Radial TRASE: 2D RF encoding through mechanical rotation and active digital decoupling

Abstract

Purpose

Two‐dimensional (2D) transmit array spatial encoding (TRASE) previously required four radiofrequency fields; however, interactions between transmit (Tx) array elements caused significant challenges for 2D imaging. Here, we present a low‐cost, 2D radial encoding scheme (Radial TRASE) using a simplified two‐coil array.

Theory and Methods

The system consists of two B₁ phase gradient coils capable of encoding any one transverse direction. By incremental mechanical rotation over a 90° range, the encoding axis can be changed, allowing a complete radial k‐space acquisition. As a first demonstration, a wrist‐sized coil pair was experimentally verified on a 2‐MHz Halbach magnet, incorporating a static B₀ slice‐selection gradient. Although a high level of isolation is achievable geometrically, for a more robust implementation, we demonstrate the capability of active digital decoupling in eliminating residual coupling through a parallel‐transmit system.

Results

Radial TRASE–encoded images of water phantoms were acquired, achieving a resolution better than 1.67 mm. Rotation of the Tx array was performed during the recovery period, which caused no imaging delays. All acquired images show minimal distortions, indicating the advantage of the simplified Tx array. The active digital decoupling technique is demonstrated to eliminate residual coupled currents, effectively increasing the isolation of the two‐coil array to −50 dB. Sequential axial slice images were demonstrated using a uniform B₀ coil to shift the slice position.

Conclusion

Two‐coil Radial TRASE can encode a 2D slice without rapidly switched B₀ gradients. Compared with previous three‐coil or four‐coil Cartesian TRASE, the design and isolation of the Tx array are significantly simplified.

1. INTRODUCTION

MRI is a powerful diagnostic tool for many soft‐tissue diseases; however, it has limited accessibility compared with other imaging modalities due to its cost and footprint, among other factors. ¹ , ² , ³ , ⁴ Modern commercially available 1.5T MRI units weigh several tons and can cost more than $1 million USD, with the superconducting magnet (B₀ magnet) and its field gradients significantly contributing to the costs. ¹ Recently, there has been growing interest in low‐cost, low‐field MRI for improved accessibility. ⁵ , ⁶ , ⁷ , ⁸ , ⁹ , ¹⁰ , ¹¹ , ¹² Commercially, Hyperfine has developed the first point‐of‐care, low‐field clinical system (SWOOP; Hyperfine Research, Guilford, CT, USA), approved by the Food and Drug Administration in 2021. ¹³ Similarly, the motivation of this project was the construction of a low‐field, low‐cost MRI system by replacing conventional B₀ encoding with radiofrequency (RF) transmit‐field (B₁) encoding principles. Here, we present an MRI methodology to B₁ spatially encode a two‐dimensional (2D) plane by projection acquisition with mechanical coil rotation and a static B₀ slice‐selection gradient.

Transmit array spatial encoding (TRASE) is an MRI technique that spatially encodes with B₁ phase gradient fields. ¹⁴ The TRASE sequence consists of long refocusing trains in which the phase gradients provide an incremental phase modulation, similar to conventional B₀ phase encoding. For one‐dimensional (1D) encoding, two such B₁ fields are required, and four for the highest‐resolution 2D Cartesian sampling. ¹⁴ , ¹⁵ , ¹⁶ The phase gradient fields are produced by an RF transmit (Tx) array, requiring a high level of isolation between each element. ¹⁷ , ¹⁸ A previous study based on Bloch simulations suggested that an isolation of at least −30 dB is necessary between each element of the 2D Tx array. ¹⁸ This isolation is impractical to achieve geometrically with four‐volume coils, whereas active PIN‐diode techniques have proven difficult due to the speed of switching required under high power conditions. ¹⁷ An alternative isolation strategy is required.

In 2013, 2D Cartesian TRASE imaging of the wrist was performed for the first time using four B₁ fields with a Helmholtz‐Maxwell coil array. ¹⁶ Although successful, this switched four‐coil Tx array was inefficient (300 W for 500‐μs refocusing pulses) and provided a small imaging volume relative to the coil formers. In 2019, the twisted solenoid was introduced as a more efficient TRASE coil, capable of producing a single‐phase gradient field in any transverse direction. ¹⁹ Using two geometrically decoupled twisted solenoids with opposing phase gradient fields (e.g., −G _1,y and +G _1,y ), 1D encoding was performed in phantom studies ¹⁹ , ²⁰ ; however, 2D imaging with four twisted solenoids has not been demonstrated. To avoid these complexities, we present a two‐coil solution termed “Radial TRASE.” This method encodes 2D k‐space radially, effectively replacing any additional RF power electronics and coils otherwise required for 2D TRASE with a simple mechanical rotation system. The two‐coil array is geometrically decoupled, with additional isolation achieved through active digital decoupling using a parallel‐transmit (pTx) system, effectively eliminating residual coupling currents.

2. THEORY

2.1. TRASE encoding principle

The TRASE encoding principle uses phase gradients ${\mathit{G}}_1$ (in radians per meter) within specially designed B₁ Tx fields for spatial encoding. ¹⁴ These fields are of the form ${\mathit{B}}_1=\left|{\mathit{B}}_1\right|e^{i{\varphi }_1(\mathit{r})}$ (spatial phase ${\varphi }_1(\mathit{r})={\mathit{G}}_1\mathit{·}\mathit{r}$) and are entirely created through the unique geometry of the RF coil(s). ¹⁴ , ¹⁵ , ¹⁶ , ¹⁷ In k‐space, the phase gradient defines a vector k $\left({\mathit{G}}_1=2\pi \mathit{k}\right)$ in cycles per meter, termed the “k‐space origin” of the coil, which itself represents the phase variation in the B₁ field. ¹⁴ Excitation with this phase gradient field excites the spins to the k‐space origin k , rather than the center of k‐space. Similarly, 180° refocusing reflects the spin states about the coil's k‐space origin.

