Purpose:

The high cost of MRI systems relegates their clinical impact to a minority of the world. MRI remains inaccessible to ~90% of the world’s population¹. Frequency-modulated Rabi Encoded Echoes (FREE) are a family of B₁ encoding pulse sequences^2-3. FREE aims to develop cheaper, quieter, and smaller MRI systems without B₀-gradient coils. We present Frequency-Encoded (FE-) FREE which performs frequency encoding using a B₁ gradient, samples using gapped pulses⁴, and promotes the removal of dedicated B₀-gradient coils.

Theory:

FREE uses adiabatic full passage (AFP) pulses to produce spin echoes insensitive to B₀ and B₁ inhomogeneity⁵. The accumulated phases of the transverse magnetization during the AFPs are spatially dependent and described by ψ(r,t)⁶:
$$\psi(r,t) = \int_{0}^{t} \omega_{eff}(r,\tau) d\tau$$
$$\omega_{eff}(r,t) = \sqrt{(\omega_{1}^{max}(r)\cdot AM(t))^{2} + (A\cdot FM(t))^{2}} $$
where τ is a dummy integration variable; t is time; r is spatial distance; r_max is the field-of-view (FOV); T_p is pulse length; AM(t) and FM(t) are the AFP’s pulse amplitude- and frequency- modulation functions; A = BW_Ω/2, where BW_Ω is the pulse excitation bandwidth; and ω₁^max(r) is the peak value of γB₁ at position r. ψ(r,t) is approximately linearly dependent² on ω₁^max(r). In the current implementation, FE-FREE consists of two successive AFPs following an excitation pulse (Fig 1). The resulting spatial phase is defined by
$$\psi_{net} (r) = \psi_{dp} (r, T_{p}^{dp}) - \psi_{ro} (r, T_{p}^{ro})$$
T_p^dp and T_p^ro define the lengths of the first and second AFPs. ψ_dp and ψ_ro define the phase from the first and second AFP. Both pulses are transmitted with the same BW_Ω, FM(t), and B₁ gradient. By setting T_p^ro = 2T_p^dp, a rotary echo centered at the middle of the second AFP is created. Hyperbolic secant (HS) pulses are used for both AFPs due to their smooth inversion profiles across B₀ inhomogeneities^7-8. Δk, defined as
$$ \Delta k(t) = \frac{d\psi_{ro}(r,t)}{dr}$$
varies throughout the HS pulses (Fig 2). Figure 2 shows that higher-order HS pulses (e.g. HS10) present lower variations in Δk. Delays of length τ_rd + τ_ro are placed in between pulse elements (τ_p) of the second AFP. τ_rd is inserted to allow coil ring-down (Fig. 1), and data acquired during τ_ro are used for reconstruction. Gapping minimally affects⁹ pulse performance and permits sampling of the rotary echo. In FE-FREE, signal is sampled while the second AFP executes a 180º rotation of the transverse magnetization. In the first (frequency-modulated) rotating frame, the transverse magnetization lacks the conjugate relationship found in a typical echo. The rotary echo splits into M_x, M_y, and M_z magnetization components. In a second reference frame rotating at the angular velocity defined by the effective field angle, α(t), the M_x and M_y contain a conjugate relationship (Fig 3), where α(t) is defined as
$$ \alpha(t) =arctan\left( \frac{AM(t)}{FM(t)}\right)$$
To reach the second rotating frame, a y-rotation matrix using α(t) of the peak γB₁ is applied to the data. To apply the rotation, the transverse signal (S_xꞌ and S_yꞌ) is sampled through traditional approaches, and the longitudinal signal, S_zꞌ(t), by calculation,
$$ S_{z'} = S_{max} - S_{xy'}(t)$$
where S_max is the maximum of the transverse signal, S_xyꞌ(t). The object is reconstructed by Fourier transformation of the transverse signal in the second rotating frame (S_xꞌꞌ and S_yꞌꞌ).

