Feasibility of undersampled spiral trajectories in MREPT for fast conductivity imaging

To investigate spiral‐based imaging including trajectories with undersampling as a fast and robust alternative for phase‐based magnetic resonance electrical properties tomography (MREPT) techniques.

(TMS), 7 transcranial direct current stimulation, 8 and RF ablation. 9Hence, MREPT represents a promising tool for non-invasive diagnosis and treatment of various pathological conditions.
The MREPT technique obtains information on the electrical properties of biological tissues by measuring the complex excitation field, B 1 .To obtain this complex B 1 field, the magnitude and phase of B 1 are acquired separately. 2 As an alternative, various "phase-based" MREPT methods have been developed, 10,11 where the magnitude of the B 1 field is not needed, since the conductivity is dominated by the phase of the B 1 field. 12The standard phase-based MREPT method utilizes the following formulation: 10

𝜎
where  ± is the B 1 transceive phase, which is the sum of the B 1 transmit phase and receive phase,  is the Larmor frequency,  0 is the vacuum permeability, and ∇ 2 is the Laplacian operator.Eq. ( 1) highlights the nature of the relationship between conductivity and transceive phase, i.e., conductivity is determined by the curvature (second derivative) of the transceive phase.However, the presence of the Laplacian operator poses a challenge for phase-based MREPT techniques as it is highly sensitive to noise. 13Therefore, achieving a high SNR within clinically acceptable scanning times is crucial for MREPT techniques.
To obtain the necessary information for MREPT, the B 1 transceive phase of almost any MRI pulse sequence can be utilized, with spin-echo 2 and balanced SSFP (bSSFP)-based 14 pulse sequences being the most common.However, the utilization of each pulse sequence has its own trade-offs.Spin-echo based sequences provide accurate transceiver phase information but require a longer acquisition time, typically ranging between 15 and 30 min. 15,16In contrast, bSSFP-based sequences offer a shorter acquisition time, but can suffer from "banding artifacts" for specific off-resonance values, which substantially distorts the phase images. 14][21][22] Nevertheless, these approaches can increase acquisition time and suffer from registration problems, highlighting the importance of selecting the appropriate pulse sequence for MREPT, to minimize artifacts and optimize measurement accuracy.
Over the past decade, spiral imaging has become increasingly prevalent in numerous MRI applications, including cardiac flow imaging, 23 diffusion-weighted imaging, 24 real-time imaging, 25 functional MRI, 26 and MR fingerprinting. 27This is owing to its inherent advantages over Cartesian imaging, 28 which can also be harnessed in the context of MREPT. 29Spiral imaging offers high acquisition speed, which is critical for achieving volumetric coverage within a clinically acceptable scanning time.In particular, center-out spiral trajectories begin covering the k-space from the center, prioritizing low frequencies compared to standard Cartesian trajectories.This property is particularly beneficial for MREPT, as the obtained B 1 phase is generally smooth and has primarily low-frequency content.Moreover, center-out spiral trajectories have high SNR efficiency, which may be crucial for addressing the detrimental effects of the Laplacian operation especially in noisy cases.Last, spiral trajectories can be efficiently designed to incorporate undersampling to accelerate image acquisition. 30,31he objectives of this study are twofold.First, we investigate the potential of spiral based imaging in the context of phase-based MREPT techniques.Second, we explore the feasibility of undersampled spiral acquisitions for conductivity imaging.To achieve these goals, we evaluate the conductivity results obtained via phase-based cr-MREPT technique in phantom and in vivo brain acquisitions.

