We performed a study on the effects of the angular position of the capacitor along the circumference of a loop coil. In the simulations and MRI experiments, it was expected that the non-uniform current distribution caused by the non-symmetrical capacitor position in the coil would improve the RF excitation (|B1+|)-field, and that this concept can be used to optimize the |B1+|-field for particular applications. These studies included both simulations and MRI experiments. Because the amount of data to present was large, we used extracted line profiles, and rather than showing each |B1|-map, we plotted statistical values. For simplicity, we labeled the coils based on the angular positions of the capacitors; for example, the coil that had its capacitor at 45° was labeled as C45.
FDTD simulations of single element in empty space. A total of 24 simulations were performed with loop coils placed in an empty space. Each coil was excited to 300 MHz using a Gaussian pulse with a bandwidth of 600 MHz, and all simulations were normalized to an input power of 1 W. Figure 1 shows the computed surface current densities (J) of the selected coils. The absolute vector surface current density of the reference coil is shown in Fig. 1a. The maximum current density was produced by C15, as shown in Fig. 1b; this current density was higher than that of the reference coil. The surface current exhibited two bands of low intensity at 90° and 270°, which were caused by the change in the phase of the x component of the phase of the current density. Figure 1c–d shows the current densities for C90 and C180, respectively. C90 exhibited J with a higher intensity on the opposite side of the capacitor location. Despite the asymmetry of the J distribution, this coil produced the most uniform |B1+|-field in the XY plane. On the other hand, a lower J distribution was computed for C180, which also had the lowest average |B1+|-field. This coil had a J intensity that was lower than that of the reference coil; however, C180 was also a coil with a symmetrical capacitor-wire distribution.
A comparison between lower and high frequencies has been included in Fig. 1e–f, showing the J distributions for C45 when it was tuned to 300 MHz and 100 MHz, respectively. These simulations show the frequency dependency: because higher frequencies correspond to shorter wavelengths, they result in greater susceptibilities to changes in the position of the capacitor. At the same time, this is an opportunity to optimize the coil with focus on the |B1+|-field in a desired area.
We computed the |B1+|-field in empty space, which depicts slices at the center of the coil in the XY plane (Fig. 2a-i) and in the ZY plane (Fig. 2a-ii). To better visualize the performance of each coil, we obtained three-line profiles of the computed |B1|-field. The first profile (white line in Fig. 2a-i) was set 50 mm from the coil, and the results are shown in Fig. 2b-i. The second line profile (green line in Fig. 2a-i) was set at a larger distance, i.e., 200 mm, and the results are shown in Fig. 2b-ii. The third line profile is taken in the ZY plane (green line in Fig. 2a-ii) 50 mm from the coil, and the results are shown in Fig. 2b-iii. These line profiles show that the reference coil (blue solid line) had a less-than-average field strength compared with the other coils. At short and large distances, the coils with capacitors positioned at 15° and 345° exhibited higher field intensities, as depicted by the red solid, and dotted lines. The line profile graphs show an interesting characteristic, where the field intensity varied in accordance with the position of the capacitor: a higher intensity was produced when the position of the capacitor was closer to the source, whereas the lowest value was produced when the capacitor was at 180°. If these values are to be modeled, they will be approximated as a sum of cosine and sine:
where \(\theta\) is the position of the capacitor given in degrees, and the coefficients a0, a1, a2, and w are dependent on the size of the coil, distance, and orientation of the line profile.
Another important aspect of this graph is how the reference coil compared with other coils; the performance of the reference coil was comparable to those of the coils with the capacitors positioned in the range 105° to 135°. It should be noted that, at this point, these inferences have been based only on empty-space simulations; in the presence of a dielectric material (loading case), the field pattern will be affected by the composition of that object. Nevertheless, this empty-space analysis revealed that it is possible to achieve a higher field intensity using coils of the same size and through simple modifications in the position of the capacitor.
