Rf Surface Receive Array Coils the Art of an Lc Circuit
ane Introduction
Sodium magnetic resonance imaging (23Na-MRI) has the potential to become a valuable tool in the clinical setting, by profitable physicians through the diagnosis, prognosis, and monitoring of a variety of pathologies including cancers, degenerative brain disorders, concussion, and osteoarthritis [1, 2]. However, due to quantum mechanical limitations such equally rapid spin dephasing and a small-scale gyromagnetic ratio, and biological restrictions including low in vivo sodium concentrations, the signal-to-noise ratio (SNR) of the resultant 23Na-MRI images tin be upwardly to thousands of times lower than traditional clinical proton MRI images [2, 3]. The point-of-source in MRI signal generation and acquisition is the radiofrequency (RF) coil, and the quality of whorl performance volition propagate throughout the remainder of the organisation. The quality of the performance of an MRI RF coil is based on the contribution of several factors intrinsic to the scroll such as the tune and match of the gyre, the quality factor (Q), the homogeneity of the produced RF field (Bone +), and the homogeneity of the induced RF signal (B1 −) [4, 5].
The simplest RF coil is a one element transmit/receive (Tx/Rx) surface gyre, consisting usually of a single conductive loop with capacitors for tuning and matching [four, 5]. The Tx/Rx surface coil has great bespeak sensitivity in a local expanse; withal, it produces an inherently heterogeneous Bone + field (and due to reciprocity, the B1 −) because the field force decreases with distance from the scroll. Multiple surface coils tin be arranged together to form a multi-aqueduct phased array whorl. Phased assortment coils tin produce a more homogeneous B1 +,− field over a larger volume than a single surface curl; all the same, the multiple elements interact with each other, which creates challenges when designing and constructing the coil. This study looks to come across if a fractal geometry can play a role in improving the SNR of 23Na-MRI past overcoming some of the shortcomings of traditional RF coils.
The term fractal originated from Benoit Mandelbrot's Fractal Geometry of Nature published in 1983 [6], even so these structures accept been studied by mathematicians since the early 1900's. Fractal geometries are patterns that tin be decomposed into self-similar elements. As such, they do not conform to standard Euclidean geometry and instead behave idiosyncratically. These patterns can exist in not integer dimensions and take unique infinite filling power [seven, 8]. Fractal geometries have been explored earlier in electromagnetic applications [9], equally fractal antennas have been used in telecommunication systems for years, with the common benefit beingness compact size which allows for a greater constructive antenna length within a smaller space. This is not the only benefit to having a fractal (or fractal-like) shape, as it has been shown that these so-called "shaped antennas" can produce college gain, directivity, and field forcefulness than a standard loop or monopole/dipole antenna of a comparable size [eight, 10].
Telecommunication systems are by and large focused on far field applications, nevertheless most MRI RF coils exist in the nearly field region, so the question arises: how do far field fractal antennas translate to near field MRI applications? A single fractal antenna can human activity as though information technology is comprised of multiple elements due to each self-similar subsection of the coil radiating as a unmarried antenna [8]. This may produce some interesting constructive and destructive interference patterns more locally around the antenna and could more evenly distribute the radiated energy. The aforementioned space filling ability of a fractal antenna has been shown to result in higher Q values and better impedance matching than standard antennas that have up the same amount of infinite, which would benefit MRI past maximizing the signal transfer [xi, 12]. It has also been shown that a multi-loop fractal-like surface whorl geometry in proton MRI applications produces higher sensitivity [13].
There exist thousands of fractal and fractal-like geometries and a smaller subset are the focus of near antenna research and applications, merely here only i pattern was studied: the Koch snowflake (Figure 1). This geometry was 1 of the fractals that was explored in [11, 12] and showed a higher Q, better impedance friction match, and less mutual inductance when in arrays. In proton MRI applications, information technology has been shown that multiple Koch snowflake fractal elements when in an overlapping array have higher sensitivity, Q values, and SNR along with reduced mutual inductance than circular elements in a similar array [14]. Previous simulations of a Koch snowflake whorl performed past Dona Lemus et al. [15] and Nowikow et al. [xvi] have shown that concrete construction and implementation is warranted.
