From a macroscopic perspective, the net magnetization vector precesses at the Larmor frequency,

\[\omega_0=\gamma B_0.\]

Often the rotating frame is considered for simplifying the analysis. The behavior or dynamics of the magnetization is described by the Bloch equation

\[\frac{d\mathbf{M}}{dt} = \mathbf{M} \times \gamma \mathbf{B}\]

where the relaxation is ignored, \(\mathbf{M}\) is magnetization, \(\gamma\) is gyromagnetic ratio, and \(\mathbf{B}\) is magnetic field or effective magnetic field.

The magnetization \(\mathbf{M}\) is considered to be aligned with the z-direction (main field direction) in equilibrium state. When a very short pulse is applied in the transverse plane, the magnetization will be flipped towards the transverse plane.

Magnetization and the rotation generated by the RF pulse

In practice, all the pulses have certain durations, i.e., the rectangular pulse.

Rectangular pulse
The flip angle from a rectangular pulse can be computed using \(\theta = \gamma B_1 T.\) The square pulse is nonselective ideally, but it does have a bandwidth in frequency because it is not infinitely short in practice.

rectangular pulse
rectangular pulse

Not everything is “on resonance”, but the voxel with better matched resonance frequency as the RF pulse gets more influence from the applied pulse. So two basic characteristics of the RF pulse are duration and bandwidth.

Gaussian pulse
Gaussian pulse has a Gaussian shape RF envolope and also a Gaussian shape frequency band. For example, Gaussian pulse can be used for fat signal saturation by shifting its center frequency.

gaussian pulse
gaussian pulse

SINC pulse
The off-resonance is not always something we want to avoid; it can be used to create selective RF excitation. SINC pulse combined with a slice selective gradient can be used to excite the signal of a certain slice. It is often mupltiplied by a window function (apodization). Things to control a sinc pulse: 1) time-bandwidth product, pulse duration (then the bandwidth can be computed) 2) slice thickness (then the slice gradient can be determined).

sinc pulse

One way to analyze the excitation profile of the pulse is small-tip-angle analysis1. Assume the flip angle small, which also means \(dM_z/dt \approx 0,\) then the Bloch equation can be simplified as

\[\frac{d \mathbf{M}}{dt} = \begin{bmatrix}0 & \Delta \omega & 0 \\ -\Delta\omega & 0 & \gamma B_1(t) \\ 0 & { 0} & 0\end{bmatrix} \begin{bmatrix} M_x \\ M_y \\ M_z \end{bmatrix},\]

where \(\Delta\omega\) is the off-resonance. This gives the result

\[M_{xy}(t) = i M_0 e^{-i\omega t} \int_0^t e^{i\omega s} \gamma B_1(t) ds.\]

which indicates the excitated transverse magnetization is related to the frequency spectrum of the RF profile \(B_1(t).\) The small-tip-angle approximation also introduces the concept of excitation k-space.

While, it always needs an rephasing gradient to refocus the magnetization within the excited slice. The need for rephasing gradient can also be understood using the excitation k-space.

sinc pulse

Shinnar-Le Roux pulse
SLR pulse2 produce more desired excitation profile compared to the SINC pulse when the large-flip angle is required. The analysis of the SLR pulse design uses the spin-domain representation, which represents the rotation by \((\alpha,\beta)\)

\[\begin{aligned}\alpha = \cos \phi/2 - i n_{z}\sin\phi/2, \\ \beta = - i (n_{x} + i n_{y}) \sin \phi/2, \end{aligned}\]

where \((n_x,n_y,n_z)\) is the rotation axis and \(\phi\) is the rotation angle. And SLR translates the pulse design problems as digital filter design problems.

SLR pulse

Multidimensional excitation pulse
Using the concept of excitation k-space, 2D and 3D spatially selective pulse can be designed.

Spatial-Spectral pulse

Adibatic pulse
…

reference: 3

Reference

  1. Pauly, John, Dwight Nishimura, and Albert Macovski. “A k-space analysis of small-tip-angle excitation.” Journal of Magnetic Resonance (1969) 81.1 (1989): 43-56. ↩

  2. Pauly, John, et al. “Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm (NMR imaging).” IEEE transactions on medical imaging 10.1 (1991): 53-65. ↩

  3. Nishimura, D. (1996). Principles of magnetic resonance imaging: Dwight d. Nishimura. ↩