In Fig. 2c, we describe the statistical results for the |B1|-field in empty space. We summarize the mean |B1+|-field for each line profile. These figures show better correlations between the field intensity and the angular position of the capacitor. At short distances (Fig. 2c–i and Fig. 2c-iii), the reference coil performed better than eight other coil configurations only in terms of field strength. At large distances (Fig. 2c-ii), the reference coil performed better than only five of the coil configurations. We computed the coefficient of variance (CV), which is the standard deviation divided by the mean value, for each case, to determine the uniformity of the field. A lower CV value indicated that the samples were more uniform. For visualization purposes, we scaled the CV based on the maximum value for each distance such that the values could be fixed in the same plot as shown in Fig. 2d-i. This analysis is also interesting because it shows that depending on the application or desired field distribution, one can choose a coil configuration that would yield a more uniform field, either at short distances or at long distances. For example, C90 exhibited better field uniformity in the XY plane both at short and long distances from the coil. We also measured the focus of the field produced by each coil, and the difference in distance between the maximum |B1+|-field value and the center of the coil, as shown in Fig. 2d-ii. Positive values indicated that the maximum value of the field was oriented to the right, negative values indicated that the field was focused to the left, and a value close to 0 indicated a well-focused field. At short distances in the XY plane, the fields produced by coils C15 and C165 were more focused toward the center. At larger distances, the field produced by the coil with its capacitor at 90° was more focused toward the center compared to the fields produced by the other coils.
FDTD phantom simulations. We performed EM simulations on the same coil configurations but in the presence of the phantom. All simulations were performed using an excitation Gaussian pulse with a center frequency of 300 MHz and a BW of 600 MHz. Each coil was re-tuned and matched to 50 Ω. The distance between the coils and phantom was 10 mm. Individual EM simulations were performed on each coil to acquire the |B1+|-field. Figure 3 shows the |B1+|-field for the selected coil configurations. Data analysis was performed using MATLAB (The MathWorks, Inc., Natick, MA, USA), and for data processing, the field maps were resampled to have the same matrix sizes, i.e., to 256 × 256 × 90.
We compared the |B1+|-field in the XY plane at the center of the coil between the reference coil (Fig. 3a–i left) and C315 (Fig. 3a–i right). We used the marked green line to create a field line profile, such that it would be easier to compare the coils with each other, as shown in Fig. 3b-i. The reference coil is indicated by the blue line. This graph shows an interesting pattern: the coils with their capacitors positioned between 15° to 150° exhibited higher field intensities focused on the left part of the field (negative x), with the highest intensity achieved by the coil with its capacitor at 75°. The coils with capacitors at 15° and 150° exhibited field intensities in this region similar to that of the reference coil. The coils that performed better in the right part of the field (positive x) were those with capacitors between 210° and 345°, with the highest field intensity in this region produced by the coil with its capacitor at 285°. The coil configuration that produced the best mean value for this line profile was C315. This indicated that it is possible to control the field focus through simple changes in the capacitor position. The |B1+|-fields in the ZY plane for the reference coil (left) and C315 (right) are shown in Fig. 3a-ii, respectively. The line profile in Fig. 3b-ii shows that C285 and C270 exhibited the highest field intensities, whereas the reference coil exhibited a field intensity that was slightly better than the average.
A statistical analysis of the simulations involving the phantom is shown in Fig. 3c and 3d. The mean and standard deviation computed for the whole phantom (volume) for each coil configuration are shown in Fig. 3c-i and 3c-ii, respectively. This analysis revealed that coil C315 exhibited the highest field intensity, and that a total of 10 coil configurations resulted in better field intensities than that of the reference, with the lowest value achieved using C150. On the other hand, coil C240 resulted in the lowest standard deviation; moreover, eight coils resulted in better standard deviations than that of the reference coil. We segmented and created ROIs at each rectangle of the phantom to visualize the field patterns for each coil configuration. Figure 3d shows a representation of the mean value at each ROI for each coil configuration. The ROI number is shown on the y-axis, whereas the coil type is shown on the x-axis. We normalized each ROI (row) based on the maximum mean value among the coils to make it easy to visualize which coil had the best performance at each ROI. We used a heat-color map to visualize these values, with a value of 100 representing the maximum field intensity on that row. This type of analysis could be used for designing loop coils with specific target zones or patterns. For example, the coil configurations from C15 to C150 exhibited higher field intensities toward the left, whereas the fields of C210 to C330 were more focused toward the right; this result also demonstrated that higher field penetration can be achieved using coil configurations C270 to C315. Of particular interest is ROI number 8, which was assigned electrical properties simulating those of CSF; the mean values at this ROI for all tested coil configurations are shown in Fig. 3c-iii. In this ROI, the coil with its capacitor positioned at 285° exhibited the highest mean value, which was 17% higher than that of the reference coil. This coil configuration was also demonstrated to be capable of producing the highest field intensity over a long distance from the coil, with most of the maximum values in the ROIs from 15 to 20, corresponding to the last row of the phantom.