This leads to the hypothesis that a Koch snowflake fractal geometry surface curlicue can improve the quality of 23Na-MRI images by increasing the resultant SNR due to a more than homogeneous B1 + (and Bane −) field, a superior filling cistron, and more robust impedance matching than a typical circular geometry surface coil. In addition, the lower common inductance for the Koch snowflake geometry will facilitate implementation in phased array coils.
2 Materials and Methods
2.1 Scroll Design
Design parameters for the initial simulations were set with fabrication in mind. Due to the complex geometry of the Koch snowflake fractal, it was decided that the coils would be manufactured on a printed excursion board (PCB) to permit for consistency and accuracy. This restricted the size of the curlicue designs to fit within a 100 mm by 100 mm area. As such, the coil bore chosen was ninety mm (∼3.5″) which falls within the iii–six″ size of many standard commercial surface coils. The option of conductor was copper equally it is a standard etching material. The width of the copper was selected to exist 3 mm as information technology was narrow plenty to allow for a Koch design only wide enough for ease of capacitor soldering for the eventual construction. With restrictions on ringlet bore and copper width, it was determined early that the simply viable Koch snowflake fractal generation that could be manufactured with a level of practicality was a third-generation Koch. Thus, two dissimilar Tx/Rx surface coils were simulated and synthetic, the first was circular in geometry to human activity as a reference for a standard "typical" surface coil and the second was a third-generation Koch snowflake fractal geometry (Figure 2).
FIGURE 2. The geometry of the (A) round, and (B) fractal coil. Each were copper etched on a 100 × 100 mm FR4 substrate with breaks in loop for the tuning capacitors (C due south1−4) and the matching capacitor (C p ). One stop of the loop for both coils was connected to the T/R switch via a coaxial cable while the other end of the loop was grounded using the same coaxial cable to course a one port network.
ii.two Simulations
The 2 coils were faux using ANSYS HFSS (Ansys Inc., Canonsburg, PA, United States), using the design constraints laid out previously. The coils were simulated as copper sheet conductor (infintesimally thin) on a 1 mm thick FR4 substrate. Each design incorporated five breaks, four in the main loop for tuning capacitors and one in between the legs for a matching capacitor. These were used to tune the coils to 33.viii MHz, the Larmor frequency of sodium at three Tesla (T), and to match the coils to 50Ω. The coils were loaded by a rectangular box of 0.9% wt/5 saline to mimic a standard 23Na-MRI phantom.
Ii sets of fields data were caused via simulation. The kickoff prepare included the electrical (E) and magnetic (H) fields, besides as the surface current density (J), obtained at unity power, where both coils were fed with the same power level to human activity as a straight comparing between the fields. The second set of fields were acquired at an adjusted ability level such that the coils would produce a magnetic field that would cause a ninety° tip in the sodium spins in the phantom straight below the coil. At this power level the at present denoted B1 fields were acquired, besides as the East fields and surface J fields.
The H and B1 fields were analyzed in MATLAB (Mathworks, Natick, MA, U.s.a.) to determine their homogeneity with two chief methods. First the homogeneity of the field was measured by calculating the mean and standard deviation of the signal over various regions of interest (ROIs) selected to exist located in the sensitive region of the coil [4, v], where the standard deviation of the field in each ROI was used as a representation of the homogeneity. The selected ROIs were ii cylindrical regions of different radii (25 and 45 mm) situated with their circular faces parallel to the coil plane with a height between a half radius and a radius away from the coil plane. Six spherical ROIs were likewise selected, with varying radii (6, 8, 10, 12, 14, and 18 mm). All ROIs are shown in Figure 3. It was also of interest to see how the magnetic fields behaved as distance from the coil increased, then a prepare of plots were made of the mean signal strength every bit a circular ROI (of two radii, 25 and 45 mm) moved abroad from the curlicue plane.