For 1D encoding, the TRASE sequence consists of a 90° excitation pulse followed by a long series of 180° refocusing pulses as depicted in Figure 1. ¹⁴ To encode, the refocusing pulses are alternated between two B₁ fields (produced by coils A and B ) with differing phase gradients ( k _A and k _B ). Each refocusing pulse reflects the spin state about its respective k‐space origin, resulting in a discrete and progressive spatial phase evolution. Because of this repeated reflection about k _A and k _B , the k‐space trajectory progresses in a “jumping fashion” with a sample spacing of $∆{\mathit{k}}_{\mathit{AB}}=2\left|{\mathit{k}}_A-{\mathit{k}}_B\right|$. As a Fourier‐based imaging technique, the FOV and spatial resolution are inversely proportional to the k‐space sample spacing and sampled k‐space extent, respectively, as follows ¹⁴ :

$$\begin{array}{cc}\mathit{FOV}& =1/∆{\mathit{k}}_{\mathit{AB}}=1/\left(2\left|{\mathit{k}}_A-{\mathit{k}}_B\right|\right)\\ ∆x& =1/\left(∆{\mathit{k}}_{\mathit{AB}}·\mathit{ETL}\right)=1/\left(2\left|{\mathit{k}}_A-{\mathit{k}}_B\right|·\mathit{ETL}\right)\end{array}$$ (1)

where ETL is the echo train length. In practice, a phase gradient magnitude of 500°/m ($k=1.39\mathit{\text{cycles}}/\mathrm{m}$) is typical throughout the literature, ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ corresponding to a field of view (FOV) of 180 mm and resolution of 1.41 mm for an ETL of 128.

FIGURE 1.

The 1D TRASE sequence and trajectory through k‐space. The TRASE sequence consists of a long echo train with refocusing pulses alternating between the two differing phase gradient coils ( A and B ). In k‐space, each refocusing pulse causes a reflection of the spin states about that coil's k‐space origin ( k _A and k _B ). The resulting trajectory through k‐space is a repeated jumping fashion, with each echo (e.g., e1, e2, e3) corresponding to a single point in k‐space with separation $∆{\mathit{k}}_{\mathit{AB}}=2\left|{\mathit{k}}_A-{\mathit{k}}_B\right|$.

Although not discussed in detail here, the TRASE encoding principle can be extended to 2D imaging by adding a third non‐collinear RF phase gradient field, with a fourth required to maintain the same resolution of the 1D sequence. ¹⁴ , ¹⁶

2.2. Radial TRASE imaging

2.2.1. Projection acquisition via mechanical rotation

In 2019, Sun et al. outlined the twisted solenoid as a new class of TRASE coil, ¹⁹ being a solenoidal variant with a sinusoidal modulation in the z‐direction (frequency of two per turn). Due to this modulation, in addition to the axial solenoidal field, the coil produces a quadrupole field in the transverse plane. ¹⁹ , ²¹ , ²² Although one quadrupole component is parallel to B₀, the perpendicular component contributes to a single B₁ phase gradient field. Depending on the coil's axial rotational orientation relative to the vertical B₀ field, the perpendicular component changes, which determines the transverse direction of this phase gradient. ¹⁹ Specifically, due to the 2‐fold rotational symmetry of the quadrupole, by rotating the coil by $+\alpha$ about its axis, the phase gradient direction changes by an angle $2\alpha$, as depicted in Figure 2. For further details, the phase gradient direction and effect of rotation are discussed by Sun et al. ¹⁹

FIGURE 2.

Illustration of the twisted solenoid phase gradient direction changing under rotation in a vertical B₀ field. (A) The 1D TRASE Tx array consisting of a twisted solenoid pair ( A and B ) with opposite phase gradients (G _1,A and G _1,B) along the x‐axis. After a mechanical rotation of $\alpha$ about the coil's axis (z‐direction), the two phase gradients are shifted by an angle $2\alpha$. (B) TRASE k‐space acquisition from the initial (I) and rotated configuration (II). In the initial configuration, the 1D‐TRASE sequence encodes the x‐axis. After the rotation by $\alpha$, the shifted phase gradient fields (and corresponding k‐space origins k _A and k _B ) encode in the direction $2\alpha$. To fully sample two‐dimensional k‐space radially, the phase gradients need to be incrementally shifted over a 180° span, which is achieved by physical incremental rotation over a 90° range.

As the transverse (radial) encoding direction can be changed, this rotatable phase gradient can form the basis for projection acquisition imaging. Typically, a twisted solenoid pair can 1D encode one entire k‐space line (i.e., full “radial” spoke). Through incremental mechanical rotation over a 90° angular range, the coil pair phase gradients can be stepped through a 180° range. In this way, 2D k‐space can be fully sampled in a radial spoke fashion using the two‐field 1D TRASE sequence. Similar to conventional projection imaging, reconstruction of the radial data set can be performed through filtered back projection or a 2D Fourier transform with a Cartesian gridding algorithm. ²³ In reconstruction, the expected spatial resolution of a fully sampled data set is determined by the 1D radial spoke resolution ²³ (i.e., the 1D‐TRASE sequence [Eq. (1)]). In principle, this approach is similar to the earliest projection MRI methods, where B₀ gradients were rotated about the sample, such as the work done in 1973 by Paul Lauterbur. ²⁴ More recently, other groups have also investigated physically rotating B₀ gradients for their benefits in silent, low‐cost imaging systems. ²⁵ , ²⁶

