A Flexible Array for Cardiac 31P MR Spectroscopy at 7 T

Purpose: The simulation optimization and implementation of a flexible 31P transmit/receive coil array, under the geometrical constraint of fitting into the housing of an already existing 12-channel proton array, to enable localized cardiac 31P MRS at 7 T is presented. Methods: The performance in terms of homogeneity, power and SAR efficiency, and receive benchmark of 32 potential array designs was compared by full wave 3D electromagnetic simulation considering the respective optimal static B1+ shims. The design with the best performance was built and compared to a commercially available single loop in simulation and measurement. Results: Simulation revealed an optimal array design comprising three overlapping elements, each sized 94 × 141 mm2. Simulation comparison with a single loop coil predicted a performance increase due to increased power efficiency and lower SAR values. This was verified by phantom measurements, where an SNR increase of 46% could be observed for localized 31P spectroscopy in a voxel positioned comparable to an in vivo cardiac measurement scenario. Conclusion: A flexible 31P/1H RF coil array with improved SNR is presented, enabling localized in vivo cardiac 31P spectroscopy at 7 T.


INTRODUCTION
Phosphorus ( 31 P) magnetic resonance spectroscopy (MRS) is known to be a powerful tool in the assessment of cell energy metabolism [1][2][3]. Coronary heart disease is one of the most common causes of death in the western hemisphere. A common reason for cardiac dysfunction is a deficit of the myocardial metabolism [4][5][6] for which cardiac 31 P MRS is a direct and non-invasive assessment method [7][8][9]. The relative and absolute concentrations of ATP and PCr, and especially their ratio are strong indicators of cardiac dysfunction [10,11]. The technique is very specific but suffers from inherently low sensitivity.
The low gyromagnetic ratio and in vivo concentration of 31 P results in a low intrinsic signal to noise ratio (SNR) ( 31 P-MRS has 100,000 × lower SNR than 1 H MRI [12]), which leads to low spatial and temporal resolution. The nuclear magnetization increases proportionally with the main magnetic field strength (B-0 ) results in significant SNR increase for all nuclei detectable by MR. At ultra-high field (≥ 7 T) the proton ( 1 H) B-+ 1 homogeneity becomes more challenging in larger anatomic regions due to the short wavelength, however, this applies less strongly to 31 P MRS because of the lower Larmor frequency. 31 P cardiac MRS additionally benefits from increased B 0 since the chemical-shift anisotropy helps to shorten the cardiac T 1 relaxation times for at least PCr and ATP, therefore, increasing SNR [12]. Comparison of spectral quality between 3 T and 7 T cardiac 31 P MRS, showed the significant benefit that comes with increasing B 0 , i.e., a 2.8 fold increase in PCr SNR [13].
To acquire as much of the theoretically available signal available, optimized RF coils need to be employed. The use of multiple receiving and/or transmitting elements in coil arrays offers SNR [1,14] and/or acquisition speed advantages on the receive side [15,16] and-if combined with adequate simulation and pulse design-greatly improved data quality and lower specific absorption rate on the transmit side [17,18]. Electromagnetic coupling of the RF coil to the tissue and, therefore, SNR, increases when the coil is conformed to the anatomy [19]. Hence, most coils are assembled on anatomically form-fitted rigid housings. However, for applications where large anatomical inter-subject variability is expected, flexible RF coils are favorable to account for this heterogeneity in anatomy. Due to the longer wavelength for 31 P it is still possible to use single loop coils for 7 T 31 P MRS on the torso, which is the RF coil of choice in most published studies [13,20,21]. More elaborate designs include a 2-element Tx/Rx overlap array [9] and combinations of a Tx volume coil with Rx arrays [22][23][24].
The goal of this study was to design, build and evaluate a dedicated flexible 31 P RF coil for phosphorus cardiac MR spectroscopy at 7 T to be integrated into a 12-channel transmission line resonator (TLR) array for torso MRI [25]. Due to the already existing coil housing, possible design dimensions for the 31 P array were limited. Suitable designs with various element sizes and arrangements were investigated via 3D electromagnetic simulation in a comprehensive study. By definition of a performance measure that takes into account power efficiency, SAR efficiency, and homogeneity of the resulting transmit field B + 1 , and the receive performance based on the resulting B − 1 field, the best performing design was identified and eventually realized. A novel concept for floating dual-tuned cable traps working at both frequencies of operation (297.2 MHz for 1 H and 120.3 MHz for 31 P) was developed and integrated into the coil housings. The performance of the proposed array was compared with a commercially available standard single loop 31 P RF coil for cardiac applications in simulation and measurement. Finally, the feasibility of acquiring localized 31 P spectra in vitro was demonstrated.

