Gradient-free spatial encoding is attractive for completely silent scanning, simplified MR system design, reduced heat generation and power consumption, and elimination of interference induced by eddy currents. A number of studies have focused on spatial encoding using radio frequency (RF) coils instead of gradient coils (e.g. rotating framing imaging¹, TRASE², etc). Recently, a novel RF encoding method based on Bloch-Siegert shift was developed by Kartasch et al³. They demonstrated the feasibility of 1D linear encoding with a specially designed B₁ field. In this study we show that with various independent B₁ fields from multiple RF coils, the spaces can be encoded nonlinearly and images can be reconstructed with such nonlinear encoding.

The Bloch-Siegert shift can be described as follows: for a spin at location r, when an RF pulse is applied with a frequency shift ω_RF from the Larmor frequency, the spin’s precession frequency in the rotating frame is $$$\omega_{BS}=\frac{(\gamma{B_1})^2}{2\omega_{RF}}$$$, if $$$\omega_{RF}>>\gamma{B_1}$$$⁴. Because the RF transmit field B₁(r) varies spatially, the precession frequency changes accordingly, which is used for spatial encoding. Due to the inherent nonlinearity of the B₁ field, this approach is well suited to nonlinear encoding strategies (similar to O-space⁵, Null space⁶, single echo⁷, and PatLoc⁸ imaging) combined with parallel imaging methods⁹ and iterative arithmetic reconstruction technique (ART)¹⁰ to accelerate image acquisition.

The measured signal is the sum of all spins modulated by encoding phasors: $$$S(t)=\int{m(r)e^{-j\int{\omega_{BS}(r)dt}}}dV$$$, where m(r) is the spin density (image contrast) at location r (all the relaxation terms are neglected). In a matrix form: $$$S=E\cdot{I}$$$, S is a vector of all the signals in time series and E is the encoding matrix. I is a vector of spin density in the imaging space⁶, which can be solved to obtain the image.

A single RF coil cannot provide sufficient information to distinguish voxels at iso-B₁ regions. Therefore, multiple transmit channels are used to increase the encoding variations. Specifically, the B₁ field at location r is the linear combination of contributions from all the transmit channels: $$${B_1}(r)=\sum{{A_c}\cdot {B_{1c}}(r)}$$$, where A_c is a complex number depicting the amplitude and phase of the transmit voltage on coil c, and B_1c is the unit B₁ field from coil c. Numerous configurations of transmit voltages can be applied on the transmit coils, resulting in many B₁ patterns to ensure that voxels will have unique frequency encodings throughout all patterns.

To illustrate the encoding, we modeled an 8-channel microstrip RF array in Comsol (Burlington, MA) (Fig.1a). The unit B1 fields from individual channels were simulated (Fig.1b). Figure 1c demonstrates that in-phase and out-of-phase voltages were applied to channel 1 and 5, resulting in completely different B₁ encoding fields. Additionally, two voxels in the imaging space (m and n) had an identical encoding frequency when channel 1 was excited, but during other encoding patterns, the two voxels experienced entirely different encoding frequencies. Therefore, they can be correctly distinguished in image reconstruction. We simulated signal measurements using a numerical phantom¹¹ with a pulse sequence depicted in Fig.2. A prephasing RF pulse with frequency $$$-\omega_{RF}$$$ was applied prior to the encoding RF to ensure the peak occurs at the center for each echo. Each TR represented one encoding pattern. The simulation used 36 encoding patterns. We tested image reconstruction with averaged B₁ fields ranging from 30 to 120 μT, and off-resonance frequencies being 4 and 8 kHz. The image (128×128) was reconstructed with 512 simulated readout points from individual receive channels (8 total), with a dwelt time of 0.1ms for each pattern. The simulated scan time is approximately 3 s for a 2D slice. The Kaczmarz algorithm was used for the image reconstruction.

Fig.3 shows the reconstructed images under different simulation conditions. The mean square error (MSE) indicated that with a higher B₁ field, the reconstructed image was more accurate due to improved encoding (Fig.3a-c). Another factor affecting the encoding is the off-resonance frequency; since ω_RF is in the denominator, the encoding efficacy is reduced with a higher frequency offset (Fig.3d-e). One limitation of the Bloch-Siegert shift is the specific absorption rate (SAR) due to prolonged RF pulses, but it can be potentially reduced with an increased TR at the cost of longer scanning time.In this study, we demonstrate that with multiple RF coils, it is feasible to nonlinearly encode the imaging space using the Bloch-Siegert shift. Simulated reconstructions showed that higher B₁ fields and lower off-resonance frequency shift improves reconstruction quality. This approach is potentially promising as a replacement for conventional gradient encoding providing excellent spatial encoding with essentially silent imaging.