Methods

Simulations: A linear γB₁ gradient of 8.5 kHz ∙ (5 cm)^-1 was used. The first and second HS10 time-bandwidths were 0.012 s ∙ 2000 Hz and 0.024 s ∙ 2000 Hz, respectively. τ_p = 50 μs and τ_ro = 50 μs, and data collection occurred during the middle 75% of the pulse. 300 points were reconstructed. Reconstructions of the central slice of the Shepp-Logan phantom¹⁰ were evaluated for on- and off-resonant cases.
Experiments: A 5-cm surface coil transmitted a non-linear γB₁ gradient of ~10 kHz ∙ (5 cm)^-1. A 100 μs square pulse excited the spins. The first and second HS10 time-bandwidth products were 0.020 s ∙ 2,000 Hz and 0.040 s ∙ 2,000 Hz, respectively. τ_p = 80 μs, τ_rd = 50 μs, and τ_ro = 5 μs.1 point was acquired per τ_ro. The total acquisition period, including pulse elements and gaps, was 0.0675 s. α(t) was defined using a γB₁ of 10 kHz. Data were acquired with a single Tx/Rx CIERMag Digital Magnetic Resonance Spectrometer operating at 0.5 T^11-14. The 0.5 T Halbach magnet had no B₀ gradient coils and a Larmor frequency variation of ± 5 kHz across the phantom. 450 points were reconstructed.

Results

Simulations demonstrated excellent agreement with the true object (Fig 4). Experiments demonstrated the ability to produce the necessary rotary echo in a highly inhomogeneous magnet with no B₀ gradient coils, leading to a good reconstruction of the 1D object (Fig 5).

Discussion

As shown here, FE-FREE requires only a B₁ gradient and is tolerant of B₀ inhomogeneity. At lower magnetic fields, merging FE- and phase-encoded (PE) - FREE may permit volumetric imaging without B₀ gradient coils. The ability to use an inhomogeneous (small) magnet together with the elimination of B₀ gradients promises to significantly cut costs and lead to a radically new class of MRI scanner in the future.

Conclusions

FE-FREE presents a promising B₁-encoding approach that enables the removal of dedicated frequency-encoding B₀ gradient coils.

Acknowledgements

This research was supported by the Minnesota Lions Diabetes Foundation, Schott Family, and Malcolm B. Hanson Endowed Chair in Radiology. Additional support was provided by the National Institutes of Health grants U01 EB025153 and P41 EB027061. E. Torres is supported by the NSF GRFP and CIC Fellowships.

References

1. Geethanath S, Vaughan JT. Accessible magnetic resonance imaging: a review. J Magn Reson Imaging. 2019;49:e65-e77

2. Torres E, Froelich T, Wang P, et al. B1-gradient–based MRI using frequency-modulated Rabi-encoded echoes. Magn Reson Med. 2021;00:1–12.

3. Hoult DI. Rotating frame zeugmatography. J Magn Reson.1979;33:183-197. Geethanath S, Vaughan JT.

4. Garwood M. Et Al. Fast and quiet MRI using a swept radiofrequency. J Magn Reson. 2006;181:342-349.

5.DelaBarre L., Garwood M. The return of the frequency sweep: designing adiabatic pulses for contemporary NMR. J Magn Reson. 2001;153:155-177.

6. Lee Y, Et Al. New phase-based B1 mapping method using two-dimensional spin-echo imaging with hyperbolic secant pulses. Magn Reson Med. 2015;73:170-181.

7. Tannus A, Garwood M. Adiabatic pulses. NMR Biomed.1997;10:423-434.

8. Silver MS, Joseph RI, Hoult DI. Highly selective π/2 and π pulse generation. J Magn Reson. 1984;59:347-351

9. Djaudat Et Al. Gapped pulses for frequency-swept MRI. J Magn Reson. 2008;193:267-273.

10. Shepp LA, Logan BF. The Fourier reconstruction of a head section. IEEE Trans Nucl Sci. 1974;21:21-43.

11. Pizetta D, Lourenço G, Silva D, Vidoto E, Martins M, Tannús A. “Magnetic resonance system configuration and editing tools”. (2015); Toronto,Canada, p667-668

12.Tannús A, Pizetta D, Silva D, Vidoto E, Lourenço G, Martins M, inventors; Universidade de São Paulo, assignee. ToRM-IDE - Integrated development environment for magnetic resonance applications. Brazil patent BR512015001484-6. 2015.

13.Pizetta D, Shimada D, Falvo M, et al, inventors; PyMR - a framework for programming magnetic resonance systems. Brazil patent BR512019001829-0. 2017.

14. Pizetta D, Lourenço G, Silva D, Vidoto E, Martins M, Tannús A. Magnetic resonance system configuration and editing tools. presented at: IUPESM 2015: Wold congress on medical physics & biomedical engineering; Jun 7-12 2015; Toronto, Canada.