Effigy 3. The ROIs used for the homogeneity measurements of the simulated fields and the 3D radial book information, as well as the SNR calculations from the 3D radial data of both the circular and fractal coils. The cylindrical ROIs are of radius 25 mm and ∼45 mm and are used to represent the coil's optimal sensitivity region between r/2 and r deep into the phantom [iv]. The simulations also used 6 spherical ROIs of radii six, viii, 10, 12, fourteen, and 18 mm, whereas the measured 3D radial information used just 3 spherical ROIs of radii nine, 12.5, and eighteen.75 mm. The noise ROI for SNR calculations using the measured data is shown equally the spherical ROI above the curlicue plane for the 3D radial data.
The East and J fields were used as a fashion to run across if the rubber measures betwixt the two coils would demand to be modified. Every bit the fractal coil has a greater effective length of conductor in the same expanse information technology was thought information technology may require more than capacitive breaks to help disperse the Eastward field and current density.
2.three Curl Construction and Demote Measurements
After simulation the two coils were and then synthetic. The coils were manufactured on a PCB, as copper carving on an FR4 substrate by Elecrow (https://www.elecrow.com/). The etched copper was 3 mm in width and 35 um in thickness, and the coils themselves replicated the simulated designs: 90 mm in bore to fit on the 100 × 100 mm substrate. The same number of breaks in each loop equally the simulations were incorporated into the pattern to allow for tuning and matching capacitors (American Technical Ceramics, NY) with values ranging between 10 and 470 pF. Ideally, the coils would be fed by a λ/two coaxial cable; all the same, due to sodium resonating at such a low frequency at 3T, the coaxial cable required would have to be 4.45 m in length which would non be feasible to implement in the MRI without introducing loops and bends. Instead, a 0.iv m coaxial cablevision was used to feed the 2 coils, the shortest length that immune the coil to attain both the phantom and transmit/receive (T/R) switch.
All S-parameter and Q measurements of the fabricated coils were performed using an Agilent 4395A network analyzer and an Agilent 87511A Due south-parameter examination gear up. Both the circular and fractal surface coil were tuned and matched, while beingness loaded past a saline phantom, to 33.viii MHz, the Larmor frequency of sodium at three T. To explore how effective each geometry was at matching to diverse parts of anatomy, once their original tune and friction match (to the phantom) were set up, the coils were loaded with different parts of the torso and the respective S 11 parameters were recorded and reported in decibels (dB): xx logx(|S 11|). The chosen regions of anatomy were the correct knee, abdomen (slightly correct from midline, below the ribcage to emulate a possible liver scan), and the dorsum of the caput as these are three potential clinically relevant regions for 23Na-MRI scans. 11 subjects were used for the human knee, belly, and head loading experiment. A ratio of unloaded to loaded Q (Q U and Q L ) was calculated for each actual load [denoted as (Q ratio )].
Additional round and fractal coils were synthetic, and tuned and matched to the phantom, to explore how the scroll geometry afflicted their mutual inductance. The S 21 of each curlicue pair while loaded past the phantom were measured as 20 logx(|Due south 21|) (dB) using the aforementioned network analyzer and S-parameter examination prepare at multiple separation distances between 4 and 9 cm (measured middle to center), as well every bit at the optimal separation for a unmarried, not split up, Southward 21 elevation. Every bit two coils interact with each other their resonant frequency splits into two modes which can be visualized as a split up in Southward 21 bend. The geometry of the ringlet dictates at what caste of curl separation geometric decoupling occurs (i.e. where the common inductance is eliminated, or at least minimized) which tin exist visualized as this dissever S 21 curve merges back into a unmarried resonant peak. The fractal coil'southward Southward 21 for each altitude was measured in three configurations: overlap on the long axis (denoted LA), overlap on the brusque axis (denoted SA), and overlap between the long axis of ane fractal and the curt axis of the other (denoted MA, for "mixed axis"). The Q U and Q L were measured for the coils at their optimal altitude of separation/overlap.