Data acquisition
In accordance with the previous literature, 32 we developed a custom MRI pulse sequence incorporating several variable density spiral trajectories, as illustrated in Figure 1.These particular sampling schemes allowed for full sampling of the inner portion of k-space that was radially below the first limit (FL), while the outer portion that was radially above the second limit (SL) was undersampled at varying ratios (UR) of 2, 3, or 4. The area between the inner and outer portions was sampled with a linearly decreasing density.The limits are defined as a factor of maximum spatial frequency.
To test the developed pulse sequences, an experimental phantom was constructed with two identical structures and a background region.The phantom had a cylindrical shape with a diameter of 16 cm and a height of 20 cm, and had homogeneous conductivity in the z-direction.The background region of the phantom was filled with an agar-saline gel containing 20 g/L agar, 2 g/L NaCl, and 0.2 g/L CuSO 4 .The structures within the phantom were created by filling longitudinal holes with a saline solution containing 6 g/L NaCl and 0.2 g/L CuSO 4 , resulting in a higher conductivity than the background region.The structures had a diameter of 3.5 cm.Overall, the expected conductivities were 0.35 S/m for k-Space trajectories of different sampling schemes used in this study.Here, different colored circles represent different radii as a factor of the maximum spatial frequency, being red = 0.25, black = 0.5, and blue = 0.75.For each case, k-space is fully sampled within the FL, and various URs are prescribed outside the SL.In the transition region between FL and SL, the undersampling ratio is varied linearly between full sampling and the prescribed UR.
the background region and 1.04 S/m for the structures, respectively. 33ext, the performance of the pulse sequences was demonstrated through in vivo measurements.The in vivo experiments were conducted on a healthy 30-y-old male volunteer, after obtaining written informed consent in accordance with the approval of the local ethics committee.
Experiments were conducted on a 3T scanner (Magnetom Skyra, Siemens Healthineers, Germany) using a 16-channel head coil array.In the experiments, spiral gradient-echo based pulse sequences with uniform and variable density patterns were utilized.The maximum TR for both the phantom and the in vivo experiments is 6.9 ms, which decreases even further with undersampling.With 16 spiral interleaves and around 10 averages, a single slice can be obtained in approximately 1 s.In the in vivo experiments, with 32 slices and multiple averages, the entire brain can be imaged in less than a minute.Sequence parameters for spiral acquisitions can be found in the Table S1.Last, to comparatively demonstrate the robustness of spiral acquisitions, an additional bSSFP acquisition with Cartesian readout was obtained from the same volunteer using the following parameters: TE/TR = 1.55/3.10ms, FOV = 270 × 270 mm 2 , matrix size = 128 × 128, FA = 15 • , number of averages = 4, number of slices = 32, total duration = 50.8s.Standard volumetric shimming provided by the manufacturer was performed at the beginning of the session, and was unaltered throughout the acquisitions.

Data processing and image analysis
To robustly reconstruct the spiral acquisitions, we employed the SPIRiT reconstruction framework. 34Specifically, we implemented a conjugate gradient algorithm for non-Cartesian SPIRiT in the image domain.Here, a total of 15 iterations were prescribed with a 5 × 5 kernel and 20 × 20 calibration area.To combine the resulting phase data obtained from various receive channels, we utilized the Virtual Reference Coil Approach. 35Furthermore, we applied a Gaussian filter with a kernel size of 5 × 5 × 3 (or 5 × 5 in case of 2D acquisition) to the transceive phase data.
To obtain conductivity information, phase-based cr-MREPT method was employed. 11This formulation is based on convection-reaction-diffusion PDE and can be shown as: where  = 1∕ and (∇ ± ⋅ ∇) is a convection term which aims to tackle the boundary artifacts.In order to prevent spurious oscillations, an artificial diffusion term −c∇ 2 , with a constant diffusion term c, is also added.For the phantom measurements, a 2D formulation of the phase-based cr-MREPT technique that omits variations in the z-direction was utilized; whereas for the in vivo measurements, the full 3D formulation was utilized.
To evaluate the performance of spiral based conductivity images, regions of interest (ROIs) encapsulating structures in the phantom images, and cerebrospinal fluid in the in vivo measurements were segmented with magnitude thresholding for each acquisition.The mean and SD of conductivity values were calculated over the ROIs.

