A Brief Primer on Adiabatic Pulses

Adiabatic Pulses


Adiabatic pulses are a special category of pulses where both the amplitude and frequency (hence phase) of the pulses are modulated. They are designed to be insensitive to B1 inhomogeneities and miscalibrations, as long as the amplitude is above a certain threshold. They are ideal for uniform inversion or refocusing across large bandwidths. For more on adiabatic pulses, please refer to Chapter 5 in Ref. [1] and Chapter 12 in Ref. [2].

Some of these pulses have specific requirements that need to be met for them to behave ideally. In this section, we will discuss a few different types of adiabatic pulses and the ideal parameters needed for their optimal behavior.

The Adiabatic Condition

Adiabatic pulses need to satisfy the ‘adiabatic condition’ in order to achieve efficient inversion. The adiabatic condition is met when the effective B1 field is sufficiently larger than the rate of frequency sweep. That is, if the sweep is slow enough, the magnetization gets ‘dragged’ along with the effective field. Once the adiabatic condition is met, i.e. the minimum required amplitude is used, efficient inversion is achieved regardless of the magnitude of the applied RF field. Take the instance of a 4-ms R=80 HS-8 pulse shown below (R = time-bandwidth product). The magnetization vs amplitude simulation shows that while a minimum amplitude of ~2 kHz is needed to achieve complete inversion, using any amplitude higher than this will achieve the same inversion. This makes adiabatic pulses very robust to pulse miscalibrations.

4-ms R=80 HS-8 pulse

Magnetization Vs RF amplitude simulation for the 4-ms, R=80 HS-8 pulse


In practice, the B1max (maximum RF pulse amplitude) for an adiabatic pulse is determined by using Equation 1:

Equation 1

Where Q is the on-resonance adiabaticity factor (at the middle of the pulse), ΔF is the sweep-bandwidth (kHz) and T is the pulse duration (ms). ΔF/T represents the linear sweep rate of the pulse. For inversion pulses, Q=5 is a widely used value. However, the minimum required Q has a strong dependence on the modulation function itself. For instance, an HS1 pulse requires a Q that is 4 times that of an HS8, WURST or CHIRP pulse to satisfy the adiabatic condition, whereas a tanh/tan pulse can use Q values much lower than 5 and still get adequate inversion.

Modulation Functions

For an adiabatic pulse, the modulation function of an adiabatic pulse is the primary factor that determines the minimum RF amplitude threshold, frequency selectivity, and attainable bandwidth. Some commonly used simple modulation functions include those for Chirp and WURST pulses.

The tanh/tan modulation generally gives the best performance for nonselective excitation and inversion of wide frequency ranges.

Tannus and Garwood [3] used the adiabatic condition to derive a set of so-called offset‐independent adiabaticity (OIA) modulation functions that satisfy the adiabatic condition equally for all frequencies within the sweep range. Among these, the HSn family of pulses is particularly useful. The n in HSn is an integer and generally takes on the values 1 - 8, called HS1, HS8 and so on. It determines the shape of the amplitude modulation, which is a key factor in determining the amplitude at which the adiabatic condition is satisfied. Below I compare the inversion performance of 3 different AFP pulses, HS1, HS8 and tanh/tan, for different R values. The HS1 and HS8 pulses both achieve OIA at R ≥ 40, but HS1 requires twice the amplitude of HS8 (or 4 times the Q-factor). However, the HS1 pulse does provide a sharper inversion profile with narrower transition widths. At low R‐values, the HS8 pulse does not adequately satisfy the adiabatic condition, making the HS1 pulse the only valid choice. However, HS8 pulses are a good choice when the maximum available RF amplitude is low.


HS1, Q=20

R=10, B1max = 1.4 kHz

R=20, B1max = 2 kHz

R=40, B1max = 2.8 khz

HS8, Q=5

R=10, B1max = 0.7 khz

R=20, B1max = 1 khz

R=40, B1max = 1.4 khz

tanh/tan, Q=1.5

R=100, B1max = 1.22 khz

R=200, B1max = 1.73 khz

R=400, B1max = 2.44 khz


The tanh/tan pulse on the other hand behaves quite differently. While it requires a much higher minimum R-value for adequate performance, it works quite well at lower amplitudes. Additionally, tanh/tan does not exhibit OIA, which means that its inversion bandwidth increases with increasing B1max. This makes them great choices for inversion over wide bandwidths, while HS1, HS8 are more suitable frequency selective inversion as required in spatial localization.

These factors should be kept in mind when creating adiabatic pulses. It is always a good idea to use simulation tools and verify whether the performance is satisfactory for the  given R and Q values.

Python scripts for generating these pulses and Bloch simulations can be made available upon request, until I get around to releasing them as a GUI tool some day!

References

  1. De Graaf, R. A. (2019). In vivo NMR spectroscopy: principles and techniques. John Wiley & Sons.

  2. Claridge, T. D. (2016). High-resolution NMR techniques in organic chemistry (Vol. 27). Elsevier.

  3. Tannüs, A., & Garwood, M. (1996). Improved performance of frequency-swept pulses using offset-independent adiabaticity. In Journal of Magnetic Resonance - Series A (Vol. 120, Issue 1)

Location: Toronto, ON, Canada

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