Bench to bore ramifications of inter-subject head differences on RF shimming and specific absorption rates at 7T

Abstract

Purpose:

This study shows how inter-subject variation over a dataset of 72 head models results in specific absorption rate (SAR) and ${\mathrm{B}}_1^{+}$ field homogeneity differences using common shim scenarios.

Methods:

MR-CT datasets were used to segment 71 head models into 10 tissue compartments. These head models were affixed to the shoulders and neck of the virtual family Duke model and placed within an 8 channel transmit surface-loop array to simulate the electromagnetic fields of a 7T imaging experiment. Radio frequency (RF) shimming using the Gerchberg-Saxton algorithm and Circularly Polarized shim weights over the entire brain and select slices of each model was simulated. Various SAR metrics and ${\mathrm{B}}_1^{+}$ maps were calculated to demonstrate the contribution of head variation to transmit inhomogeneity and SAR variability.

Results:

With varying head geometries the loading for each transmit loop changes as evidenced by changes in S-parameters. The varying shim conditions and head geometries are shown to affect excitation uniformity, spatial distributions of local SAR, and SAR averaging over different pulse sequences. The Gerchberg-Saxton RF shimming algorithm outperforms circularly polarized shimming for all head models. Peak local SAR within the coil most often occurs nearest the coil on the periphery of the body. Shim conditions vary the spatial distribution of SAR.

Conclusion:

The work gives further support to the need for fast and more subject specific SAR calculations to maintain safety. Local SAR_10g is shown to vary spatially given shim conditions, subject geometry and composition, and position within the coil.

1. Introduction

Ultrahigh field (UHF) MRI provides greater spin polarization [1] and SNR with a cost of higher RF power absorption and more inhomogeneous excitation [2]. Along with shorter wavelengths and higher conductivity, intersubject differences contribute profoundly to transmit inhomogeneity and increased specific absorption rates (SAR) at UHF [3]. Parallel transmission (pTx) and RF shimming are a few ways to lessen inhomogeneity and control SAR. Methods like universal pulses [4], dense transmission arrays [5], and dielectric padding [6], all attempt to improve the versatility and feasibility of >3 T imaging of each patient.

Understanding the contribution of intersubject variability to electric and magnetic (E&M) field inhomogeneity and therefore B₁ and SAR variation is an important step towards creating universal or patient-tailored hardware and pulse design. The magnetic field inhomogeneity results in flip angle non-uniformity whereas the electric field inhomogeneity results in spatially varying local SAR. The current standard for accommodating intersubject variation in local SAR is the use of a vendor dependent safety margin [7-9] which depends on specific models and patient positioning. The safety margin may not account for patients that differ from standard models and potentially over-compensates for tissue heating, thereby limiting pulse sequence performance. Many studies have focused on predicting the appropriate safety margin from values spanning 1.5 to 5 or designing entirely different safety concepts [8-12].

Recently, studies have focused on predicting or monitoring local SAR while the patient is on the table [13,14]. Meliado and coworkers used a large data set of prostate electromagnetic field maps to design a Convolutional Neural Network (CNN) capable of predicting local SAR with ${\mathrm{B}}_1^{+}$ maps as the input [9]. Without a model for prediction, local SAR calculations are on the order of 10s of minutes and require the electric field data to assemble Q-matrices for individual SAR calculations or use the virtual observation point algorithm [15]. To address this, Milshteyn and coworkers designed a workflow to quickly segment subjects in the scanner and solve for electric field maps using a fast E&M solver to calculate local SAR within an average of 8 min [13]. The subjects were segmented into 5 tissue classes with tissue below the shoulders approximated as a uniform rectangular block to allow the electric field to dissipate through the body rather than loop near an abrupt truncation [14].

Accounting for SAR in the scanner is difficult since many factors influence the final electric field distribution in the body including the RF coil, the circuit model, coil-coil coupling, the RF shield, patient position, patient composition, and the channels pulse weights. Beginning with the RF coil, ideal current distributions and coil placement have been investigated in depth by Lattanzi, Deniz, and coworkers [16-18]. Simulating accurate RF coil models complete with decoupling circuits and multiple radiating elements can take 10s of hours to converge. Fast optimization of the S-matrix, usually separate from the simulation space, are typically achieved with state of the art co-simulation methods introduced by Kozlov and Turner [19] and improved on by variants thereof [20-22]. Coil-coil coupling also influences the shape of the E&M field [23] and should be considered in final RF coil safety assessments [20].