RF Array Design
Potential RF coil designs were intended to cover the average human heart size of 12 × 8 × 6 cm 3 and its location ∼2 cm below the sternum [26]. The developed 31 P array acts as an extension to an existing 1 H RF coil [25], to enable acquisition of additional metabolic information of the heart muscle. The proton coil array consists of 12 TLR elements that were fabricated on a flexible substrate with a rigid PCB part in the center of each TLR element connected to their tuning and matching components. The PCB is connected to a rigid housing box incorporating each elements interface board, including T/R switches, 1:3 splitters and cable traps. The considered designs are to fit into the rigid housing boxes of the TLR elements, which poses a hard constraint on the maximum number of elements, coil sizes, and shapes. Figure 1 shows all considered RF coil array configurations, ranging from 1-to 4-channel arrays differing in size, arrangement and position, yielding a total number of 32 simulated array designs. The 12 TLR elements and their respective shields are depicted in gray. To discretely sample the possible configurations, the element size was varied in multiples of the 1 H TLR element dimensions of 94 × 94 mm 2 . Regarding the position, the respective array center matches either the 1 H array's center (corresponding to the center of the body) or is shifted by one half 1 H element width to the patient's left. The flexibility of the 1 H coil leads to a bending of the leftmost and rightmost elements. By shifting the center of the 31 P coil the array experiences a different degree of bending which has an influence on the overall produced field. Those positions are denoted body-centered (bc) and heart-centered (hc), respectively.

Electromagnetic Simulation
All coil designs were modeled in XFdtd 7.5 (Remcom, State College, PA, USA) using 1 mm thick wire as perfect conductors. The 3D EM simulations and their post-processing were computed on a workstation equipped with 4 GPUs (Tesla C2070, Nvidia, Santa Clara, CA, USA) enabling GPU acceleration, 12 CPUs (Intel R Xeon R X5690, 12 M Cache, 3.46 GHz, 6.40 GT/s Intel R QPI, Santa Clara, CA, USA), and 190 GB RAM. Each coil element was cut into equally long copper stubs connected by a capacitor to limit the electrical length of the coil. Depending on the configuration it was used in, the number of gaps was 8, 6, and 4 for the 1, 3, and 4 element arrays, respectively. This corresponds to stublengths between ≈ λ/15 and λ/42 for the 2 × 1.5 elements (4 channel array) and 1.5 × 1 element (1 element array), respectively, preventing any wavelength effects for all presented designs. All capacitors were eventually replaced by 50 voltage sources to enable a fast RF co-simulation approach [27] in ADS (Keysight Technologies, Santa Rosa, CA, USA). All designs were simulated as overlap-decoupled arrays. An overlap factor of 0.86 was used [28]; as this factor only applies to quadratic elements, for non-quadratic elements decoupling was corrected by additional counter-wound inductances (CWI) [29] during RF co-simulation. Realistic loss incorporation was implemented by assigning capacitors their realistic equivalent series resistances by extrapolating an ESR model for the ATC 100 E capacitor series (http://www.atceramics.com/multilayer_capacitors.html). Solder joint losses were modeled as series resistances extrapolated to 120.3 MHz from literature [30]. Counter-wound inductances were modeled lossless since they were solely used to mimic sufficient overlap decoupling. Losses of the power splitter (−0.36 dB/channel) and the transmit/receive switches (−0.8 dB/channel) were measured on the bench and incorporated into the simulation. Each group is divided in body-centered (bc) and heart-centered (hc) arrays. bc-arrays are centered above the sternum, hc-arrays are shifted by half a housing box to the left, to be centered above the heart, as indicated on the torso on the bottom right. Colors represent the size of the individual elements, as stated in the legend on the bottom, the numbers are the element sizes in multiples of the underlying 12-ch 1 H array housing boxes (grid on torso, bottom right); one unit cell has a length of 94 mm. Each phase set results in a certain value for PE, RH, SE, and fϕ . The phase set that maximizes fϕ is the designated optimal phase set of the corresponding RF coil. The total number of phase optimization simulations for each voxel model equals 477508.
The proposed array designs were loaded with realistic human body models ("Duke" and "Ella, " Virtual Family, IT'IS Foundation, Zurich, Switzerland), yielding a total of 64 3D simulation setups to be compared. Combination of 3D EM field data and co-simulation results and further post-processing was performed in Matlab 2017b (Mathworks, Natick, MA, USA). In order to compare the performance of the designs, optimal static B 1 + shimming was obtained by varying the relative phase shift ( ϕ) between the elements in 5 • steps for the 2-and 3-element arrays and in 10 • steps for the 4 element arrays, respectively (see Table 1 for the total number of phase sets | i |).
The optimal phases were determined for each design by maximizing a merit function f ϕ that is an equally weighted combination of power efficiency (PE), SAR efficiency (SE), and relative homogeneity (RH): In Equation (1) the maximum value is evaluated over all simulated phase combinations for one specific design. The mean values were averaged over an ROI comprising the heart lumen and muscle and normalized with respect to the maximum value for the respective array. To identify the best design (d i , i = 1,..., 32, i.e., all considered coil designs), an extended merit function f tot , additionally taking into account the receive efficiency in terms of SNR [31] was evaluated for all phase-optimized designs, but now normalized with respect to the maximum values for SE, PE, RH, and SNR over all investigated designs, respectively: The values for PE, SE, RH, and SNR were averaged over "Duke" and "Ella, " both equipped with the same design.
To evaluate the influence of the CWI decoupling, another set of simulations with 2 elements of dimensions 1 × 1 ( ∧ = 94 × 94 mm 2 ) and 1 × 3 ( ∧ = 94 × 282 mm 2 ) and overlap factors between 0.76 to 0.92 in steps of 0.02 was performed. The arrays were loaded with a rectangular phantom filled with a material mimicking tissue (σ = 0.55 S/m, ε = 51) and were tuned, matched and decoupled in co-simulation using CWI where necessary. Static B + 1 shimming for a spherical ROI with a diameter of 125 mm, located 35 mm below the RF coils was derived in the same manner as described in the previous paragraph using Equation (1). The arrays with the overlap factor resulting in best decoupling were compared to the corresponding arrays with overlap factor 0.86 with additional CWI in terms of S 12 , RH, PE, SE, SNR, and maximum 10 g-SAR. For a theoretical comparison of a commercially available single loop 31 P coil (RAPID Biomedical GmbH, Rimpar, Germany) with the best design identified above, the performance of both RF coils was investigated using the aforementioned simulation workflow. The single loop has a diameter of 140 mm and an assumed wire thickness of 1.5 mm. The loss of the transmit/receive switch was set to the same as for the array (−0.8 dB) and incorporated in the evaluation. Both setups were positioned on the voxel models as closely to reality as possible in terms of distance and curvature in order to obtain results comparable to the measured data.

