Designing RF Pulses: From Square to Adiabatic to Multi-Slice
Rudolf Stollberger1
1Graz University of Technology, Austria

Synopsis

Keywords: Physics & Engineering: Pulse design

This section of the course covers the concepts involved in understanding the theory and implementation of radio frequency (RF) pulses. The presentation covers the different basic types of RF pulses, explains essential definitions for RF pulses, used simplifications and their limitations, parameters to check implementation properties, and several strategies to find RF pulses that meet different requirements as well as possible.

Introduction and Background

High-frequency pulses are the driving force of magnetic resonance. Depending on the respective task, a distinction is made between excitation pulses (non-selective, spatial/spectral selective), inversion and saturation pulses (magnetization preparation), and refocusing pulses (spin echo). The Bloch equations describe the motion of macroscopic magnetization under the influence of the high-frequency field B1+ and the resonance offset Dw. Therefore, they are the basis for calculating the effect of a certain RF pulse on the magnetization and for the inverse problem - the search of an RF pulse B1+(t) to transform the magnetization from the initial state M(t0) to a target state M(tp).
For a short pulse duration, the relaxation can be neglected to a good approximation and the simplest solution for flipping the magnetization by a certain angle a consists of a short RF pulse with constant amplitude (rectangular, squear or block pulse). The effect of such a pulse can be described by a rotation matrix [Mansfield]. This is also a basic element for the implementation of more complex amplitude-modulated pulses as used for slice selection.
Another way to manipulate the magnetization exists in adiapatic RF pulses. For these pulses the amplitude and the frequency are modulated simultaneously whereby the effective field Beff in the rotating coordinate system is initially oriented parallel to the magnetization and then drags the magnetization rotating around Beff into a new position. The hyperbolic secant pulse is a typical representative of an adiapatic inversion pulse found as a solution of the Bloch-Riccardi equation, neglecting also relaxation effects [Sliver]. A particular advantage of these pulses is their robustness to field inhomogeneities, but they typically have higher pulse energy and pulse duration.

RF pulses, pulse properties and design strategies

A wide variety of RF pulse types are used for different MR applications. For this, depending on the application, typical pulse requirements such as short echo time, homogeneous layer profile, robust inversion or simultaneous excitation of several layers must be met, but also the device-specific constraints such as the maximum B1 amplitude or limitations by maximum pulse energy or pulse power.
In this presentation, the relationship between pulse parameters, pulse characteristics, and given restrictions will be discussed for different pulse types, and available design strategies for different pulse categories will be explained.
The starting point of the discussions are simple block and composite pulses. These are well suited to introduce and present important basic concepts and relationships.
For amplitude modulated, slice-selective pulses with small flip angles, the design is guided by the Fourier relationship between the RF pulse shape and the slice profile (pulse spectrum) for a constant slice encoding gradient. This relationship can be used to describe multidimensional spatially selective pulses with the k-space formalism known from imaging.
For larger flip angles, the nonlinearity/bilinearity of the Bloch equation must be considered. For this, numerical enhancement strategies (Shinnar-LeRoux algorithm) are presented to improve the slice profile for a specific larger flip angle. For Sinc-function based pulses, the influence of the time-bandwidth product (TBW) parameter on the slice profile will be explained. To reduce pulse power, the concept of variable-rate selective excitation can be applied.
Important further developments for selective pulses are spatial-spectral pulses and simultaneous multi-slice excitations and refocusing. The basic principles for designing with the associated effects are part of this block.
Another class of pulses that will be shown are saturation and inversion pulses with their required properties.
The robustness of RF pulses to B1 and B0 inhomogeneities is a particularly important issue in the field of high-field systems. For this purpose, the possibilities of adiapatic pulses are presented and the principle of parallel transmission is briefly explained.
Finally, it will be shown how optimal control can be used to design RF pulses that are able to use the entire parameter space available while outperforming various state-of-the-art pulses in terms of their properties.

Acknowledgements

Austrian Science Fonds FWF, # I 4870

References

Books

· Mansfield P, Morris PG. NMR Imaging in Biomedicine. Supplement 2 Advances in Magnetic Resonance. Academic Press, 1982 (Ch. 3 Basic imaging principles)

· Bernstein M A, King K F, Zhou X J. Handbook of MRI pulse sequences. Elsevier, 2004.

· de Graaf RA, In Vivo NMR Spectroscopy: Principles and Techniques. John Wiley and Sons, Ltd, 2013 (Ch 5. Radiofrequency pulses)

Paper

1. Hahn EL, Spin echoes, Phys Rev, 80, 580–594, (1950)

2. E. T. Jaynes. Matrix treatment of nuclear induction. Phys Rev, 98(4):1099–1105, (1955)

3. Levitt MH, Freeman R, Compensation for pulse imperfection in NMR spin echo experiments, J Magn Reson, 43, 65-80 (1981)

4. Levitt MH, Ernst RR, Composite pulses constructed by a recursive expansion procedure, J Magn Reson, 55, 247-254 (1983)

5. Mansfield P, Maudsley AA, Morris PG, Pykett IL, Selective pulses in NMR imaging: A reply to criticism, J Magn Reson, 33, 261-274 (1979)

6. Hoult DI, The solution of the Bloch equations in the presence of a varying B1 field – an approach to selective pulse analysis, JMR 35, 69-86 (1979)

7. Young IR, Bryant DJ, Payne JA, Variations in slice shape and absorption as artifacts in the determination of tissue parameter in NMR imaging, Magn Reson Med 2, 355-389 (1985)

8. Ngo J Th, Morris PG, General Solution to the NMR Excitation Problem for Noninteracting Spins, Magn Reson Med 5, 217-237 (1987)

9. Runge VM, Wood ML, Kaufman D M, Silver MS. MR imaging section profile optimization: Improved contrast and detection of lesions. Radiology 167:831-834 (1988.)