2.4 Experimental Setup
The remaining experiments were performed using a GE MR750 3 T magnet (General Electric Healthcare, WI). The phantom was elevated on the bed of the MRI by a foam riser such that when the selected coil was placed atop, the field-of-view (FOV) of the coil was at magnet isocenter. The coil in employ was continued to the scanner via a single-channel sodium T/R switch. The roll that was existence used was positioned on top of the phantom, substrate down, such that the coaxial feed was aligned downwards the bore of the magnet. The coil was secured in place using a weighted purse. Each coil had a vitamin E capsule taped to the center of the whorl to allow for localization with a generic proton three-plane localizing browse using the body coil. The phantom used to load the coils during their tune and match was the phantom that was used for imaging. The phantom was a rectangular plastic box of dimensions 160 × 270 mm with a depth of 90 mm containing 0.9% wt/five saline (aqueous NaCl). The size of the phantom was chosen to let for a B1 + map of the total FOV of each scroll if a xc° tip bending was applied directly below the curl aeroplane.
ii.5 MNS Prescan
To determine the power required to attain a 90°tip angle, a GE Healthcare sequence, implemented by Schulte et al. [17] chosen the MNS prescan, was used. The power was calibrated to a 10 mm thick airplane at the surface of the phantom, directly beneath, and parallel to the aeroplane of the coil, and the lid of the phantom. Before the MNS prescan was run, an initial 10dB attenuation was applied to the RF ability, so that the transmit proceeds (TG) could be effectively chosen (no initial attenuation would result in too much ability for a 90°tip, even if the TG was 0).
2.6 B1 + Mapping
The Bloch-Siegert shift method, as outlined by Sacolick et al. [18], was selected as the way to map the B1 + fields of the two coils. Four maps of the B1 + field were calculated for each ringlet: 1 perpendicular to the coil airplane and through the center of the coil, and iii maps parallel to the roll plane, at differing depths (15, 25, 35 mm) from the whorl. The data was caused using a 2D, four-arm spiral sequence over a 150 mm FOV. The tip bending of the ten mm thick slice selective pulse was xc°. The repetition fourth dimension (TR) of the sequence was 84 ms, and fifty indicate averages (NEX) were used. Yard-space was reconstructed using the algorithm described by Beatty et al. [19] into an 80 × eighty matrix. The hateful and standard divergence of the field force over each map's FOV was calculated to determine the homogeneity of the field in that slice.
2.7 Homogeneity and Point-to-Noise Ratio Measurements
A 3D radial sequence was used to obtain a 48 × 48 × 48 paradigm matrix of the phantom. A ninety°difficult pulse was used to excite a 150 × 150 × 150 mm volume, and 7,333 spokes were sampled—one per every 23 ms TR.
Two NEX were used, and 1000-space was reconstructed using the previously stated algorithm [nineteen]. The volume selected was centered on the curl in the transverse plane, and encompassed the entire depth of the phantom below the curlicue, and an additional air space above the coil. This book was called such that the entire FOV of the coil would be imaged while leaving enough "empty space" available for a noise measurement.
The Bi + mapping sequence acquired the data in a 10 mm thick slice, and as such did not allow for reliable measurements of field homogeneity over a volume. Withal, since the acquired bespeak is proportional to the sine of the flip angle, and the flip angle is proportional to the magnitude of the B1 + field, the point acquired past the 3D radial imaging sequence over its fifteen cm FOV is representative of the field strength over that same volume. The homogeneity of the field was then measured using this 3D book, by computing the hateful and standard deviation of the betoken over various ROIs selected to be every bit similar as possible to the ROIs used to clarify the simulation results. The cylindrical ROIs for the experimental data were situated in the same coil sensitivity region with radii of 25 and 45 mm. Due to a reduction in resolution of the experimental information versus the simulated fields, only three spherical ROIs could exist obtained with radii of nine, 12.5 and 18.75 mm. The ROIs used for this 3D radial data can be found alongside the fake fields' ROIs in Effigy 3 (Note the FOV of the simulated data is 11 cm equally opposed to the xv cm FOV of the experimental data which explains the size discrepancy in the figure). As with the simulations it was of interest to see how the Bane + field behaved as distance from the scroll increased. Thus, plots were made of the hateful indicate forcefulness every bit a circular ROI (of ii radii, 25 and 44 mm) moved away from the coil plane. Much like the analysis of the simulated data, all experimental data was analyzed using MATLAB.