2.2.2. Slice selection with a static axial B₀ gradient

Radial TRASE offers a method for in‐plane encoding but leaves the axial direction unencoded. For the third dimension, a static axial slice‐selection B₀ gradient was used, avoiding the need for a switched B₀ gradient system. Instances of imaging within static gradients can be found throughout the literature, such as STRAFI (stray field imaging) ²⁷ , ²⁸ and NMR MOUSE (mobile universal surface explorer). ²⁹ , ³⁰ Under these static conditions, this built‐in gradient causes reversible dephasing in the slice direction, similar to a readout gradient. As such, during each echo acquisition window, this static gradient frequency encodes the slice (axial) direction. Because Radial TRASE effectively phase‐encodes the transverse directions radially, a three‐dimensional 3D data set is therefore collected, consisting of (potentially) several thin partitions within the RF‐selected slice. The thickness of each partition is defined by the Fourier process (i.e., readout bandwidth per pixel), which is narrower than the RF‐selected slice. In reconstruction, this is used to select the central (on‐resonance) portion of the RF‐selected slice with the effect of eliminating slice‐edge/off‐resonance artifacts as demonstrated previously by Sedlock et al. in an inhomogeneous Halbach magnet. ²⁰ Specifically, the static readout gradient benefits short, hard pulses, which generally have poor RF bandwidths (i.e., slice profiles) but are advantageous in sequence timing.

In the readout direction, the bandwidth per pixel is inversely proportional to the acquisition window $T_{\mathit{acq}}(=\mathrm{\Delta }\mathit{t}·n)$ as ${\mathit{BW}}_{\mathit{acq}}=1/T_{\mathit{acq}},$ ²³ where $T_{\mathit{acq}}$ is the product of the dwell time Δt and samples n. For a constant sampling rate, the static B₀ readout gradient G _z then defines partitions of thickness $∆z_{\mathit{par}}$ as follows ²³ :

$$∆z_{\mathit{par}}={\mathit{BW}}_{\mathit{acq}}/\left(\gamma G_z\right)=1/\left(\gamma G_z{T}_{\mathit{acq}}\right)$$ (2)

where γ is the gyromagnetic ratio. This equation represents the resolution in the slice direction (i.e., partition thickness), with the readout forming a stack of partitions, of which we only reconstruct the central partition.

It should be noted that although a 3D data set is collected, only the central (on‐resonance) partition is reconstructed. Especially for hard pulses, the further partitions quickly show increasing off‐resonance artifacts, caused by TRASE phase errors from incomplete refocusing. ³¹ A prior 2D Cartesian TRASE study indicated that successful TRASE encoding with a hard pulse (of length $t_p$) was observed over a $0.2/t_p$ sample bandwidth. ¹⁶ For a 200‐μs hard pulse, the required receiver bandwidth of 1 kHz per pixel can be achieved with an acquisition window of 1000 μs. A partition thickness of 5 mm could then be reconstructed with a static gradient of strength 4.7 mT/m. For a permanent static gradient, this thickness can be slightly adjusted by changing the acquisition window.

3. METHODS

3.1. B₀ magnet

3.1.1. Halbach magnet and static B₀ gradient

All experiments were performed within a cylindrical Halbach magnet with a central 2‐MHz resonance frequency. The constructed magnet was based on an open‐source design by O'Reilly et al. at Leiden University Medical Center. ³² The magnet is 54 cm long with a minimum internal diameter of 27 cm. An optimized shim array was constructed and achieved a field homogeneity of 3400 ppm over a 16‐cm‐diameter spherical volume. For slice selection, a static axial gradient (7 mT/m) was produced by adding two rings of opposite polarity to the magnet ends. The gradient is reasonably linear over a 150‐mm‐diameter, 100‐mm‐long cylinder. However, the selected slices are tilted by 18° relative to the transverse plane. The magnet was placed in a copper‐shielded room to minimize electromagnetic interference.

3.1.2. Sequential slice imaging with a uniform B₀ shift coil

Sequential or interleaved slice imaging can be performed within the static axial gradient over the repetition time (TR) period. However, the number and range of slices are limited by the coil's narrow 15‐kHz bandwidth and the nonselective pulses used. For a wider range, a uniform B₀ shift coil was added inside the bore of the Halbach magnet. The geometry of this coil was based on an open‐source tool, CoilGen. ³³ The shift coil has a diameter of 200 mm, a length of 300 mm, and a mean vertical field strength of 0.0485 mT/A. With the approximately 7‐mT/m axial gradient, the slice position is expected to shift by 6.93 mm/A.

3.2. RF coil design and decoupling

3.2.1. Coil geometry

To demonstrate the Radial TRASE technique, a wrist‐sized truncated twisted solenoid coil pair from our previous study was used. ²⁰ This 200‐mm‐long coil array consists of an inner and outer coil of 100‐mm and 125‐mm diameters, respectively. In summary of its design from reference, ²⁰ the target imaging volume was 100‐mm long and 80‐mm in diameter. The optimized wire pattern of both coils was determined using Biot‐Savart field calculations with the BSMag MATLAB package, ³⁴ as described by Sedlock et al. ²⁰ Using the computer‐assisted modeling (CAM) program Fusion360 (Autodesk, California, USA), both coils were previously 3D‐printed with the optimized wire pattern modeled directly into their formers. The coil geometries and field characteristics are listed in Table 1 from Sedlock et al. ²⁰ For an ETL of 128, the expected 1D spatial resolution of this coil pair is 1.23 mm with a FOV of 158 mm (Eq. [1]). Although the coils were only optimized for one orientation, ²⁰ recent BSMag simulations of the B₁ fields found negligible change with rotational angle, as depicted in Figure 3.

TABLE 1.

Coil geometry and characteristics: The geometric parameters of the wrist‐sized truncated twisted solenoids from Ref. 20.

RF Coil	Diameter (mm)	Length (mm)	Turns	Pitch (mm)	Modulation amplitude (mm)	Mean B1 magnitude (uT/A)	Mean uniformity a	Mean phase gradient strength (deg/m)
Inner	100	200	16	15	24.4	79.7	1.044	575
Outer	125	200	15	17	33.2	70.0	1.038	501

Note: The mean B₁ field values were simulated over a central cylindrical region 100 mm long and 80% of the coil's diameter. The simulated B₁ fields do not include the influence of the decoupling solenoids or return wire. The gap between each truncated turn was 5 mm (g = 5 mm).