With respect to the patient, Kopanoglu and coworkers showed that with one head (Ella of the Virtual Family), slight variations in positioning within a surface loop array can result in up to 5-fold local SAR increases [10,24]. These studies show the need for further investigation of SAR variation due to the relationship between patient-coil coupling. Publications focusing on patient variability say little regarding the essential metric used for benchtop coil design, the scattering parameters. Simulations have the luxury of assuming perfectly tuned and matched circuits, however this might not be the case on the bench or in the bore [20]. Non-ideal tuning and matching conditions due to patient variability are well known in RF coil design [25]. Solutions for this have included real-time circuit control, however these methods are expensive and have a large footprint in the scanner [26,27]. Thus, this work attempts to include the field variations accompanying more realistic matching of surface loops in an array including coupling between elements.

With respect to patient modeling, multi-tissue compartment, high-resolution models most accurately describe the human body. However, SAR and ${\mathrm{B}}_1^{+}$ field measurements may only require a limited resolution [28] (1–3 mm for 7T) and limited number of compartments. Simplified models with only 4 clusters of gray matter, fat, cortical bone, and CSF have been shown to vary <12% from more detailed 47-compartment models [7]. B₁ fields are typically generated across parts of or the whole of the virtual population [29] to design pTx arrays. On this front, de Greef and coworkers simulated strip-line coil elements around 6 models of the virtual population and reported the worst-case SAR scenarios [30]. Work focusing on what makes RF coil elements robust to large patient populations is lacking.

It is possible to simulate hardware across multiple head models with little loss of accuracy in SAR estimation. Although the individual models of the virtual population include over 300 tissue compartments and 0.5 mm isotropic resolution [31], the database is composed of 11 models and may not represent a more general population. It is thus important to understand the extent to which E&M fields can vary in as many subjects as possible to identify characteristics that make coils appropriate for an entire population. Hence, the goal of this study is to show transmission, SAR, and coil efficiency variability across 72 unique, anatomically accurate head models in an 8-channel surface loop transmission coil array to provide an order of magnitude larger database of E&M fields for comparison. It is clearly shown that the S-matrix, typically used to quantify the efficiency of a coil’s tune, match, and decoupling circuits, vary with coil placement and subject head geometry. This is often ignored in assessments of SAR variability. Local SAR values are shown to depend on subject geometry and are more variable than global SAR. A comparison of local SAR values in the head, over a range of tissue compartments is made to correlate SAR with specific anatomical features.

2. Materials and methods

2.1. Coil design and simulation

An 8-channel (8ch.) surface loop design was chosen since it is one of the most common designs for parallel transmission methods at 7T. The coils are wrapped in an approximately elliptical shape with a major axis of 26 cm from anterior to posterior and a minor axis of 25 cm from the patient’s left to right centered around the brain. Each loop is 16.3 × 9.2 cm with a 5 mm trace width (Fig. 1a). The resonance was tuned with eleven distributed capacitors. The matching circuit consisted of 1 series inductor, 1 series and 1 parallel capacitor [32]. To achieve decoupling between neighboring coil elements, 2 capacitors made up the capacitive decoupling circuit [33] and connected neighboring elements near the center of each coil (Supplementary Fig. 1). Each of the 8 coils within the array were tuned, matched, and decoupled from neighboring elements while loaded with the median-sized patient with respect to volume (Fig. 1b). The coil was left at this circuit configuration for all other subjects, to mimic an in-scanner coil. A copper sheet of 3 mm thickness was used as an RF shield, surrounding the coil at a diameter of 30 cm and length of 30 cm with no slotting. FDTD simulations were performed with commercially available XFdtd software (Remcom Inc., State College, Pennsylvania). The meshing of the coil and each head model was 1 mm isotropic with adaptive meshing around circuit elements and the coil to ensure the circuits’ inclusion in the mesh. The domain of the FDTD mesh ended in 7 perfectly matched layers. The time-step based convergence criterion of each simulation was set to −40 dB, equivalent to the fields changing <0.01% at the last time step. Each port simulation was accelerated with 4 Maxwell architecture GPUs (NVIDIA, Santa Clara, CA). On average, each port took ≈ 50 min to converge. The total run time for the simulations was ≈ 20 days.

