Radiofrequency Pulse Shapes and Functions G16.4427 Practical MRI 1 Radiofrequency Pulse Shapes and Functions

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: = 0 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

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:

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

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

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

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

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

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

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

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

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

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?

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.

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.

Any questions?

Basic Radiofrequency (RF) Functions

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

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?

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?

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?

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°

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.

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

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

Graphical Explanation RF

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

Any questions?

See you next Thursday!