RF Pulse Design
Shaihan Malik1
1Kings College London, United Kingdom

Synopsis

RF pulses are an integral part of every MR sequence, and may take on multiple different functions (excitation, saturation, inversion, refocusing, etc…). The Bloch equation governs the interaction between RF fields and magnetization, and so RF pulse design is essentially the process of inverting the Bloch equation: we ask “What should the RF fields do given the way we want the magnetization to end up?”. This talk will cover: the small tip angle approximation (STA); Shinnar Le Roux (SLR) pulse design method; Multidimensional and multiband RF pulses; and Parallel transmission pulse design.

Target Audience

This talk is aimed at physicists and engineers who want to understand more about how RF pulses used in standard pulse sequences are designed.

Overview

RF pulses are an integral part of every MR sequence, and may take on multiple different functions (excitation, saturation, inversion, refocusing, etc…). The Bloch equation governs the interaction between RF fields and magnetization, and so RF pulse design is essentially the process of inverting the Bloch equation: we ask “What should the RF fields do given the way we want the magnetization to end up?” It turns out that this can be more complicated than it first appears and so there are a few different design methods used in practice. This talk will cover:
  • The small tip angle approximation (STA)
  • Shinnar Le Roux (SLR) pulse design method
  • Multidimensional and multiband RF pulses
  • Parallel transmission pulse design

Small Tip Angle approximation

The simplest way to understand the operation of RF pulses is a frequency domain (Fourier) picture. This can be used to understand slice selection, for example. However it is not immediately obvious how this relates to the Bloch equation. We will see that in the approximation of ‘small tip angles’ (STA) it is possible to derive a Fourier relationship from the Bloch equation, and that this can then be used to design slice or frequency selective pulses using standard Fourier domain filter design methods1,2.

Shinnar-Le Roux Method

The STA approximation cannot be used to design pulses that aim to rotate magnetization by large angles; 90° excitations, inversions and refocusing pulses are not well served by this approach. However if we divide an RF pulse into a series of smaller rotations applied one after the other (the ‘hard pulse approximation’) then it is possible to relate the RF pulse waveform to a ‘polynomial’ function which is itself related to the magnetization profile by Fourier Transformation. Mapping of an RF pulse to the ‘polynomial’ is referred to as the forward SLR transform, and the inverse SLR transform does the reverse.
To design a large tip angle slice selective pulse we first design its frequency domain representation using filter design methods, specifying parameters such as transition widths and the amount of acceptable ripple both in and out of the stop band. Once done we can map to the ‘polynomial’ representation by inverse FT, and from here we can obtain the required RF pulse via the inverse SLR transform3,4.

Multidimensional and multiband RF pulses

So far we have considered slice selective pulses, however it is possible to design pulses that are selective in 2 or 3 spatial dimensions as well. These may be useful for navigator excitations, localised saturation or inner volume imaging for example. The original derivation of the STA approximation from Pauly1 showed how to design such pulses within a ‘k-space’ representation; we will use this picture to analyse some examples. RF pulses also have frequency as well as spatial selectivity and we will see how to extend the k-space picture to cover pulses that are simultaneously spectrally and spatially selective5.
Finally we will consider the design of multiband pulses that are required for simultaneous multi-slice imaging. We will mainly consider modulated versions of single band pulses, and look into the methods that aim to get around peak power limits on such designs6,7.

Parallel Transmission

The assumption made by all of the previous design approaches is that the RF transmit field ($$$B_1^+$$$) is spatially uniform. This is not the case at ultrahigh field or when dealing with surface transmit coils. With a classic single channel transmit RF system the only degree of freedom available to the RF pulse designer is the pulse waveform – hence it is possible to control the temporal behaviour of the RF field but not its spatial distribution. On the other hand, replacing a single transmitter with a parallel array of multiple transmitters opens the possibility to control both spatial and temporal aspects. In order to use these additional degrees of freedom we require additional design tools; we will consider here only the STA approach to such designs but will point to more advanced methods8.

Further Reading

The references given in the previous sections are a minimal set for following the lecture content. Each topic covered represents a whole sub-field of work, with many areas not even covered by this short talk. Notable other areas include (but are certainly not limited to): Adiabatic RF pulses9 and Optimal control design directly via Bloch equation10.

Acknowledgements

No acknowledgement found.

References

1. Pauly, J., Nishimura, D. & Macovski, A. A k-space analysis of small-tip-angle excitation. J. Magn. Reson. 81, 43–56 (1989).

2. Pauly, J. & Nishimura, D. A Linear Class of Large-Tip-Angle Selective Excitation Pulses. J. Magn. Reson. 82, 571–587 (1989).

3. Shinnar, M., Eleff, S., Subramanian, H. & Leigh, J. The synthesis of pulse sequences yielding arbitrary magnetization vectors. Magn. Reson. … 80, 74–80 (1989).

4. Pauly, J., Le Roux, P., Nishimura, D. & Macovski, A. Parameter relations for the Shinnar-Le Roux Selective Excitation Pulse Design Algorithm. IEEE Trans. Med. Imaging 10, 53–65 (1991).

5. Meyer, C. H., Pauly, J. M., Macovski, a & Nishimura, D. G. Simultaneous spatial and spectral selective excitation. Magn. Reson. Med. 15, 287–304 (1990).

6. Seada, S. A., Price, A. N., Hajnal, J. V & Malik, S. J. Optimized amplitude modulated multiband {RF} pulse design. Magn. Reson. Med. (2017). doi:10.1002/mrm.26610

7. Auerbach, E. J., Xu, J., Yacoub, E., Moeller, S. & Uğurbil, K. Multiband accelerated spin-echo echo planar imaging with reduced peak RF power using time-shifted RF pulses. Magn. Reson. Med. 69, 1261–7 (2013).

8. Padormo, F., Beqiri, A., Hajnal, J. V & Malik, S. J. Parallel transmission for ultrahigh-field imaging. {NMR} Biomed. 29, 1145–1161 (2015).

Proc. Intl. Soc. Mag. Reson. Med. 28 (2020)