^a The mean uniformity was calculated as the mean of $\mid B_1\mid /\mid B_{1,\mathit{min}}\mid$, where $\mid B_{1,\mathit{min}}\mid$ is the minimum ${\mathrm{B}}_1$ magnitude in the imaging volume.

FIGURE 3.

B₁ field simulations in the center of the outer coil (z = 0 plane) for four physical rotation angles (0°, 30°, 60°, and 90°) showing the B₁ magnitude (A), B₁ phase (B), and B₁ phase gradient (C). The horizontal axis in each plot is along the encoding direction for each rotational angle. The black dotted lines designate the boundaries of the optimized imaging volume (80‐mm diameter). Within these bounds, the B₁ phase and magnitude are relatively unchanged during rotation in their respective encoding directions. The small variation that does exist is primarily due to the pitch of the radiofrequency coil (i.e., not tightly wound), resulting in the zero spatial phase location being slightly off‐center.

3.2.2. Geometric decoupling

Decoupling of the coil pair was primarily geometrical using the two‐element solenoid array method described in the reference. ¹⁷ This was achieved by attaching a co‐rotating and counter‐rotating regular solenoid (decoupling solenoids) in series with the inner and outer twisted solenoids, respectively. When positioned concentrically, the mutual inductance of these decoupling solenoids opposes that of the twisted solenoid pair. This configuration provides a Maxwell‐like and Helmholtz‐like coil combination, allowing geometric isolation with an appropriate choice of decoupling solenoid turns.

In construction, the decoupling solenoids were wound around their own 3D‐printed former with the same diameter as their respective coils and a 4‐mm pitch. As suggested by our previous work, ²⁰ the decoupling solenoids were redesigned to be taken further away from the twisted solenoids (>200 mm) to reduce B₁ field perturbations. The number of turns was initially determined using inductance simulations with the FastHenry2 toolbox. ³⁵ To account for other coupling mechanisms (e.g., capacitive), the number was modified based on constructed S₁₂ measurements. Maximum isolation was achieved with 11 turns for both coils. Under careful positioning, isolation levels better than $S_{12}=-23$ dB were achieved. To prevent shifting scatter parameters, the decoupling solenoids were rotated coaxially with the twisted solenoids.

3.2.3. Active digital decoupling using a pTx array

For accurate imaging, TRASE requires precise RF field generation, which is limited by the isolation of the RF‐Tx array. ¹⁸ The combination of partial geometric decoupling with active digital decoupling (ADD) can achieve the required performance. ADD is a method in which the input of a pTx array is adjusted to compensate for mutual coupling. ³⁶ ADD was first described by Vernickel et al. in 2007 ³⁶ and again by Hoult et al. in 2008. ³⁷ For two‐coil TRASE, each refocusing pulse uses a single coil (i.e., one coil is idle), which simplifies the implementation of ADD. In this case, the induced current within the coupled idle coil can be canceled with one simultaneous compensation ADD pulse of equal magnitude and opposite phase, effectively increasing the isolation of the Tx array. Similar to B₁ shimming at ultrahigh field strengths, ³⁸ , ³⁹ ADD requires that each Tx array element can be driven simultaneously and independently. Although this necessitates two independent/synchronized RF channels with precise pulse adjustments, it avoids all complications with high‐power switching during the tightly packed echo train, ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ , ²⁰ including RF power amplifier (RFPA) switching, phase shifters, and coil enable/disable circuits.

In practice, settings for the ADD compensation pulses are determined with a PicoScope (2405A; Pico Technology, Cambridgeshire, UK), a PC‐based oscilloscope. The current within both coils is measured directly from a small current‐measuring transformer. The primary windings are added in series within the RF coil resonance circuit (see Figure 4). The secondary windings are connected to the PicoScope, which directly monitors the RF coil current as an electromagnetic field (Faraday's law). Calibration consists of a single refocusing pulse transmitted on each coil individually, measuring the coupled current amplitude and phase within the idle coil. From this PicoScope measurement, a compensation pulse is generated with equal amplitude and opposite phase to the coupled current. If sufficient cancellation is not achieved, an iterative approach is used by repeating the calibration with the previous compensation pulse. The remaining residual coupled current is measured, and a second waveform is generated, which is added to the prior.

FIGURE 4.

Circuit diagram depicting the current monitoring transformer and a photo of the coil electronics. (A) Circuit diagram of the outer coil. The values of C_M and C_T (2 to 120 pF) represent the matching and tuning capacitors, respectively. The two fixed capacitors (C₁ and C₂) both had a value of 470 pF. The values of L_T and L_D (combined 37.2 μH) represent the outer twisted solenoid and its decoupling solenoid, respectively. A transformer was placed within the resonance circuit to measure the current amplitude and phase in the RF coils. (B) Photograph of the coil electronics outside the magnet. Both decoupling solenoids are wound around their own formers, with their exact position used to control the isolation (S₁₂). The current monitoring transformers are kept on the circuit boards away from the twisted and decoupling solenoids.

3.3. Coil array rotation system

Coil pair rotation is performed directly on the central axis. To prevent any RF interference, the motor was located outside the shielded room. A rigid 550‐mm‐long, 25.4‐mm‐diameter G10 drive rod connects the coils to the motor through a 300‐mm‐long, 55‐mm‐diameter copper waveguide. Within the magnet, the coil pair rests on a 97‐mm inner diameter 3D‐printed bore tube, which guides rotation. A 3D‐printed apparatus connects the coils axially to the drive rod. The decoupling solenoid formers are tightly connected to the drive rod outside the magnet (see Figure 4B). Through slight translation of the decoupling solenoids alone, the isolation of the coil pair can be adjusted. A complete 3D model of the rotation system is shown in Figure 5.