The 3D volume acquired by the radial sequence was as well used for SNR measurements. To allow for a homogeneity/SNR comparison, multiple SNR values were calculated, using the five ROIs described previously (Effigy 3) as the signal region in the SNR calculation. The noise region used in the calculations was kept abiding and tin can be seen in Effigy three (spherical region to a higher place the coil). SNR was calculated as μ signalROI /σ noiseROI .
3 Results
three.i Coil Loading and Matching
The Due south eleven and Q values of the two coils when unloaded, loaded by the phantom, and loaded by the varying anatomical regions are given in Tabular array i. The values associated with anatomical regions (due north = 11 subjects) are reported as a mean ± standard difference. The Q ratio (Q U /Q Fifty ) is besides reported in the table. The coils were tuned and matched while being loaded by the phantom, and equally such the respective S 11 (at 33.viii MHz) of both are below a respectable −28 dB. At that place are three notable differences between the two coils when information technology comes to coil friction match and tissue coupling. Starting time, the Koch curl has a significantly meliorate match than the round coil when being loaded by the anatomical regions, which can be observed in the lower S xi values. The 2d notable difference is the measured values when the coils are unloaded. The circular coil "unmatches" more so than the Koch scroll (i.e. a higher S eleven) which results in a college Q U . These combine for a notable difference in Q ratio when loaded by the phantom, and less notable difference in Q ratio when loaded past human being subjects.
TABLE ane. The S eleven values (in dB), the Q values (unitless), and the ratio of unloaded to loaded Q are reported in this tabular array for both the circular and fractal scroll over various loads. A mean and standard difference are reported for the values calculated over the 11 subjects.
3.two Mutual Inductance
The S 21 curves of the circular coil pair, and three configurations of the fractal coil pair, are shown in Effigy four over a 20 MHz bandwidth. At a separation distance of 9 cm both coil pairs had a separate Southward 21 peak, and as coils began to overlap, the peaks merged into i, each at a unique distance. It tin be seen that depending on the rotational orientation of the fractal pair, the optimal altitude to eliminate mutual inductance was varied, ranging betwixt 5 and 6 cm, whereas the circular pair merely had 1 optimal distance of 6.5 cm. Information technology can be noted likewise that the fractal coil pairs, regardless of orientation, were less responsive in terms of an S 21 carve up peak as a office of distance, which could pb to the assumption that the interaction between the 2 fractal coils is less than the interaction between the two circular coils. The Q Fifty and the Q U values are reported in Table 2 for the different coil pair configurations. There were ii instances, when the curl pair was unloaded, that the Southward 21 top carve up and the Q U was not measurable—and is indicated every bit "northward/a" in the table.
FIGURE iv. The S 21 curves (in dB) of (A) the circular gyre pair, (B) long axis (LA) overlap of the fractal pair, (C) mixed axis (MA) overlap of the fractal pair, and (D) short centrality (SA) overlap of the fractal pair. The roll diagrams in the top left corner of each plot bear witness the ringlet configurations. The black "OP" curve is the curve at an optimal distance of separation where the mutual inductance goes to nothing and the curve merges into one atypical peak.
TABLE two. The Q values for each coil pair when arranged at their optimal distance to eliminate mutual inductance. Those distances are respectively half dozen.five, v, five.v, and 6 cm. The Q U was reported every bit "n/a" for the configurations where the Southward 21 peak was split at the optimal distance when unloaded.
3.iii B1 + Field Homogeneity and Signal-to-Noise Ratio
The four B1 + maps fabricated for each roll are shown in Figure 5, alongside their fake counterparts. The experimentally determined maps were overlayed onto their respective slices taken from the 3D volume acquired using the radial sequence. The mean and standard deviation of the B1 + maps are given in Table iii. The caused maps were quite noisy, specially the 3rd parallel slice 35 mm from the coil, which explains why the field strength appeared to increase at a larger depth. Even so, the standard deviation of the round coil'due south Bane + field for each map was consistently lower.