RF Coil Implementation
The design determined by simulation, i.e., the 3 channel 1 × 1.5 heart-centered array, was implemented. For flexibility, the 31 P array was constructed out of flexible stranded wire (Ø = 2 mm). Crosstalk between the 1 H and 31 P arrays was minimized by replacing every second segmenting capacitor of the 31 P loops by an LCC trap [32], resulting in three traps per element. To ease the handling of the whole coil and to keep it as flexible as possible, a separate interface box for the coil was avoided by placing transmit-receive switches, preamplifiers and power splitters inside the 12 separated 3D-printed housing boxes of the 1 H array. Performance of the 31 P array was tested on the bench, measuring S-parameters for five human volunteers (3 male, 2 female) using a vector network analyzer (E5071C, Agilent, Santa Clara, CA, USA).
In order to prevent common mode currents on the cables at both Larmor frequencies, double tuned floating cable traps were implemented [33]. By nesting two floating traps [34] into one another, blocking at two different frequencies can be achieved. Two hollow dielectric cylinders are split in half along their axes and are covered by conductive copper layers on the inside and outside walls. The inner trap shares its outer copper layer with the inner copper layer of the outer trap. At one end, all three concentric copper layers are short-circuited, while tuning capacitors are connecting the outer to the middle and the middle to the inner layer on the other side. The capacitors (CHB series, Exxelia Ceramics, Pessac, France) across the outer (inner) shell coarsely control the first (second) resonance frequency of the Frontiers in Physics | www.frontiersin.org trap, respectively. By varying the distance between the two halves of the cylinders, the frequencies can be finely adjusted, however not independently. The trap body was 3D-printed (Rebel 2, Petr Zahradník Computer Laboratory, Ústí nad Labem, Czech Republic) from ABS plastic material (ε ≈ 2). The dimensions of the hollow cylinders were Ø = 8 mm, 13 mm, and 20 mm, respectively, all with a length of 55 mm.