FIGURE 5.

Depiction of the mechanical rotation system. (A) Image of the connected motor outside the RF shielding. The drive rod connects to the motor and is passed directly through a waveguide to the coils. (B) A Fusion360 model of the coil pair within the magnet. The RF coils (red and yellow) rest on the bore tube (green), which positions the coils centrally and guides rotation. A three‐dimensional‐printed connection (gray) affixes the coils to the drive rod (blue) axially and houses their circuit boards.

The rotation electronics consist of a NEMA 24‐hybrid stepper motor (24HS39‐3008D; Stepper Online, Nicosia, Cyprus) with a two‐phase hybrid stepper‐servo driver (HSS60; Smart Automation, Changzhou, China). ⁴⁰ , ⁴¹ A Raspberry Pi computer was used to control the stepper motor automatically. Although the motor has a step angle of 1.8° (200 steps/revolution), micro‐stepping the driver allows 800 steps per revolution (0.45° increments).

3.4. Radial TRASE experimental description

3.4.1. Experimental setup and image reconstruction

A dual‐channel, high‐duty‐cycle RFPA designed for TRASE imaging was used to drive the coils. ⁴² The forward RF power was measured from a dual‐directional coupler (DDS‐1; PreciseRF, Aurora, Oregon, USA) with an oscilloscope (DPO2024B; Tektronix, Beaverton, Oregon, USA). The inner coil was connected to a passive crossed‐diode transmit‐receive (T/R) switch, allowing it to serve as the receive coil. For data acquisition/processing and to generate the TRASE sequence, a custom‐built console termed “DNMR” was used. The imaging sequence consisted of the XY‐4 phase cycling scheme ⁴³ with the following parameters: 200‐μs hard excitation and refocusing pulses, echo time = 2000 μs, ETL = 128, acquisition window = 1000 μs, TR = 3000 ms, and no averaging. During the TR recovery period, the coils were rotated in 0.45° increments up to 90°, giving 201 radial k‐space spokes separated by 0.9°. Incremental rotation steps were completed in 60 ms, resulting in no delays for this sequence.

Acquired data were processed to generate a sinogram (stack of 1D profiles), from which reconstruction used filtered back projection. TRASE 1D profiles typically show spike artifacts at the edge of the FOV due to unencoded spins resulting from B₀ and B₁ inhomogeneities. ³¹ , ⁴³ These were removed by truncating each profile before constructing the sinogram. For each profile, a smooth center of mass (COM) correction shift was applied. This compensated for any isocenter drift during rotation by shifting the COM of each 1D spoke to the center of the FOV, as is done for some CT and radial MRI applications. ⁴⁴ , ⁴⁵ k‐Space filtering was applied to reduce noise, and a zero‐filling factor of four was used in reconstruction. ⁴⁶

3.4.2. Rotational RF calibration

After each setup, a 5‐min calibration was performed to adjust the RF‐Tx parameters. The Tx power was determined from maximal echo amplitude of a Carr‐Purcell‐Meiboom‐Gill sequence. The Tx phase offset between the inner and outer coil was determined with the PicoScope directly from their current monitoring transformers. Similarly, the ADD settings were determined by transmitting on one coil and measuring the induced current in the idle coil. Due to the cylindrical design of the Halbach magnet, the scatter parameters S₁₁ and S₂₂ do not significantly change during rotation, so it was sufficient to calibrate only one angle for the Tx power and phase. Despite this, the isolation (S₁₂) did slightly change with rotation. To compensate for this shift, the ADD calibration was performed at seven angles between 0° and 90° (15° increments). The calibrated data were fit to a polynomial curve from which the ADD settings for all rotation angles were extracted.

3.4.3. Active digital decoupling demonstration

To demonstrate the capabilities of ADD, the current induced within the idle coil is displayed with and without the compensation pulse when refocusing on the active coil. The coil pair was tuned and matched to a resonance frequency of 2 MHz with scattering parameters: S₁₁ (inner) = −29.7 dB and S₂₂ (outer) = −26.4 dB. Isolation was geometrically set to S₁₂ = −20.0 dB, indicating a 10% coupled current from the transmitted coil.

Under these conditions, Radial TRASE is performed with and without ADD. To show any geometric distortions caused by coil coupling, a large phantom consisting of seventeen 70‐mm‐long, 8‐mm‐diameter tap water vials was imaged. One vial is placed in the center, and eight are evenly spaced around 60‐mm‐diameter and 90‐mm‐diameter circles (Figure 7A). For the ADD image, a complete calibration was performed to maintain an equivalent isolation of at least −40 dB throughout the rotation.

FIGURE 7.

TRASE experimental results with ADD and a geometric isolation of $S_{12}=-20\mathrm{dB}$. The pulse sequence consisted of 200‐μs hard pulses, echo train length of 128, echo time of 2000 μs, repetition time of 3000 ms, and 201 radial spokes. (A) The imaged phantom contains seventeen 8‐mm‐diameter water vials. One vial is placed centrally with eight vials each positioned in rings of diameter 60 mm and 90 mm. The two Radial TRASE images were obtained without (B) and with (C) an ADD compensation pulse. Both images have identical displays and wide windowing. In comparison, ADD effectively reduced the blurring of most outer vials.

