G16.4427 Practical MRI 1
Radiofrequency Pulse Shapes and Functions
G16.4427 Practical MRI 1 – 12th February 2015
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:
æ dM z æ
M0 - M z
æ dt æ = -w rf M yæ +
T1
æ
ærot
=0
Why ?
æ dM xææ
M xæ No Off-Resonance Effects  ω = ω
rf
0
æ dt æ = w 0 - w rf M yæ - T
æ
ærot
2
(
)
We also turn off the RF field
æ dM yææ
M yæ
before observing the evolution
æ
æ = - w 0 - w rf M xæ + w rf M z T2 of the magnetization
æ dt ærot
(
)
G16.4427 Practical MRI 1 – 12th February 2015
Solution of the Bloch Equation
• The transverse and longitudinal components are
decoupled:
æ dM æ
M
xæ
=
æ dt æ
T2
æ
ærot
xæ
æ dM z æ
=0
æ
æ
dt
æ
æ
rot
æ dM yææ
M yæ
æ
æ =dt
T2
æ
ærot
• We are usually interested in the transverse component,
as it determines the time signal detected:
M xæyæ
æ dM xæyææ
1
M xæ + iM yæ æ æ
=
M xæyæ
æ
T2
æ dt ærot
Þ M x¢y¢ = q e-t T2
G16.4427 Practical MRI 1 – 12th February 2015
A k-Space Analysis of Small-Tip-Angle
Excitation
G16.4427 Practical MRI 1 – 12th February 2015
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
G16.4427 Practical MRI 1 – 12th February 2015
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
G16.4427 Practical MRI 1 – 12th February 2015
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
G16.4427 Practical MRI 1 – 12th February 2015
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
G16.4427 Practical MRI 1 – 12th February 2015
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
G16.4427 Practical MRI 1 – 12th February 2015
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
G16.4427 Practical MRI 1 – 12th February 2015
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 sliceselection gradient lobe, which improves slice profile
G16.4427 Practical MRI 1 – 12th February 2015
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
G16.4427 Practical MRI 1 – 12th February 2015
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?
G16.4427 Practical MRI 1 – 12th February 2015
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.
G16.4427 Practical MRI 1 – 12th February 2015
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
On-Resonance
Profile
Off-Resonance
Profile
(Original Pulse)
Off-Resonance
Profile
(VR Pulse)
G16.4427 Practical MRI 1 – 12th February 2015
Bernstein et al.
(2004)
Handbook of MRI
Pulse Sequences.
Any questions?
G16.4427 Practical MRI 1 – 12th February 2015
Basic Radiofrequency (RF)
Functions
G16.4427 Practical MRI 1 – 12th February 2015
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
G16.4427 Practical MRI 1 – 12th February 2015
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?
G16.4427 Practical MRI 1 – 12th February 2015
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?
G16.4427 Practical MRI 1 – 12th February 2015
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 (onresonance) is 90°. What is its amplitude?
G16.4427 Practical MRI 1 – 12th February 2015
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°
G16.4427 Practical MRI 1 – 12th February 2015
Examples of Inversion Pulses
SLR inversion pulse
Slice profile
SINC inversion pulse
with Hamming window
Slice profile
Bernstein et al. (2004)
Handbook of MRI
Pulse Sequences.
G16.4427 Practical MRI 1 – 12th February 2015
Application: T1 Measurement
• One popular method to measure T1 is the inversionrecovery 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
M0
Mz
t
(
-TI T1
M z = M 0 1- 2e
TI = T1 ln 2
G16.4427 Practical MRI 1 – 12th February 2015
)
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
G16.4427 Practical MRI 1 – 12th February 2015
Graphical Explanation
RF
t
G16.4427 Practical MRI 1 – 12th February 2015
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
M xy
M0
-2t T2
M xy = M 0 e
t
G16.4427 Practical MRI 1 – 12th February 2015
2t = TE
Any questions?
G16.4427 Practical MRI 1 – 12th February 2015
See you next Thursday!
G16.4427 Practical MRI 1 – 12th February 2015