RF Array Design
The computational cost of each individual full wave simulation depends on the size of the array, the number of gaps, and the voxel model used as load. CPU/GPU RAM requirements and the total computation time for a single EMS were between 0.38/0.25 GB and 1.47 h (design: Ella 1 element 1.5 × 1 hc) and 2.49/1.52 GB and 49.9 h (design: Duke four element 2 × 1.5 bc). Postprocessing of the individual designs is highly dependent on the number of channels, ergo the number of different phase sets that need to be calculated in order to evaluate the static B-+ 1 shimming, and was between 0.46 and 13.6 min for the Ella 1 element 1.5 × 1 hc and Duke 4 element 2 × 1.5 bc, respectively.
Static B + 1 shimming was optimized for each of the 64 simulated designs (32 for "Duke, " and 32 for "Ella"). Figure 2A shows the resulting point cloud for the B + 1 shimming procedure for an exemplary dataset. Each point represents the result in terms of RH, PE, and SE for a certain phase set. The phase sets that result in maximum RH, PE, SE, and f ϕ , are marked by blue, black, green, and red circles, respectively. Figure 2B shows the resulting B + 1 / √ P in maps achieved with the optimal phase set for best RH, best PE, and best SE (which is equal to best f ϕ in the shown case). Evaluating the extended merit function f tot over all designs resulted in the final design choice of a 3 element array with element sizes of 94 × 141 mm 2 , centered above the heart. Figure 3 depicts the mean values over the heart ROI for f tot , RH, SE, PE, and SNR for all simulated designs for Ella, Duke, and the average over both (black).
In the 2-channel array simulation for determining the influence of the CWI decoupling, the optimal overlap factor was determined to be 0.88 for the 1 × 1 and 0.78 for the 1 × 3 sized arrays. S 21 for optimized overlap decoupling (OL) only and fixed overlap plus additional CWI decoupling (OL + CWI) were always below −17.1 db. For both array types, i.e., 1 × 1 and 1 × 3, the highest deviation between OL + CW and OL designs was found in the maximum 10 g SAR value, with an increase of 1.29% ( ∧ = 0.02 1/kg) and 4.13 % ( ∧ = 0.03 1/kg) for the 1 × 1 and 1 × 3 arrays, respectively. All results are summarized in Table 2. These findings support the hypothesis that simplifying the simulations of all array designs with a fixed overlap + CWI to mimic optimal overlap decoupling is reasonable.
Bench measurements of the implemented array in loaded condition before and after incorporation into the 1 H housing were conducted on 5 human volunteers (3 male, 2 female, 30 ± 3.6 years) and show sufficient matching and isolation between array elements, i.e., the reflection coefficients (S 11 , S 22 , S 33 ) were always below −17.5 dB and −17.2 dB, respectively, while transmission coefficients (S 12 , S 23 , S 31 ) were below −13.1 and −13.6 dB. The array needed to be slightly retuned and rematched after integration due to slight position changes and distance to the sample. The measured Q ratio (Q u /Q l ) for all three elements prior and after incorporation was above 5.5 and 5.7, indicating sample loss dominance and negligible additional losses due to the 1 H coil and housing. The floating double tuned traps were correctly tuned, with a blocking of −10.5 dB/−34 dB and a bandwidth of 3 MHz/6 MHz at 120 MHz/297.2 MHz respectively. The tuning range for both blocking frequencies by changing the gap size between the halfcylinders was ±10%, which was sufficient to tune the traps to the desired frequencies.

Performance Comparison With Single Loop
In simulation the proposed three element array yields a mean power efficiency in the heart ROI that is 58% higher than the respective values for the single loop reference coil. In terms of SAR efficiency, the array performs 124% better than the loop; the 10 g averaged SAR values decrease by 51%. Relative homogeneity is 30% better. The results are presented in detail in Table 3 and

MR Measurements
A maximum deviation in B + 1 acquired with and without the 31 P array present of <20% was found (see Figure 5). Before acquiring CSI data, a series of localized spectra were obtained in order to find the reference voltage for a voxel in a location similar  Arrays with two elements of dimensions 1 × 1 and 1 × 3 were simulated once with an optimized overlap factor (OL) and with a fixed overlap factor of 0.86 and additional counter-wound inductances (OL + CWI). RH, PE, SE, and SNR values are averaged over a spherical ROI volume. Performance difference of the OL + CWI vs. OL arrays can be seen in the bottom row and is negligible for both dimensions, supporting the CWI's use to mimic optimal overlap decoupling.
to the human heart, i.e., 7 cm from phantom surface wall in ydirection. Figure 6A shows all spectra plotted as signal amplitude vs. reference voltage. The reference voltage is the voltage that would be required to achieve a 90 • flip angle using a 1 ms block pulse. The signal amplitudes were fitted with a sin 3 function, corresponding to the signal equation for STEAM sequences. The reference voltage for the single loop is 400 V, whereas the array needs 880 V in the same voxel. The localized spectra of the acquisitions where 90 • were reached are shown in Figure 6B for the array and the single loop. From these spectra SNR values of 52 for the array, and 35 for the single loop were calculated. In