3.4.4. Radial TRASE experiment

As a first demonstration of image quality, Radial TRASE was performed with a custom 3D‐printed line‐pair (lp) phantom. For this phantom, the units “mm/lp” represent the width of a single line‐pair (bright and dark line); for example, 2 mm/lp indicates a 1‐mm bright and 1‐mm dark line. The phantom consists of six line‐pair spacings: 5, 4, 3.33, 3, 2.5, and 2 mm/lp. Each line is 20 mm in length and depth and was filled with tap water. The Fusion360 3D phantom model is shown in Figure 8A. The phantom was positioned within the magnet's center with its lines orientated vertically to minimize partial volume effects caused by the tilted slice selection. The coil pair was tuned and matched to a resonance frequency of 2 MHz with a scattering matrix of S₁₁ (inner) = −32.5 dB, S₂₂ (outer) = −26.8 dB, and S₁₂ (geometric isolation) = −23.3 dB. ADD was used to achieve a minimum equivalent isolation of −40 dB throughout rotation.

FIGURE 8.

Radial TRASE resolution experimental results of a custom line‐pair phantom. The pulse sequence consisted of 200‐μs hard pulses, echo train length of 128, echo time of 2000 μs, repetition time of 3000 ms, and 201 radial spokes. (A) The three‐dimensional‐printed line‐pair phantom consisting of 5, 4, 3.33, 3, 2.5, and 2‐mm/lp water‐filled segments. (B) The two‐dimensional filtered back‐projection image. The 3.33‐mm/lp segment is clearly resolvable, indicating an in‐plane spatial resolution of at least 1.67 mm, which is similar to the expected resolution of 1.23 mm. (C) The constructed sinogram from the radial one‐dimensional (1D) TRASE profiles, showing the TRASE spatial axis (vertical) and radial projections (horizontal). Although a smooth center of mass correction was applied to the sinogram, some spoke alignment errors remain.

3.4.5. Sequential slice experiment

To demonstrate sequential slice imaging, an angled phantom consisting of nine 8‐mm‐diameter, 70‐mm‐long water vials was used. One vial is positioned centrally, and the other eight are angled toward the center by 24° (see Figure 9A). Five sequential slices of this phantom were obtained with a nominal spacing of 10.4 mm by varying the current within the shift coil by 0, ±1.5, and ± 3 A. The resonance frequency of the coil (and RF pulses) was kept at 2 MHz for all slices. Calibration was performed for the central slice, and the coils were set up with a similar scattering matrix as in the previous experiment with ADD.

FIGURE 9.

Demonstration of sequential slice Radial TRASE imaging. The pulse sequence for each slice consisted of 200‐μs hard pulses, echo train length of 128, echo time of 2000 μs, repetition time of 1000 ms, acquisition window of 1000 μs, four averages, and 201 radial spokes. A uniform B₀ coil within the magnet bore is used to shift the resonance slice due to the static axial gradient. Imaging was performed with a phantom containing nine 8‐mm‐diameter water vials. One vial is positioned centrally, with the remaining eight angled toward the center by 24°. (A) Sequential slice images were obtained for B₀ shift coil currents of −3, −1.5, 0, 1.5, and 3 A. The expected shift between each slice is 10.4 mm with a reconstructed partition thickness of 3.36 mm. From left to right, the ring of vials spreads outward, indicating successful shifting of the resonance slice position by the uniform B₀ coil. (B) Fusion360 model with a different perspective of the imaged vial phantom.

4. RESULTS

4.1. Active digital decoupling results

The effect of ADD current cancellation in the idle outer coil is shown in Figure 6. PicoScope transformer measurements were made in the center of the transmitted pulse and reported in mV, which is proportional to current. Initially, the magnitude of the outer coil coupled current was 232 mV (Figure 6A), induced by a transmitted refocusing pulse on the inner coil (measured 2900 mV peak). From this measurement, an outer‐coil ADD compensation pulse was transmitted with equal amplitude and opposite phase. This effectively compensated for the coupled current, which now measures with a magnitude of 8 mV for the same inner coil transmitted pulse. This decrease in coupled current is equivalent to increasing the geometric isolation from −20 to −49.3 dB.

FIGURE 6.

Demonstration of the ADD compensation pulse in eliminating residual coil coupling. The coils were geometrically isolated to $S_{12}=-20\mathrm{dB}$. Transformers within the resonance circuits monitored the coil currents as an induced electromagnetic field (EMF) (Faraday's law). For a refocusing pulse transmitted on the inner coil (yellow), the coupled induced current was measured on the outer coil (blue) without (A) and with (B) an ADD compensation pulse. Initially, an EMF magnitude of 232 mV was induced on the outer coil. From this measurement (magnitude and phase), an ADD pulse was applied to the outer coil under the same conditions. This ADD pulse reduced the EMF magnitude from 232 to 8 mV, effectively improving the coil isolation by a further −29.3 dB.

Images with and without ADD are shown in Figure 7 for identical displays, wide windowing, and no k‐space filtering. In comparison, the image obtained with ADD shows less blurring of the vials, especially in the outermost ring. In this example, the mean corrected RFPA pulse output with ADD can be written as

$$\left[\begin{array}{l}{V}_{\mathit{\text{Inner}},\mathit{\text{Corr}}}\\ V_{\mathit{\text{Outer}},\mathit{\text{Corr}}}\end{array}\right]=\left[\begin{array}{ll}1& 0.09e^{i\left({ø}_i-{ø}_o+173°\right)}\\ 0.12e^{i\left({ø}_o-{ø}_i+116°\right)}& 1\end{array}\right]\left[\begin{array}{l}{V}_{\mathit{\text{Inner}}}\\ V_{\mathit{\text{Outer}}}\end{array}\right]$$ (3)

where ${ø}_i$, $\left|V_{\mathit{\text{Inner}}}\right|$, and ${ø}_o$, $\left|V_{\mathit{\text{Outer}}}\right|$ are the Tx pulse phase and voltage on the inner and outer coils, respectively. The mean ADD pulse phase offsets (173 and 116) were determined from the PicoScope and varied by less than ±20° throughout the rotation. Conversely, the mean ADD pulse voltage was approximately 10% of the primary Tx pulse on both coils and varied by less than ±4% over the rotation.