When a transmission coil is analyzed, it is often desirable to show the SAR, and the performance of the magnetic field in relation to the SAR. In Fig. 4a, we show the 10 g-averaged SARs computed for the reference coil (left) and coil C180 (right), respectively. Figure 4b-i shows the maximum SAR for each of the coil configurations. The lowest SAR was exhibited by C180, whereas the highest was exhibited by C60. Normalization of the |B1+|-field was performed via division by the square root of the maximum SAR. Figure 4a-ii show the normalized |B1+|-fields for the reference and coil C345, respectively, whereas the mean value of the normalized |B1+|-field is shown in Fig. 4b-ii. The results show that coil C345 exhibited the best performance, especially when compared with that of the reference coil.
FDTD simulations of array coil with phantom and Duke model. We also performed EM simulations on combinations of two coils to analyze the performance in terms of coupling [25, 26], as shown in Fig. 4c. For this analysis, we intentionally positioned the coils such that the distance between the coil centers was 65 mm. The objective was to determine if, at this distance, it is possible to obtain a better decoupling than that of the reference coil. We performed simulations with the following combinations: first, we performed simulations on pairs of coils with the same capacitor positions, and then performed simulations with pairs of coils with capacitors in opposite positions, e.g., 90° and 270°. Figure 4c-i shows the S21 parameters between coils with the same capacitor positions. As shown in the figure, the coils with capacitors positioned between 135° and 225° exhibited better decoupling than that of the reference coil. Specifically, coil C180 exhibited an S21 of − 22.26 dB, whereas the reference coil exhibited an S21 of − 17 dB. On the other hand, for the coils with capacitors in opposite positions, including the pairings with coils C90 and C180, the S21 parameters are shown in Fig. 4c-ii. For this type of configuration, coils C195 and C165 resulted in better decoupling than those of the other combinations; however, the values were still lower than that of the coil pair with C180.
We also performed EM simulations with a male human model (Duke), which consisted of more than 100 tissues, and positioned pairs of coils along the spinal cord. All simulations were performed using a Gaussian pulse with a central frequency of 300 MHz and a BW of 600 MHz, with a normalized input power of 1 W. We performed a total of 24 simulations; each simulation was conducted on a coil array consisting of 12 elements. Each simulation involved one of the 23 coil configurations with capacitor positions ranging from 15° to 345° in 15° intervals, or the reference coil. For each simulation, the array was composed of coils of the same configuration, e.g., one simulation involved an array consisting only of coils in the C15 configuration, whereas another simulation involved an array consisting only of coils in the C90 configuration. The distance between the centers of the coils in the X direction was 65 mm. Figure 5a shows the |B1+|-field normalized by the square root of the maximum SAR. For the |B1+|-field in the XY plane, that of the reference coil array (Fig. 5a-i) is shown in the top row, whereas that of the C75 coil array (Fig. 5a-ii) is shown at the bottom; Fig. 5a show the XY planes at the centers of each pair of coils, as illustrated by the green lines. Coil C75 exhibited a higher field intensity than that of the reference coil array, both in the XY and ZY planes. In Fig. 5b, we summarize the statistical analysis for the entire volume and resized the matrix sizes of the simulations such that they are depicted with the same dimensions. Figure 5b-i shows the |B1+|-field mean value for each of the coil arrays. The C75 coil array exhibited a higher field intensity than those of the other coil arrays, whereas the reference coil array exhibited the lowest field intensity. Furthermore, C180 exhibited the lowest mean value among the coils equipped with a single capacitor; incidentally, C180 was also a symmetrical type of coil. With regard to the uniformity, we computed the CV for each coil array and plotted the corresponding values in Fig. 5b-ii, which shows that C285 was the coil array that produced the most uniform field, which was approximately 60% better than that produced by the reference coil array. We also computed the maximum SAR10g for the whole body, as shown in Fig. 5b-iii, which shows that C60 and C300 had the highest SAR10g values, and that C180 had the lowest maximum SAR10g. The results show that, in terms of the SAR, at least six of the coil arrays performed better than the reference coil array.