Figure 5. The experimental B1 + field maps of the circular and fractal coil, alongside each maps simulated analogue at the same slice location. The simulated B1 field maps and the measured Bane + field maps are given in μT and the simulated H field maps are given in A/m.
Table iii. The mean and standard deviation of each of the measured B1 + maps shown in Figure 5, given in μT over the FOV shown in the map. The three parallel slices are listed in order of depth from the coil: fifteen, 25, and 35 mm respectively.
The bespeak strength and homogeneity calculated over the various ROIs from the 3D book are given in Figure 6, again, aslope their faux counterparts. In the simulations the field strength of the fractal scroll over the ROIs was consistantly higher, whereas experimentally the coil with the higher signal for each ROI fluctuated; yet, the standard deviation of the signal acquired by the circular whorl is consistently lower than that of the Koch coil, over all ROIs—false and measured. The plots of signal behaviour with distance from the coil are shown in Effigy vii. The indicate from the Koch scroll decays (in space) more rapidly than that from the circular coil, with signal variation of the latter being less likewise. The SNR, calculated over diverse ROIs (Figure 3 and Tabular array 4) shows the circular gyre consistently had greater SNR than the fractal coil.
Figure 6. The ways and standard deviations over various ROIs shown in Figure 3 for the simulated H field (given in A/m), the power compensated simulated B i field (given in μT), and the measured signal from the 3D radial volume data (given in a.u.) for both the round and fractal ringlet. As the standard deviation of the ROIs was used as the main metric for determining a given coil's homogeneity, the standard deviations are shown for comparison as their own bar in the plot. The radii of each ROI is the label for each set of bars.
Effigy 7. The plots of field force and variation of the simulated H field (given in A/thousand), the power compensated simulated B one field (given in μT), and the measured bespeak from the 3D radial volume data (given in a.u.) over a circular ROI (of radii 25 and 45 mm) as the ROI moves abroad from the curlicue and deeper into the phantom for both the round and fractal coils. The mean of the field strength is given by the bold curves in cherry and blackness, and the lighter bands in gray and pink that encompass the bold curves represent the range of indicate values over the ROI at each distance from the whorl.
Tabular array four. The SNR values for each 3D radial experimental information ROI shown in Figure 3. The first 3 ROIs in the table are the spherical ROIs, and the last two are the cylindrical ROIs. The noise ROI used for calculations is also given in Effigy 3 as the spherical ROI higher up the gyre plane.
3.4 Curlicue Rubber
The false roll surface electric current densities, scroll surface electric fields, and the simulated electric fields in the phantom can exist found in Figure 8. It tin be seen that the fractal coil'due south conductive surface had both a lower current density and electric field than that of the circular whorl at both power levels. The Eastward fields deposited in the phantom are quite similar for both coils, nevertheless the round coil's E field appears to penetrate deeper into the phantom, whereas the fractal coil's E field is more full-bodied into a smaller expanse (on a slice per piece basis).
FIGURE eight. The faux surface current density, the surface electric field, and the electric field depositied in the phantom for both the circular and fractal curlicue.
4 Discussion
This report explored the network parameter and Bane + field characteristics of a Koch snowflake fractal geometry RF surface roll compared to that of a standard circular surface roll for sodium MRI. The hypothesis was that the fractal coil would provide increased SNR in sodium MRI. Coil loading and matching, common inductance, RF field homogeneity and SNR was evaluated. However, the fractal pattern merely excelled in being less prone to object-induced match variability and in reducing mutual inductance between loops, as compared to circular coils of the aforementioned diameter.