4.2. Radial TRASE imaging results

The 2D radial‐encoded TRASE image of the line‐pair phantom is shown in Figure 8. The constructed sinogram in Figure 8C shows the series of 201 1D radial projections through the phantom. In the reconstructed image (Figure 8B), 3.33‐mm/lp resolution is clearly visible, with the 3‐mm/lp segment being partially resolved. This is a similar level of resolution expected from this coil pair for an ETL of 128. Although no significant distortions exist in the image, some apparent alignment errors within the sinogram might contribute to image blurring. The calibration and acquisition times were 5 and 10 min, respectively. The measured Tx power for the inner and outer refocusing pulses was 5.10 and 6.12 W, respectively.

4.3. Sequential slice imaging results

Five slices obtained from the angled vial phantom are shown in Figure 9. An increased level of noise was observed with the B₀ shift coil active due to incomplete filtering of the DC lines, so four averages were used for each slice with a 1000‐ms TR. An echo acquisition window of 1000 μs (receiver bandwidth of 1 kHz/pixel) was used, giving a Fourier‐defined partition thickness of 3.36 mm for the static 7‐mT/m axial gradient (Eq. [2]). Each slice is theoretically separated by 10.4 mm, and due to the angled vials, the diameter of the ring increases from slice to slice.

5. DISCUSSION

We have introduced a method to radially TRASE encode 2D k‐space by mechanical rotation of a two‐coil Tx array over a 90° angular range. In tandem with a static axial B₀ slice‐selection gradient, Radial TRASE offers a 2D‐RF imaging technique without any requirements from rapidly ramped B₀ gradients. Through the addition of a slowlyramped, low‐voltage uniform B₀ coil, sequential slice imaging can be performed. Compared with previous 2D Cartesian TRASE, Radial TRASE offers a significantly less complex Tx system, with no requirement for RF switching. Specifically, Radial TRASE imaging requires only two RF‐Tx coil channels, compared with four for Cartesian TRASE. To achieve the high level of isolation necessary, Cartesian TRASE requires the addition of coil‐switching infrastructure, whereas that complication is avoided here by the use of geometric and ADD techniques. Regarding spatial resolution, Radial TRASE is a series of 1D‐TRASE scans (projections); therefore, it maintains the spatial resolution achieved by the 1D‐TRASE sequence with two opposing phase gradient coils (Eq. [1]).

As a compromise, Radial TRASE imaging does require a fast and accurate rotation system. Here, mechanical rotation was performed within the TR, causing no delay to the overall sequence timing. A full 90° rotation of the coil pair can be completed in 10 s or a 0.45° incremental rotation in 60 ms. In our experiments, the motor accuracy over a full 90° range (one complete rotation) was within ±0.5°. In implementation, the motor should be shielded to minimize EMI, and the rotating coils must be housed outside a stationary bore tube. Although such mechanical rotation is not typical of MRI systems, achieving incremental rotations of less than 1 degree per TR period is straightforward using a low‐torque motor.

Our first Radial TRASE demonstration succeeded in 2D‐imaging several different water phantoms. Significantly, the images are free from major TRASE‐encodingartifacts, which have previously prevented 2D Cartesian TRASE with the twisted solenoid. ¹⁸ From inspection of our line‐pair study, when comparing Figure 8A,B, it is clear that the top compartment of the 3‐mm/lp segment was empty due to an error with our custom 3D‐printed phantom. Regarding resolution, the 3.33‐mm/lp segment is clearly resolvable, indicating an in‐plane resolution of at least 1.67 mm (i.e., two pixels per lp for this segment). In comparison, the theoretical mean spatial resolution of this coil pair for an ETL of 128 is 1.23 mm. Although portions of the 3‐mm and 2.5‐mm/lp segments are discernible, a reduced spatial resolution was expected. General causes for degraded 1D resolution have been reported in the literature, ¹⁷ including mechanical inconsistencies and relative coil pair rotation orientation. For Radial TRASE specifically, despite the smooth COM correction applied, the reconstructed sinogram (Figure 8C) has clear alignment errors of several spokes, which is known to cause image blurring. ⁴⁷ Methods to reduce these misalignments, such as spoke‐to‐spoke correlation, ⁴⁷ are common in computed tomography imaging and could be adapted for future iterations of Radial TRASE imaging.

The introduction of ADD proved successful in eliminating residual coupled currents. As shown in Figure 6, ADD reduced the coupled currents by a factor of 29, effectively increasing the coil isolation from the geometrically achieved −20.0 dB to −49.3 dB. Achieving better than −40 dB geometrically is challenging, whereas this isolation is easily enabled by ADD for a weakly coupled Tx array (i.e., no resonance splitting). ADD thereby eliminates high‐power switching but requires two synchronized RF channels (including electronics such as RFPAs) and a method to accurately measure coil currents. Here, the coil currents were measured using in‐circuit transformers (Figure 4) and a PC‐based oscilloscope. For Radial TRASE, the ADD settings can be calibrated rapidly, requiring only a single RF pulse to be measured on each coil at several rotational angles. However, we note that the T/R inner coil required two calibration iterations to achieve the same cancellation as the Tx‐only outer coil, which is attributed to the nonlinear components in the passive T/R switch.