As an example of how this study can be applied to coil array design, we present a simple coil selection with the objective of improving the |B1+|-field distribution on the spinal cord. The goal of this selection was to obtain a similar field intensity along the entirety of the spinal cord. The reason for this objective is that when a planar linear coil array is used, the field intensity is higher at the regions of the body that are closer to the coil, usually at approximately T6 and T11, whereas C1 to C4 would be subjected to lower field intensities because of the larger distance to the coils. The |B1|-field along the spinal cord could be optimized to be of a similar amplitude through selection of coil configurations that would yield a similar field intensity at every region of the spinal cord. To demonstrate this example, we performed a statistical analysis for each XY plane marked. We divided each region and computed the mean |B1|-field for each of the coil arrays, the results of which are shown in Fig. 5c-i. In this figure, the y-axis identifies the six regions corresponding to the position of the pair of coils, whereas the x-axis identifies the type of coil. For region 1, which corresponds to C1 to C4, the coil that exhibited the highest mean field value was C75, with an average of 0.033 µT. If we use this value as a reference, we can determine other coils that would have similar mean field values in the other regions. For regions 2 to 6, we can select the pairs of coils with C60, C240, C150, the reference coil, and C75, respectively. Each of these coils would have an approximate field value of 0.033 µT. We performed EM simulations using this selection of coil pairs to create an array. The resulting |B1+|-field is shown in Fig. 5c-ii and 5c-iii, including a comparison with the reference coil array. In this field map, we highlighted the value of the field equal to 0.02 uT, which was constant along the spinal cord for the case of the array of selected coils in comparison with the reference coil array. The field standard deviations computed for only the spinal cord for the reference and optimized coil arrays were 0.024 and 0.010 µT, respectively. This simple example demonstrated that it was possible to enhance the field pattern through the selection of coils with different capacitor positions.
Different capacitor placement in loop coils. We developed circular coils based on the dimensions of the simulations. We made one pair of coils for each capacitor position from 15° to 345° in 15° intervals, plus a pair of reference coils that had four capacitors evenly distributed. We tuned and matched the coils as shown in Fig. 6a-i. To simplify tuning and matching, we used variable capacitors. The coils were tuned to 297.3 MHz. The coils were placed on top of the phantom at the loading conditions. After tuning and matching, we tested the coupling between each pair of coils with the same capacitor positions. The S21 parameters are shown in Fig. 6a-i. We performed the test on three separation distances: 55, 60, and 65 mm between the centers of the coils, as shown in the top, middle, and bottom panels, respectively. Through this test, we intended to demonstrate that a number of the coils will exhibit better decoupling for different overlapping distances. The left column of Fig. 6a-i shows the S21 parameters for the reference coil, whereas the right column of Fig. 6a-i shows those for C45. Figure 6a-ii shows a summary of the S21 parameters for all the coils. According to the figure, for a distance of 55 mm, which corresponded to a high overlapping distance, the coils with the capacitor at 315° exhibited to a low coupling of approximately − 30 dB, in comparison to the reference coil, which demonstrated a coupling of − 7 dB coupling. Most of the coils exhibited a lower coupling than that of the reference coils. When the distance between the coils was 60 mm (middle row in the figure), coils C45 and C255 exhibited a coupling of − 20 dB, whereas when the distance was 65 mm (lower row), coil C30 demonstrated a coupling of − 17 dB. In addition, we observed a number of special cases, where coils C105 and C285 exhibited couplings of − 21 and − 27 dB, respectively, when the distance between the coupled coils was 50 mm. Meanwhile, C300 demonstrated a coupling of − 24.9 dB when the coil separation distance was 45 mm, whereas C90 exhibited a coupling of − 20.6 dB when the coil separation distance was 40 mm.
Experiment implementation. We tested the performance of the coils by acquiring MR images in the presence of a phantom. The images were obtained using each coil equipped with a capacitor at each of the listed angular positions and with the reference coil. The images were captured using a 7T MRI scanner, with a gradient echo (GRE) pulse sequence, a repetition time of 300 ms, echo time of 4 ms, slice thickness of 3 mm, acquisition matrix of 256 × 256, and flip angle of 30°. We repeated the experiment with each coil using the phantom, as described in the Method section. The images acquired using the selected coils are shown in Fig. 6b, showing the axial and sagittal views, respectively. We performed a comparison between the reference coil (Fig. 6b-ii), C45 (Fig. 6b-iii), and C240° (Fig. 6b-iv). The images are visualized with the same intensity window. For the reference coil, the images for ROIs 2 and 4 have lower intensities than those for C45. Meanwhile, the bottom row of the image, depicting the observations for C240, shows a higher intensity compared with those of the other two coils. We examined the image intensity in each ROI of the phantom and visualized the summarized values in Fig. 6c-i, where the x-axis denotes the coil type, starting with the reference, and the-y axis denotes the ROI region. We used a heatmap representing the mean values; for easy visualization, we normalized the values to the maximum for each ROI, such that a row or ROI will show which coil had the best performance. According to the figure, the coil with its capacitor at 45° exhibited higher mean values than those of the other coils, whereas coil C240 also exhibited high intensity in its ROIs, according to the lower row of the figure. The statistical analysis for the sagittal view is shown in Fig. 6c-ii, which visualizes four regions and shows that C240 exhibited the highest mean value.