DISCUSSION
In this work we show the successful integration of an optimized 3-channel 31 P array for cardiac MRS at 7 T into a flexible 12 channel 1 H coil. The bold values state the averaged values over Duke and Ella for the heart and VOI respectively. RH, PE, SE, and SNR values are averaged over the whole heart volume and over the VOI used in measurement. A set of 32 31 P array layouts, each evaluated using two different voxel models (one male, one female) to incorporate inter-subject variability was compared via full wave 3D electromagnetic simulation with realistic loss estimations to find the best performing array design. Static B + 1 shim phase sets optimized for a combination of  homogeneity, power and SAR efficiency were calculated for all investigated arrays. All array layouts used a fixed overlap and additional counterwound inductances to decouple the array elements in order to save simulation time, since finding the optimal overlap for differently sized array elements is very time consuming due to the necessity to rerun the 3D simulation for each setup multiple times while changing the overlap factor slightly, until optimal decoupling is achieved. It was exemplarily shown for two elements that the differences between optimal overlap only as compared to a fixed overlap factor and the additional CWIs are negligible.
A new way of visualizing B + 1 shimming results in the entire phase shift parameter space was introduced, allowing for quick and easy visual inspection of the variation of performance parameters on the chosen phase set. A figure of merit taking into account an equally weighted combination of homogeneity, power and SAR efficiency was employed. This approach can be universally employed for any transmit array. Depending on the requirements of the application, the weights for the figure of merit could be changed to favor a specific performance parameter. This could be useful e.g., to optimize the B + 1 shim more strongly for SAR efficiency in applications that are SAR demanding, or for homogeneity where a uniform flip angle distribution is essential, or for power efficiency where the available transmit power is limiting.
The best performing design was a 3-element array centered above the heart with individual elements of 94 × 141 mm 2 . It was integrated into the housings of the proton coil, including performance tests on the bench and in the MR scanner.
Maximum B + 1 deviations for the 1 H array alone vs. the combination with the 31 P array were found in superficial areas and were below 20%, indicating sufficient decoupling between the two frequencies.
Despite the higher calculated power efficiency, in the experiment higher pulse amplitudes were necessary in the VOI for the array when compared to the single loop ( Figure 6A). The main cause for this behavior is that the array was simulated and constructed to be optimal for human subjects, but the measurement was performed on a homogeneous phantom. Firstly, this led to a mismatch of the RF coil to the phantom load, resulting in a significant decrease of the effective input voltage at the coil ports. Secondly, the power efficiency was simulated for the array bent on a human load, but since the phantom did not allow for bending, the coil was used in flat configuration, leading to lower efficiency in depth. In addition, shielding effects from the conductive structures of the 1 H coil elements and interfaces were not considered in simulation and could possibly also reduce transmit efficiency. Losses associated to imperfect decoupling from the 1 H array, and induced common mode currents on the cable shields further contribute to the difference, although to a lesser extent, since an effort was made to keep them as small as possible.
Nevertheless, an SNR increase of + 46% in the VOI was demonstrated with identical flip angle as in the reference coil, which shows the advantages of the array in terms of receive sensitivity and supports the above reasoning for suboptimal transmit performance on the phantom.
Because of the mentioned limitations of the phantom measurement, an even stronger increase for in vivo measurements can be expected. In a next step, the required tests and documentation of the coil for approval of the ethics board will be established to enable the usage of the coil in an in vivo study.

DATA AVAILABILITY STATEMENT
The datasets generated for this study are available on request to the corresponding author.

AUTHOR CONTRIBUTIONS
SR and EL designed the study. SR did the 3D simulations and drafted the manuscript. SR and MV implemented and measured the coil on the bench. SR, SW, and AS did the MR measurements and the post-processing. EL, SW, and AS revised the manuscript.

FUNDING
This work was funded by the Austrian Science Fund (FWF) grants P28059-N36 and P28867-B30.