Phantom measurements
The feasibility of the proposed method was first demonstrated using phantom measurements.Figure S1 depicts the phase difference maps obtained from acquisitions with different undersampling schemes.Here, the reference image was designated as the phase map obtained from the fully sampled acquisition.The differences observed between each case are relatively small, with a maximum mean squared error below 10 −3 .Figure 2 displays conductivity images obtained through the phase-based cr-MREPT technique.A magnitude image from the fully sampled case is also included as a visual reference.The conductivity differences between the undersampling schemes and the fully sampled case are depicted in Figure S2.Notably, the fully sampled case exhibits a halo artifact that becomes less prominent by introducing undersampling.However, as the undersampling ratio increases, residual aliasing artifacts hinder the overall image quality, as observed for the case of UR/FL/SL = 4/0.25/0.50.
Table S2 displays the mean and SD values of the conductivity images obtained through the phase-based cr-MREPT technique for the phantom experiment.The mean values for each case are in close proximity to the expected values of 0.35 S/m for the background and 1.04 S/m for the structures.The reduction of the halo artifact improved the accuracy of the conductivity measurements and resulted in lower SD values especially for the cases where this artifact is less prominent (like UR/FL/SL = 3,4/0.5/0.75).However, as clearly seen in Figure 2, in cases of extreme undersampling, such as with low FL/SL and high UR, the SD values increased once again due to the distortions introduced in the conductivity images.
The conductivity images obtained with varying numbers of averages are shown in Figure 3.Here the sampling scheme of UR/FL/SL = 3/0.5/0.75 was selected for illustration purposes.It is noteworthy that even with a single average with an acquisition time below 0.1 s, the structures are clearly discernible.While the addition of more averages leads to the expected improvement in quality, there is almost no observable difference in image quality after eight averages.In Table S3, we report the mean and SD values for the various numbers of averages.Notably, the SD does not vary after eight averages.Consequently, this acquisition scheme enables the acquisition of a single slice in well under 1 s.It is observed that the fully sampled spiral acquisition is afflicted with halo artifact (depicted with a white arrow), which are effectively mitigated by the implementation of undersampling.However, it should be noted that in cases of extreme undersampling, such as those with low FL/SL and high UR, the entire conductivity image is susceptible to distortions.

F I G U R E 3
Conductivity images obtained with varying numbers of averages (1, 2, 4, 8, 10, 16, and).Even with only one average, structures are visible, albeit with poorer image quality.As the number of averages increases, the image quality improves as expected.However, noticeable differences in image quality become minimal after eight averages.

F I G U R E 4
Conductivity maps derived from a healthy volunteer using various sampling schemes.The proposed spiral trajectories demonstrate the ability to obtain conductivity maps, albeit with residual artifacts evident across tissues and boundaries at higher undersampling ratios.Nevertheless, all measurements successfully delineate prominent structures, such as the CSF.
visual reference.The conductivity images clearly display prominent structures, such as CSF.However, similar to the conductivity images obtained from the phantom, cases with low limits and high undersampling ratios lead to image distortions.
Figure 5 shows the comparison of conductivity maps obtained from the 14th slice of bSSFP and spiral acquisitions, alongside with their magnitude and phase images as a reference.For spiral case, the sampling scheme of UR/FL/SL = 3/0.5/0.75 was chosen.While the effects stemming from the off-resonance are not visible in the magnitude image of bSSFP, they result as an artifact in the conductivity images (white arrow).On the other hand, conductivity maps obtained via spiral trajectories are free from such distortive artifacts.More importantly, these distortive artifacts arising from the off-resonance can affect a Conductivity maps derived from a healthy volunteer using bSSFP and fully sampled spiral acquisition.The selected ROIs (red dots) are shown overlaid on magnitude images and are used for calculating conductivity values as reported in Table S4.The effect of off-resonance, although not clearly visible in the bSSFP magnitude image, can be seen in the conductivity map obtained with the bSSFP (white arrow).In contrast, the conductivity map obtained with the spiral trajectory present no such artifact.large volume, as depicted in Figure S3, for two additional cross sections across the imaging volume.
The average conductivity values calculated for the CSF using ROIs (depicted with red dots in Figure 5) are listed in Table S4.The resulting conductivity values are within the close proximity of the expected value at body temperature, which is 1.794 S/m. 36