10. Shinnar M, Leigh JS. Frequency-response of soft pulses. J Magn Reson 75:502-505. (1987)

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

12. Pauly J, Leroux P, Nishimura D, Macovski A. Parameter relations for the Shinnar-Leroux selective excitation pulse design algorithm. IEEE Trans Med Imaging 10:53–65. (1991)

13. Bottomley PA and Hardy CJ. Two‐dimensional spatially selective spin inversion and spin‐echo refocusing with a single nuclear magnetic resonance pulse. J. Appl. Phys. 62:4284–4290. (1987)

14. Yip CY, Fessler JA, Noll DC. Iterative RF pulse design for multidimensional, small-tip-angle selective excitation. Magn Reson Med 54:908-917. (2005)

15. Meyer CH, Pauly JM, Macovski A, Nishimura DG. Simultaneous spatial and spectral selective excitation. Magn Reson Med, 15(2):287–304,1990.

16. Zur Y. Design of improved spectral-spatial pulses for routine clinical use. Magn Reson Med, 43(3):410–20, 2000.

17. Hargreaves BA, Cunningham CH, Nishimura DG, Conolly SM, Variable-rate selective excitation for rapid mri sequences”, Magn Reson Med, 52, 590–597 (2004)

18. S. Mueller, “Multifrequency selective RF pulses for multislice MR imaging”, Magn Reson Med, vol. 6, no. 3, pp. 364–371, 1988

19. D. G. Norris, P. J. Koopmans, R. Boyacioglu, and M. Barth, Power independent of number of slices (PINS) radiofrequency pulses for low-power simultaneous multislice excitation, Magn Reson Med, vol. 66, no. 5, pp. 1234–1240, 2011

20. D. G. Norris, P. J. Koopmans, R. Boyacio ˘ glu, and M. Barth, “Power independent of number of slices (PINS) radiofrequency pulses for low-power simultaneous multislice excitation”, Magn Reson Med, vol. 66, no. 5, pp. 1234–1240, 2011

21. Barth M, Breuer F, Koopmans P, Norris D, Poser B, Simultaneous multislice (SMS) imaging techniques, Magn Reson Med, 63–81 (2016)

22. Silver MS, Joseph RI, Hoult DI, Selective spin inversion in nuclear magnetic resonance and coherent optics through an exact solution of the Bloch-Riccati equation, Physical Review A 31, 2753 – 2755 (1985)

23. Bendall, M. R., and Pegg, D. T. Uniform sample excitation with surface coils for in vivo spectroscopy by adiabatic rapid half passage. J. Magn. Reson. 67: 376-381. 1986

24. Ugurbil, K., Garwood, M., and Bendall, R. Amplitude- and frequency-modulated pulses to achieve 90 degree plane rotation with inhomogeneous B1 fields.J. Magn. Reson. 72:177-185. (1987)

25. Conolly S, Nishimura D, Macovski A, Glover G. (1988) Variable-rate selective excitation. J Magn Reson 78:440–458. (1988)

26. De Graaf, Nicolay K, Adiabatic rf Pulses: Applications to In Vivo NMR, Concepts Magn Reson 9: 247-268 (1997)

27. Norris DG, Adiabatic Radiofrequency Pulse Forms in Biomedical Nuclear Magnetic Resonance, Concepts in Magnetic Resonance, 14(2) 89–101 (2002)

28. Katscher U, Bornert P, Leussler C, van den Brink J. Transmit SENSE. Magn Reson Med 49:144-150. 2003

29. Grissom WA, Xu D, Kerr AB, Fessler JA, Noll, DC Fast large-tip-angle multidimensional and parallel RF pulse design in MRI. Medical Imaging, IEEE Transactions on, 28:1548-1559. (2009)

30. Conolly S, Nishimura D, Macovski A, Optimal Control Solutions to the Magnetic Resonance Selective Excitation Problem, IEEE TMI, VOL. MI-5(1986)

31. Xu, D., King, K. F., Zhu, Y., McKinnon, G. C., & Liang, Z. P. Designing multichannel, multidimensional, arbitrary flip angle RF pulses using an optimal control approach. Magnetic Resonance in Medicine, 59(3), 547-560. (2008)

32. Van Reeth E, Ratiney H, Tesch M, Grenier D, Beuf O, Glaser SJ, Sugny D. Optimal control design of preparation pulses for contrast optimization in MRI. J Magn Reson. 2017; 279: 39- 50.

33. Aigner CS, Clason C, Rund A, Stollberger R, “Efficient high-resolution RF pulse design applied to simultaneous multi-slice excitation”, J Magn Reson, vol. 263, 33–44 (2016)

34. Rund A, Aigner CS, Kunisch K, Stollberger R, “Magnetic resonance rf pulse design by optimal control with physical constraints”, IEEE Trans Med Imaging, vol. 37, no. 2, 461–472 (2018)

35. Graf C, Soellradl M, Aigner CS , Rund A , Stollberger R, Advanced design of MRI inversion pulses for inhomogeneous field conditions by optimal control, NMR Biomed ;35(11):e4790 (2022)

36. Graf C, Rund A, Aigner CS, Stollberger R, Accuracy and performance analysis for Bloch and Bloch-McConnell simulation methods. J Magn Reson, 329, 2021, doi.org/10.1016/j.jmr.2021.107011

37. Zhao C, Wang K, Graf C, Stollberger R, Wang DJJ, Optimization and evaluation of inversion pulses for background suppressed pseudo-Continuous Arterial Spin Labeling at 7T, Proceedings of the ISMRM 2022

Proc. Intl. Soc. Mag. Reson. Med. 31 (2023)