Fig. 1.

The capacitively decoupled array can be seen in 1a surrounding the median sized subject. The red portion of the body model is the Duke neck and shoulder region composed of muscle. The two capacitors between coil elements make up the capacitive decoupling circuit. Each loop has 11 tuning capacitors with 4 fixed at 20 pF (see supplementary fig. 1). The matching circuit is a series inductor, series capacitor, and parallel capacitor. The S-matrix for the same decoupled model is plotted in 1b. The worst decoupling was S₁₇ with −11.6 dB. The worst input reflection was S₈₈ with −19.7 dB. All other input reflections were < −20 dB. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.2. Scattering parameters

A key metric to indicate a well-tuned, matched, and decoupled transmit element is the scattering matrix. S_mn, where m and n indicate the m^th and n^th coil elements, corresponding to the entries of the scattering matrix and can be calculated with the equation

$$S_{mn}=20^{\star }lo{g}_{10}\mid \frac{V_m^{-}}{V_n^{+}}\mid dB$$ (1)

where ${\mathrm{V}}_{\mathrm{m}}^{-}$ and ${\mathrm{V}}_{\mathrm{n}}^{+}$ are the m^th and n^th port’s reflected and incident voltage amplitude, respectively. For each transmit element, S_mn was quantified for each head. Each element’s S_mn value was optimized for the median sized subject (with respect to volume) in the pool. The surface loop design distributes the capacitance across 11 capacitors in series to increase the individual capacitance of each. This ensures relatively uniform current distribution along the conductor as the copper traces are separated by a capacitor every 1/20th of a wavelength. To determine optimal lumped element values for the tuned, matched, and decoupled array, a co-simulation approach was used [19]. Lumped element values were determined after running 12 lumped elements for each coil as 50-Ω ports and plugging the 96 × 96 scattering matrix into the Agilent/Keysight ADS 11 (Keysight, Santa Rosa, CA) circuit simulator. Initial values of optimal capacitor values were determined from previous experience. The S-matrix goals were S_nn < −30 dB, S_mn < −20 dB. An exception was made for S₁₈ being < −15 dB, since it was assumed coils near the face would suffer from more variation in loading (Fig. 1b). The final lumped element values can be found in supplementary Table 1. Each capacitor was modeled with an equivalent series resistance taken from vendor datasheets and was roughly 0.05 Ω.

2.3. Head models

Seventy-one head models were segmented from T₁-weighted MRI and computed tomography (CT) data. The subjects were deep brain stimulation (DBS) patients. To simulate heads without stimulation electrodes, the segmented labels of the electrode implants were replaced with their nearest neighbor tissue groups. The skull structure was calculated from the CT data, which for each patient included axial images from the top of the head down to approximately the roof of the mouth. For this reason, bone structures (mostly jaw and spinal vertebrae) below the CT data volume were replaced with air. The size of this region varies from patient to patient. MR data were used to segment soft tissue in this region, the shape of the mouth and neck is retained. Following the work of Wolf et al., a shoulder and neck region composed entirely of one tissue (muscle) was taken from the virtual population Duke model in order to avoid SAR inaccuracies due to a truncated head [34]. Muscle was chosen as the shoulder and neck tissue due to its similar properties to other tissues [34-36]. This shoulder and neck region was fixed for each head model so that any transmission or SAR variability subject to subject would originate from the differences in the head itself. It has already been shown that a simplification of the shoulder region’s resolution and tissue compartments has only a small effect on SAR estimates in brain imaging [34]. The head models were segmented into 9 compartments including gray matter, white matter, CSF, cortical bone, fat, skin, eye, thalamus, and air. These tissues were chosen as they follow the work of de Buck et al. in determining the most important tissue labels via k-means clustering [7] which include gray matter, fat, cortical bone, CSF, and air. The head library was segmented but not registered to an atlas to maintain as much subject variability as possible. Since the Duke model’s shoulder and neck region may be different in size or scale to each head in the database, the head model is moved up or down the neck of the Duke model until a reasonable difference between the top slice of the neck and bottom slice of the head appears. A linear interpolation was then applied radially to smooth out the discontinuity between the two slices. The purpose of including this neck and shoulder region is to maintain as close to an anatomically correct body model as possible while still maintaining the subject differences in the head region only. Each head was placed within the coil with the top of the coil level with the top of the head. The mesh of each coil and subject was exported to ensure the coil and model did not overlap in any voxels. Each head was also visually inspected within the simulation software to ensure the coil did not overlap with the patient.

