
Pulse width (T)
Indicates the duration of the RF pulse
Typically measured in seconds or milliseconds
RF bandwidth (∆f)
A measure of the frequency content of the pulse
FWHM of the frequency profile
Specified in hertz or kilohertz
Flip angle (θ)
Describes the nutation angle produced by the pulse
Measured in radians or degrees
Calculated by finding the area underneath the envelope of the RF pulse

6. RF Envelope Denoted with B1(t) and measured in microteslas
Relatively slowly varying function of time, with at most a few zero-crossings per millisecond
The RF pulse played at the transmit coil is a sinusoidal carrier waveform that is modulated (i.e. multiplied) by the RF envelope
The frequency of the RF carrier is typically set equal to the Larmor frequency ± the frequency offset required for the desired slice location

7. RF Envelope vs. RF Carrier
RF envelope - B1(t)
RF carrier
The RF envelope describes the pulse shape, i.e. the magnetic field in the rotating frame

8. SLR Pulses
For small flip-angles, the shape of an RF pulse can be determined by inverse Fourier transformation of the desired slice profile
The Shinnar-Le Roux (SLR) algorithm allows the inverse problem to be solved directly and efficiently without iterations
Allows the pulse designer to optimize the pulse before it’s generated
Uses the SU(2) representation for rotations and the hard pulse approximation to describe the effect of a soft pulse on the magnetization with 2 polynomials of complex coefficients
Given 2 complex polynomials corresponding to the desired magnetization, the inverse SLR transform yields the RF pulse

9. Practical Considerations For SLR Pulses
Pulses designed with SLR account for the nonlinearity of Bloch equations only at a single flip angle
If played at a different flip angle there will be deviations from the desired profile
If this is an important consideration:
A set of pulses designed for different flip angles could be stored on the MR scanner
The SLR design could be done in real time when the operator selects the flip angle

10. Variable-Rate (VR) Pulses
A one-dimensional spatially selective RF pulse that is played concurrently with a time-varying gradient is called a variable-rate (VR) pulse
Also known as VRG or VERSE pulses
The main application is to reduce SAR
Decrease RF amplitude near the peak of the pulse
VRG = variable-rate gradient
VERSE = variable-rate selective excitation

11. Variable-Rate (VR) Pulses
A one-dimensional spatially selective RF pulse that is played concurrently with a time-varying gradient is called a variable-rate (VR) pulse
Also known as VRG or VERSE pulses
The main application is to reduce SAR
Decrease RF amplitude near the peak of the pulse
Another application is to play the RF excitation concurrently with the gradient ramps
Efficient use of the entire time allotted for the slice-selection gradient lobe, which improves slice profile

12. VR-Modified SINC Pulse
To maintain the nominal flip angle when RF amplitude is reduced, the VR pulse is proportionately stretched (time delayed). Why?
Answer: flip angle is the area under the RF envelope

13. VR-Modified SINC Pulse
To maintain the nominal flip angle when RF amplitude is reduced, the VR pulse is proportionately stretched (time delayed).
What happens as a result?

14. VR-Modified SINC Pulse
To maintain the nominal flip angle when RF amplitude is reduced, the VR pulse is proportionately stretched (time delayed).
As a result the RF bandwidth is decreased
The slice selection gradient amplitude must be proportionately reduced to obtain the same slice profile
Bernstein et al. (2004)
Handbook of MRI
Pulse Sequences.

15. Off-Resonance Effects
A VR-modified pulse is designed to maintain the original slice profile for on-resonance spins (e.g. water)
The pulse designer can dilate the RF pulse and adjust the gradient, but has no control over the precession period of off-resonance spins
The slice profile of off-resonance spins (e.g. fat) is distorted
Off-Resonance
Profile
(Original Pulse)
On-Resonance
Profile
Off-Resonance
Profile
(VR Pulse)
Bernstein et al.
(2004)
Handbook of MRI
Pulse Sequences.

16. Any questions?

17. Basic Radiofrequency (RF) Functions

18. Excitation Pulses
Excitation pulses tip the magnetization vector away from the direction of B0
They are implemented by switching on B1(t) for a short time (200 μs to 5 ms)
T1 and T2 relaxation during the pulse can be neglected
They are characterized by the flip angle (θ), which is the angle between the direction of B0 and the magnetization vector after turning off RF
For non-adiabatic excitation pulses, θ is calculated as the area under the envelope of B1(t)
Typically θ = 90° for spin echo and θ = 5-70° for gradient echo

19. Slice Profile And Flip Angle
The distribution of the flip angle across the selected slice is called the slice profile
What is the ideal slice profile?

20. Slice Profile And Flip Angle
The distribution of the flip angle across the selected slice is called the slice profile
The ideal slice profile consists of a uniform flip angle within the desired slice and θ = 0° outside
Why it cannot be achieved in practice?

21. Slice Profile And Flip Angle
The distribution of the flip angle across the selected slice is called the slice profile
The ideal slice profile consists of a uniform flip angle within the desired slice and θ = 0 outside
It would require a pulse of infinite duration, so several approximations are used in practice
Problem
A hard RF pulse has a rectangular-shaped envelope. Its pulse width is 100 μs and its flip angle (on-resonance) is 90°. What is its amplitude?

22. Inversion Pulses
An inversion pulse nutates the magnetization vector from the direction of B0 to the negative B0 direction
The nominal flip angle is 180°

23. Examples of Inversion Pulses
SLR inversion pulse
Slice profile
SINC inversion pulse
with Hamming window
Slice profile
The SLR produces a more uniform slice profile than a sinc inversion pulse with identical pulse width, bandwidth and flip angle
Bernstein et al. (2004)
Handbook of MRI
Pulse Sequences.

24. Application: T1 Measurement
One popular method to measure T1 is the inversion-recovery method
The magnetization is inverted with a 180° RF pulse and then spin-lattice relaxation begins
After a time TI, the value of Mz is detected applying a 90° RF pulse and measuring the FID signal
The measurement is repeated for several TI and T1 is calculated by fitting the inversion recovery equation t

25. Refocusing Pulses
Due to the gradients, local magnetic field inhomogeneities, magnetic susceptibility variation, or chemical shift, the spins contributing to the transverse magnetization have a range of precessing frequencies
As a result there is a phase dispersion (fanning out)
A refocusing RF pulse (typically 180°) rotates the dispersing spins about an axis in the transverse plane so the the magnetization vector will rephase (or refocus) at a later time
The refocused magnetization is known as spin echo

26. Graphical ExplanationRF

27. Application: T2 Mapping
The simplest method to map T2 is the multi-echo method
Multiple images are acquired with different time delays
The resulting intensities are fitted on a pixel-by-pixel basis to extract the T2 value using the spin-spin relaxation curve t

28. Any questions?

29. See you next Thursday!