Impedance matching is important when designing, constructing, and tuning an MRI RF gyre—the better the match, the optimal the ability transfer between the coil and the organization, and hence a higher resultant SNR [xx]. Most coils are tuned and matched once to a standardized phantom or specific region of anatomy. Some in house manufactured surface coils may include a variable capacitor that can be adjusted based on the load, however most commercial coils lack this feature, and furthermore, would be impractical for phased arrays. At that place has been piece of work in tuning circuits for single and multi curl applications however they are limited [five, 21, 22]. The coil impedance match is ideal when loaded past the phantom/body office to be probed. One time the coil is loaded past something else, whether information technology be a unlike phantom or different region of the trunk, or a dissimilar person altogether, the match will suffer for it. The Southward 11 parameter is a measure of the coil match, where the lower the Due south xi (in dB) the better. Based on Due south xi measurements (Table 1) the fractal coil was more robust and insensitive to anatomical region compared to the circular coil. All the same, this advantage came at the cost of a reduced Q ratio . Thus, fractal-based coils may offer a robust solution to impedance matching (at a price) for varied anatomy when manual matching is not an option (i.e. phased array configurations). The S eleven values would need to be explored further in a larger phased assortment configuration.
When surface coils are bundled in arrays they begin to interact and interfere with each other via geometric coupling, resulting in an equivalent mutual inductance betwixt coils. Information technology is not possible to eliminate between-coil mutual inductance in phased arrays. Yet, a goal is to ideally minimize this problem in order to maximize SNR [v]. As private roll elements are moved towards each other and begin to overlap, the common inductance effect worsens. As the overlap continues the coils will reach an optimal overlap that eliminates the mutual inductance, where this optimal configuration varies based on geometry [23]. Optimal overlaps are well documented for coil array elements for standard geometries, notwithstanding it can be a challenge to adhere to those restrictions when trying to optimize coil coverage over a prepare area. The S 21 network parameter of a coil network indicates the mutual inductance between two coils. When two coils experience coupling it appears as a split Due south 21 peak and the peaks will merge into one when the mutual inductance is minimized. The S 21 measurements made (Figure 4) between the pair of circular coils, and the pair of fractal coils, revealed that the fractal pair allowed for more than configurations and spacings, such that the mutual inductance betwixt coils could be minimized or ideally eliminated. It was as well observed that the fractal pair was less sensitive to coil overlap as seen by less profound S 21 tiptop splitting which could allow for increased error when arranging large numbers of coils in assortment configurations.
The homogeneity of the coil's B1 + field is also essential in maximizing image SNR. The SNR is related to the sine of the flip angle, and the flip angle is related direct to the B1 + field strength. In gild to maximize signal from a sample, the flip angle needs to be 90°, assuming adequate T1 recovery time is permitted. This also requires a spatially homogeneous Bane + field across the sample too. Information technology is well known that surface coils produce inherently heterogeneous fields, and any increment in homogeneity would be benign to increasing SNR [4]. Measured Bi + field maps were created (Figure 5) for both circular and fractal coils, and field homogeneity was compared. The standard divergence of the field strength over each map was used every bit a metric for homogeneity, where the lower the standard deviation the higher the homogeneity (Table three). Over all the maps of the circular coil had a lower standard deviation, indicating a more homogeneous field compared to the fractal gyre. Withal, these maps were noisy even after averaging over a 10 mm thick slice. This made homogeneity assessment over a volume challenging. Thus, a 3D book was imaged and bespeak intensity variation beyond several ROIs within the curlicue FOV were used as a B1 + field homogeneity metric (Effigy half-dozen). For the experimentally adamant data, across all five ROIs selected, the circular coil had lower signal variation compared to the fractal ringlet, based on ROI standard departure measurements. Over again, lower standard deviation inside an ROI indicates a more homogeneous field, corroborating the Bane + field map findings. The 3D volume was also used to create a plot of signal intensity versus altitude from the gyre airplane, to visualize axial homogeneity (Figure 7). Surface coils have reasonable homogeneity at approximately one radius from the whorl airplane after which the betoken uniformity decreases rapidly (with distance). The fractal ringlet indicate drops faster with distance which besides supports the same conclusion as the perpendicularly sliced Bi + map that the round coil produces a more homogeneous field compared to the same diameter fractal coil.