2.4. RF-shimming

Three shim methods were applied to the electric and magnetic field sensitivity maps. Two shims, a traditional 45-degree phase offset between each coil (i.e., circularly polarized if the coil is unloaded) is referred to as CP_ul and a shim where the sum of the phases from each coil is zero at the center voxel of the region of interest (i.e., circularly polarized when the coil is loaded) is referred to CP_l. A non-linear magnitude least squares iterative phase updating algorithm known as the Gerchberg-Saxton algorithm is referred to as GS [37]. These shim conditions were selected because they are incrementally complex. CP_ul fixes the amplitude and phase, independent of the subject, whereas CP_l’s amplitude is fixed but phase is subject dependent, and the GS algorithm’s phase and amplitude are both subject dependent. Each shim was applied and analyzed over two target fields of view, the entire brain (denoted as Brain Volume) and a 4 mm thick transverse slice (denoted as Slice) in the center of the coil excluding non-brain tissue.

2.5. Transmit and SAR metrics

Using the superposition principle, the ${\mathrm{B}}_1^{+}$ and electric field for a given complex shim weight can be defined as

$${\mathit{B}}_1^{+}(r)=\sum _{n=1}^N{w}_n{\mathit{b}}_{\mathit{n}}(r)and\mathit{E}(r)=\sum _{n=1}^N{w}_n{\mathit{e}}_{\mathit{n}}(r)$$ (2)

where r denotes the spatial location and n the coil number (out of N transmit elements). The field maps b_n(r) and e_n(r) represent the complex field of the nth transmit element with unit RF power and all other transmit elements zeroed. The b_n(r) is also known as the sensitivity profile of the nth transmit element. The Coefficient of Variation (CoV) over a select region of interest (ROI) was defined as

$$CoV(\Vert {\mathit{B}}_1^{+}\Vert )={100}^{\star }\frac{std\left(\Vert {\mathit{B}}_1^{+}(ROI)\Vert \right)}{mean\left(\Vert {\mathit{B}}_1^{+}(ROI)\Vert \right)}$$ (3)

where ∣ ∣x∣ ∣ is the magnitude of the complex field x and std. denotes the standard deviation. Following the work of Deniz et al. [38], a transmit efficiency metric denoted as η was calculated and has the units of μT² W⁻¹. SAR metrics were also calculated. The SAR in each voxel is defined as

$$SAR=\frac{\sigma \Vert \mathit{E}{\Vert }^2}{2\rho }$$ (4)

where σ is the conductivity and ρ the tissue density of the voxel. By averaging SAR over global and local volumes, Global SAR (SAR_global), whole head SAR (SAR_Head), and local SAR can be calculated. SAR_global averaged over the entire subject including the Duke shoulder portion whereas SAR_Head truncated the shoulder region below the midpoint of the C5-C6 vertebrae in the Duke model. Local SAR_10g was calculated using the SAR Average tool [39] which uses a spherical masking technique around each voxel of interest to define 10-gram regions. SAR for an arbitrary pulse sequence (SAR_seq) was also calculated following the work of Collins et al. [40]. Defining a hard pulse’s flip angle as

$$\alpha =\gamma \tau \Vert B_1^{+}\Vert$$ (5)

where γ is the gyromagnetic ratio, and τ is the pulse duration. A hard pulse with a field strength of 1 μT and 6 ms duration corresponds to a 90° flip angle. SAR for an arbitrary pulse of flip angle α and duration τ can be defined using the SAR given by a 90-degree hard pulse of 1 ms duration as

$$SA{R}_{\tau ∕\alpha }=f{\left(\frac{1ms}{\tau }\right)}^2{\left(\frac{\alpha }{{90}^{°}}\right)}^{2}SA{R}_{1ms∕{90}^{{}_{°}}}$$ (6)

where f depends on the pulse shape (f = 1 for a hard pulse). The SAR levels of a sequence can then be defined through time as a sum of each pulse’s SAR times its pulse length divided by the total imaging time described mathematically as

$$SA{R}_{seq}=\frac{\sum _{p=1}^P{\tau }_{p}SA{R}_{{\tau }_p∕{\alpha }_p}}{TT}$$ (7)

where p indexes the pulses, τ_p is the p^th pulse’s duration, and TT is the total time of acquisition. This can be used to estimate SAR limits imposed by the FDA for continuous imaging during a 6-min period.