An unexpected outcome of this study were the images obtained with and without ADD in Figure 7. Notably, ADD reduced some blurring of the image, especially in the outermost vials. This is partially attributed to the uniformity of the RF coils, which worsens radially (Figure 3A). Coupling between coils adds a second perturbing RF field, which alters the RF spatial phase and can further decrease the uniformity, ¹⁸ causing less accurate refocusing. Regardless, the image without ADD compensation was far less distorted than expected. A prior Cartesian TRASE study reported that −20 dB isolation is insufficient for 2D imaging, with significant coupling‐related encoding artifacts resulting. ¹⁸ In contrast to Cartesian TRASE, the Radial TRASE sequence is far simpler, being a series of projections using the same two RF coils. In speculation, any B₁ phase errors caused by the coupled fields should be consistent between spokes, so long as the isolation is not a function of rotation angle. This might increase the tolerance of Radial TRASE to coupled fields. However, a complete analysis will require full Bloch simulations, which is beyond the scope of this introductory work. As an aside, ADD might have applications in Cartesian TRASE imaging, where, in combination with geometric isolation, the required −30 dB isolation between all Tx elements could be achieved. ¹⁸

Image reconstruction used a simple magnitude‐based filtered back‐projection algorithm. ⁴⁸ Because Radial TRASE has no in‐plane readout gradient, full radial Nyquist sampling ²³ of a NxN data matrix (where N = ETL) requires an acquisition time of TR·N·(π/2). Here, the phantom consisted of pure water with a long T₁ time (TR = 3 s), resulting in a scan duration of 10 min for an ETL of 128. For samples with more practical relaxation times, the acquisition would be greatly reduced. Furthermore, acceleration is possible through advanced reconstruction techniques commonly used in non‐Cartesian imaging. ⁴⁹ Of interest are sparsity‐based reconstruction methods, such as compressed sensing ⁵⁰ and FOCUSS (focal underdetermined system solver). ⁵¹ These methods exploit incoherent sampling, which is often satisfied by golden‐angle radial sampling where sequential spokes are separated by 111.25°. ⁴⁹ , ⁵⁰ This sampling pattern, and thus these reconstruction techniques, can be satisfied by Radial TRASE. However, the spokes should be collected in angular order from 0º to 180° to reduce the rotation increment size.

An inherent limitation of the proposed imaging scheme is that only axial slices can be defined, because the twisted solenoid encodes in transverse directions only. ¹⁹ Despite this, Radial TRASE is compatible with sequential (or interleaved) slice‐imaging techniques, which were enabled here with a uniform B₀ shift coil. In conjunction with the static axial gradient, the uniform coil shifts the slice position without change of resonant frequency, allowing all experiments to be performed on‐resonance. In comparison to conventional high‐slew rate gradients, this uniform field has the significant advantage that the field is only switched between echo trains, so long ramp times are adequate (e.g., 10 ms); hence, only very low voltage is required. The B₀ shift coil, therefore, has very modest power requirements (a few Watts).

In the sequential slice experiments (Figure 9), five different slices, each separated by 10.4 mm, were demonstrated as the outer ring of angled vials increased in diameter along the phantom's length. There does appear to be a minor distortion due to the tilt in the axial gradient, with the angled slices causing the bottom vials to appear closer together than the top vials. In this experiment, each slice was imaged sequentially due to the limitations of our DNMR system. Future implementations will interleave the slices within the TR period. Regardless, Radial TRASE encompasses a complete 2D imaging system free of any rapidly switched B₀ encoding gradients or their accompanying subsystems.

There is scope for improved signal‐to‐noise ratio. First, filtering of the shift‐coil DC lines could be significantly improved. Second, no dedicated receiver was used, which simplified the coil array but required a T/R switch. The volume coils (twisted solenoids) were not optimized as receivers and are further complicated by their decoupling solenoids' increasing resistance. In future work, the receiving quality should be considered in design (i.e., wire length and gauge) and both Tx coils should receive, as the geometric isolation is sufficient to increase signal‐to‐noise ratio. ⁵² Alternatively, a dedicated receive coil may be included, as was done in some prior TRASE studies, ¹⁵ , ¹⁶ although the Rx coil would need to be isolated from the rotating Tx array. This may be possible through a combination of geometric and pre‐amp decoupling techniques ⁵³ but was not explored here.

In this study, imaging was performed within a cylindrical Halbach magnet. Due to magnet symmetry, the frequency response of the Tx array (S₁₁ and S₂₂) negligibly changed with rotation, whereas isolation (S₁₂) varied by less than 1 dB over the full range. However, prior experiments show sensitivity to rotation in asymmetric environments (e.g., bi‐planar permanent magnet). Notably, under these conditions, the magnitude of the coupled fields becomes a function of angular position, which contributes significant spatial encoding distortions. Although ADD may be able to compensate, ideally, the RF coils would rotate around a cylindrically symmetric environment. This can be achieved through the magnet geometry itself or RF shielding within the magnet.

6. CONCLUSION

We have introduced the Radial TRASE encoding scheme, which features a significantly simplified RF‐Tx array over previous variants, requiring only two channels. Complete 2D k‐space sampling is accessible by physical rotation of the Tx coil pair over 90°, which is achieved rapidly during the recovery period without delaying the TRASE sequence. Phantom imaging with a resolution of millimeter‐sized features over a wrist‐sized volume has been demonstrated. In combination with a static axial gradient, 2D slice‐selective images can be obtained while rejecting TRASE‐encoded off‐resonance artifacts by fast Fourier transform. Coil rotation is particularly straightforward to implement in a cylindrically symmetric environment, such as a Halbach magnet. The absence of RF coil switching for each RF pulse in the echo train avoids impacts on data acquisition, such as deadtime and signal glitches. Instead, the simplified two‐coil array is decoupled geometrically, and with active digital decoupling, effective coil isolation is further improved. By both eliminating the requirement for a switched B₀ gradient subsystem and further simplifying the RF implementation, this MRI configuration is particularly promising for applications resonating at the lowest end of the NMR spectrum of technical complexity.

CONFLICT OF INTEREST

Dr. Jonathan Sharp and Dr. Tomanek Boguslaw are consultants for M Tech Research Canada Ltd. Dr. Aaron Purchase is an employee of M Tech Research Canada Ltd.

Sedlock C. J., Purchase A. R., Tomanek B., and Sharp J. C., “Radial TRASE: 2D RF encoding through mechanical rotation and active digital decoupling,” Magnetic Resonance in Medicine 95, no. 2 (2026): 987–1001, 10.1002/mrm.70104.