When comparing the simulation results to the experimental at that place are some differences that should be noted. Interestingly in the simulations, the fractal roll produced a college mean field over the ROIs (Figure 6) and this discrepancy was reduced when power was compensated for. This can be explained past the fact that the fractal'south field, as tin exist seen in Effigy 5, is more than concentrated into a smaller FOV, whereas the circular ringlet's field is more than dispersed over a larger area. And and so experimentally due to the MNS prescan (the fractal coil required less power than the round whorl), the coils produced the same field force at the surface of the phantom and so the hateful field strength'south experimentally are quite similar. This aforementioned phenomenon can exist seen in Figure 7 besides, where in simulation the fractal scroll produces a stronger field closer to the coil. As the ROI increases from 25 to 45 mm, the means of the coils at shallower depths converge, again confirming that the fractal coil's FOV is smaller with a field that is more concentrated to the center of the slice. And and so over again since the MNS Prescan calibrates the power such that the coils give equal field forcefulness, the curves for the experimental data in Figure seven are essentially the same mean value.
The main similarity between the fake magnetic fields and the experimental data is that the circular coil consistently had a lower standard deviation meaning higher homogeneity. However, it is important to note that since it's been determined that the FOV's of the two coils are different, information technology may be of some involvement to investigate other ROI'southward of different shapes and sizes to more accurately deduce if the circular coil has a more homogeneous field for all applications.
SNR was calculated from the aforementioned 3D volume, over the same 5 ROIs equally described higher up. Adjustment with expectations that the roll with the more homogeneous Bone + field would provide a higher SNR, the round roll produced a higher SNR measurement beyond all ROIs used. While this outcome is the reverse to our hypothesis there are notwithstanding potential benefits to a fractal surface coil that deserve exploration. Sodium has only i resonance elevation in vivo, but homogeneity over a wider excitation bandwidth was not explored as it is not necessary for sodium. Every bit a fractal geometry consists of cocky-like elements, each individual chemical element of the whole tin can radiate every bit its ain antenna, metaphorically. In practicality what this means is the single chemical element fractal antenna can have a wider bandwidth than a not-fractal antenna, or even be multi-band [eight, 10, 12]. A wider bandwidth could amend the spectral homogeneity of the generated B1 + field. Improved transmit homogeneity would be beneficial for imaging nuclei with broad spectral bandwidths such as 13C, nineteenF and 31P, particularly when chemical shift imaging of these nuclei is existence attempted.
In determination we found that, although a Koch snowflake shaped surface ringlet did not provide improved SNR or spatial homogeneity, the largest potential do good is with reduced mutual inductance and robust impedance matching if implemented in phased assortment configurations. Phased assortment coils almost always provide higher SNR than their surface coil counterparts [24–26]. They can advance acquisitions via parallel imaging [27, 28] or compressed sensing techniques [29]. Multi-element arrays are a claiming to blueprint and construct as geometric coupling plays a big role in the ability of the coil to role properly. In that location take been attempts to create unique geometries of elements to reduce this cantankerous-talk and mutual inductance [30], but we believe the Koch snowflake would offering a more robust solution. As shown by the measured Southward 21 parameters, the fractal design allows for more than latitude when it comes to positioning individual elements over a surface, and provides minimal mutual inductance betwixt a pair of coils. Another challenge facing phased array coils is that they are a claiming to manually rematch with each scan, and is seldom, if e'er, done. The ability of a fractal design RF ringlet to provided a lower S 11 over varying loads could too improve the betoken transfer to the organization, boosting SNR. Thus, farther exploration into larger array fractal-based phased configurations, compared to arrays of circular coils, is warranted.
Data Availability Statement
The raw information supporting the determination of this commodity will be fabricated available by the authors, without undue reservation.
Author Contributions
All authors listed have made a substantial, directly, and intellectual contribution to the work and approved it for publication.
Funding
Funding was provided through a Natural Sciences and Engineering Research Council (NSERC) Canada NSERC Discovery Grant (RGPIN-2017-06318) to MN.
Conflict of Involvement
The authors declare that the research was conducted in the absence of whatever commercial or fiscal relationships that could be construed as a potential disharmonize of involvement.
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