Purpose

The MRI scanner is an invaluable clinical tool, yet there are great health care disparities with this technology. Due to its cost and infrastructure requirements, MRI is often inaccessible to people with lower socioeconomic status. In an effort to reduce cost and increase portability, we have pursued techniques that eliminate the need for certain expensive, large components of the MRI scanner. Specifically, with only RF field (B₁) gradients for spatial encoding, B₀ gradient coils and amplifiers can be eliminated. Previous RF imaging methods^{1, 2, 3} have been limited by their intolerance to resonance offset and B₀ inhomogeneity, which is incompatible with the goal of employing small magnet configurations. Here, a family of B₁-encoded spin-echo sequences are presented as a solution to this problem. The method is dubbed FREE for Frequency-modulated Rabi Encoded Echoes.

Methods

In FREE, spatial information is encoded similar to conventional phase-encoding, although the gradient is in B₁, rather than B₀. FREE is accomplished by exploiting properties of any frequency-modulated adiabatic full-passage (AFP) pulse (e.g., Hyperbolic Secant (HS)⁴). AFP pulses produce uniform flip angle of π, even with large variations in the magnitudes of B₁ and B₀⁵. When AFP pulses of differing time-bandwidth products (R) are used in a double spin-echo sequence, the phase of the echo signal can be made to be proportional the amplitude of the B₁ field⁶. In FREE, the R-value of the generated pulse patterns are increased by increasing the pulse duration (Tp), while holding the pulse’s bandwidth (BW) constant. The maximum Rabi nutation frequency during the pulse is $$$\omega_{1}^{max} = \gamma B_{1}^{max}$$$ , where $$$\gamma$$$ is the gyromagnetic ratio. When $$$\omega_{1}^{max}$$$ is spatially dependent (i.e., in a B₁ gradient), a spatially-dependent phase is imprinted on the spin-echo signal. Due to the AFP’s sweeping effective field, the total phase accrued by the end of a given π rotation is
$$ \phi_{HS} = \pm \frac{Tp}{2}\int_{-1}^{1}\sqrt{(\omega_{1}^{max})^{2}sech^{2}(\beta \tau) + (\Omega - A \cdot tanh(\beta \tau))^{2}} d\tau \:(1)$$
where β is a truncation factor, set to sech( β ) = 0.01, Ω is resonance offset in rad/s, A = BW/2 in rad/s, τ is normalized time defined as 2t/Tp - 1 for 0 ≤ t ≤ Tp, and R = ATp/π. The sign of in Eq. 1 depends on the direction of the frequency sweep (e.g., from low to high). When varying the R of AFP pulses in a double spin-echo sequence (Fig. 1), the effect of B₀-inhomogeneity (accounted for by Ω) is negated, while only a linear dependence on remains. The dependence of $$$\phi_{HS}$$$on $$$\omega_{1}^{max}(x)$$$ for double echoes produced with different R values is shown in Fig. 2. A direct relationship exists between the difference in R values (∆R) and the amount of phase encoding achieved, as described by Eqs. 2 and 3. For given field-of-view (FOV) and known B₁ gradient, the value of ∆R that satisfies the Nyquist criterion can be determined. A k-space trajectory is achieved by repeated incrementation of ∆R. For given FOV and, ∆R for Cartesian sampling is calculated as
$$\Delta R = \frac{A}{\frac{d}{dx}[-C(\omega_{1}^{max}(x))]\cdot2 FOV} \: (2)$$
where
$$C(\omega_{1}^{max}(x)) =\left[ A^{2}\left(\frac{1}{\beta} log(\omega_{1}^{max}(x)) - 1 \right) + \frac{A}{2\beta}\left(\sqrt{(\omega_{1}^{max}(x))^2 - A^2}\right) tan^{-1}\left(\frac{2A \sqrt{(\omega_{1}^{max}(x))^2 - A^2}}{-2A^2 + (\omega_{1}^{max}(x))^2}\right) \right] \: (3)$$
Using multiple shots (Fig 1a), k-space is sampled by keeping the R of one of the pulses constant and varying the other by integer multiples of ∆R. The first shot has the greatest difference in R-values (R_o - R_min). Defining N as the number of samples to be collected, and R_o as the R value of the non-changing pulse, then
$$R_{min} = R_{o} - \frac{N}{2} \Delta R \: (4)$$
Experimental and simulated studies utilized the following parameters: linear B₁-gradient having $$$\omega_{1}^{max}(x)$$$ ranging from 0.25 to 2.5 kHz (simulations), and 0.25 kHz to 3.56 kHz (experiments), HS8 pulses (BW = 4 kHz), with maximum Tp = 10 ms, and R from 24.25 to 40, with ∆R = 0.25. Experiments were performed with the sequence seen in Fig. 1a. Data were acquired with a CIERMag Digital Magnetic Resonance Spectrometer⁸, configured for two Tx/Rx channels and operating at 1.5 Tesla (63.8 MHz), controlled by Python Magnetic Resonance Framework (PyMr^{9, 10, 11}) and ToRM-Console¹². Gradient-echo (GRE) and FREE reconstructions are compared.

Results

Simulated results (Fig. 4) demonstrate the ability of FREE to produce reconstructions which are similar to that of standard phase-encoding, until a large resonance offset (±2kHz) exists across the object. Experimental 1D images of the phantom (Fig. 5) show excellent agreement between FREE and frequency-encoded projections of the object, at least for positions nearest the coil.

Discussion

The distortions observed in the simulation with large offset (±2kHz) are due to limited pulse bandwidth, and thus, simply increasing BW would remedy this. At locations far from the coil, the phantom was not visualized well with FREE, most likely due to poor adiabatic condition in this low region. In addition, the reconstruction was affected by the non-linear B₁ of the surface coil. To remedy this, image distortion-correction algorithms might be applied in the future⁷.

Conclusion

FREE enables phase-encoding with no B₀ gradient coil. The sequences presented here show the potential for a new approach that enables the removal of gradient coils in future MRI system designs

Acknowledgements

This work was supported by National Institute of Health grants P41 EB027061 and U01 EB025153, Schott Family Foundation, and the Minnesota Lions.