DISCUSSION
In this work, we investigated the use of spiral trajectories with various undersampling strategies for conductivity imaging.Spiral trajectories were a natural candidate for this work due to their center-out sampling strategy, which efficiently covers low k-space frequencies.By employing undersampling with spiral trajectories, the total acquisition time was substantially reduced without compromising image quality, as demonstrated in both phantom and volunteer experiments.Our results suggest that spiral trajectories with undersampling can be a promising approach for fast conductivity imaging, enabling the acquisition of high-quality images in a shorter time frame.The use of spiral trajectories with undersampling strategies offers significant advantages in terms of acquisition time.The center-out sampling strategy implemented in this study prioritizes low k-space frequencies, thus allowing the acquisition of low spatial frequency information at a high temporal resolution.This property is particularly useful for conductivity imaging, as the phase data from which conductivity is calculated does not contain high frequency components.Furthermore, spiral acquisitions excel in covering k-space rapidly, particularly with long readouts.This rapid k-space coverage sets spirals apart from their Cartesian counterparts, further enhancing their suitability for applications that demand swift data acquisition.However, it is important to mention that techniques like GRAPPA can also expedite data acquisition with Cartesian trajectories, partially bridging the gap in the acquisition speed.As a result, we can obtain a single slice with multiple averages in less than 1 s, and conduct whole brain imaging in less than a minute, making spiral acquisitions with undersampling schemes particularly well-suited for clinical applications.
Spiral imaging is known for its low minimum TE, allowing for imaging of challenging tissue types like the lung 37 or heart. 38Alongside with the acquisition speed advantages, spiral imaging has already made real-time phase-contrast imaging feasible. 39The implementation of these techniques may pave the way for real-time MREPT imaging in the future.
Compared to other commonly used pulse sequences in MREPT, such as bSSFP, spiral imaging offers distinct advantages in terms of artifact reduction.bSSFP images are particularly vulnerable to off-resonance, which can substantially distort conductivity images as demonstrated in Figure 5.This phenomenon arises because the observed phase is dependent on the off-resonance. 19In particular, approximately π-radian phase differences in the vicinity of banding artifacts are further amplified by the Laplacian operation used in the conductivity calculations.In contrast, spiral imaging exhibits less coherent aliasing artifacts compared to Cartesian trajectories, especially when undersampling is utilized. 25Additionally, spiral imaging is inherently oversampled at the k-space center, which makes it more robust against motion artifacts. 25n the experiments, we identified two main sources for errors.First, there's a slight difference between the expected and actual conductivity values in the phantom experiment, particularly in the structures.This could be due to either the limitations in the assumptions used for phase-based conductivity reconstructions, 10 such as transceiver phase assumption and low B 1 magnitude gradient assumption, or simply the smoothing effect introduced by applying a Gaussian filter to the phase data.Second, we observe halo artifacts in both phantom and in vivo images.Main suspects of this well-known artifact in spiral imaging are static off-resonance and concomitant fields, resulting as blurring or ringing in the image, as well as gradients imperfections, causing artifacts near the edge of the imaged object. 40While these artifacts can be mitigated with the utilization of undersampling, there are additional approaches to further eliminate these unwanted effects, including calculating the delays of the gradients, 41 correction of B 0 eddy currents, 42 or even simultaneous correction of off-resonance, trajectory errors, and concomitant field effects together. 43These approaches can be integrated together with undersampling strategies to further suppress these artifacts.
The utilization of spiral trajectories in MREPT presents a promising prospect for clinical applications.MREPT has been previously applied in various clinical scenarios, such as brain tumors, ischemia, and hemorrhage cases.In these scenarios, the phrase "time is brain" rings particularly true, making fast acquisition techniques like spiral imaging crucial for the successful implementation of MREPT.The rapid acquisition provided by spiral imaging can enable clinicians to obtain high-quality conductivity images within clinically feasible timescales, making it an attractive option for future clinical studies.However, further validation is required to establish the clinical utility of spiral MREPT and its superiority over other imaging techniques in various clinical scenarios.

CONCLUSIONS
We have presented a spiral-based sampling framework for conductivity imaging.In both phantom and volunteer experiments, we utilized spiral trajectories with and without undersampling in conjunction with SPIRiT reconstructions.Compared to conductivity images obtained with Cartesian bSSFP sequences, spiral trajectories demonstrate improved robustness against field inhomogeneity artifacts and substantially reduced the acquisition time.Overall, spiral trajectories are demonstrated to be a viable option for conductivity imaging, and can further increase the clinical utility of these methods.

Figure 4
Figure 4 illustrates the in vivo conductivity values obtained through the phase-based cr-MREPT method, where the magnitude image of the fully sampled case is used as a