2.6. Pulse sequences

To demonstrate how Average SAR depends on patient, sequence, number and type of pulses, Eq. (7) was evaluated using each subject’s unique SAR_Head with the CP_ul shim weights. Two sequences, a 3D Gradient Echo (3D GRE) and 3D Fast Spin Echo (3D FSE), were considered for comparison, employing commonly used imaging parameters. Each sequence had a 250 × 200 × 200 mm field of view and a total scan time of 6 min. Every pulse is assumed to be a hard pulse and is based on each individual’s SAR_Head calculated for a 2 ms 90-degree hard pulse (Eq. 6). The 3D GRE sequence had TR/TE = 24 ms/ 10 ms, and a 2 ms excitation pulse with a flip angle of roughly 10°. After 14,884 RF pulses within the 6-min sequence, the nominal resolution was 1.64 × 1.64 × 1.64 mm (Fig. 5a). The 3D FSE’s sequence had an echo train length (ETL) of 40, TR/TE = 11,500 ms/ 13 ms, 90-degree excitation pulse with 2 ms duration, and 180° refocusing pulse with 4 ms duration. After 12,874 RF pulses within the 6-min scan, the nominal resolution was 1.8 × 1.8 × 1.8 mm (Fig. 5b). All pulses were assumed to be 180-degree for refocusing, and 90-degrees for excitation except the GRE excitation pulse which was calculated as the Ernst angle given an average T₁ across the head. Given a fixed scan time of 6 min for each FSE and GRE, the SAR_seq (Eq. 7) was calculated for every subject and variable number of pulses.

Fig. 5.

Two conventional MRI sequences are played out with a fixed scan time of 6 min. Each line represents a specific subjects SAR_seq value over the number of pulses within a 6-min scan. The 3D GRE sequence has a much lower SAR_seq since the flip angles are small and pulse durations short (5a). The 3D FSE sequence used 90- and 180-degree excitation and refocusing pules, therefore the SAR_seq is higher (5b). The pulses are assumed to be block pulses where f = 1 in eq. 6. If more elaborate pulses such as SLR or HS pulses f becomes >1.

3. Results

3.1. Scattering parameters

The median-sized subject that the coil was tuned, matched, and decoupled for is denoted as T&M. The S_nn values for the T&M subject for coils 1–8 were −21.8, −23.4, −20.5, −27.2, −27.3, −21.5, −25.9, and — 19.7 dB respectively. For the worst tune, match, and decoupled subject (denoted Worst T&M), calculated as the lowest percent power transmitted summed over all 8 coils, the S_nn values were −9.7, −3.7, −5.8, −7.2, −5.0, −3.6, −6.9, and − 1.3 dB respectively (Fig. 2a). The mean and standard deviation of each S-matrix coordinate was computed over all 72 models (Fig. 2b, c). The worst mean input reflection was coil 8 with S₈₈ = −4.8 dB. The worst mean of the off-diagonal coupling coefficients was S₆₇ = −13.5 dB. The worst standard deviation on the diagonal was coil 2 with 8.1 dB.

Fig. 2.

S_nn was calculated to measure the response of each channel to variations in the loading of the head model in 2a. The Duke model is included in the dataset as the 72nd head model. Each patient is color coded as a circle except for the legend entries. The range of the S_nn values demonstrates subject to subject variability purely based on the impedance of the load from varying tissue composition and positioning. In 2b and c it is clear the diagonal of the S-matrix suffers from changing loading conditions more than the off-diagonal entries.

3.2. RF-shim methods

With respect to slice-wise shimming (Fig. 3), the GS shim weights achieved the best solution for field homogeneity with an average field homogeneity of 14.2% across the 72 models. This is to be expected as the solution has more degrees of freedom than the simple phase assignments of CP_ul and CP_l. Two subjects with coil 8 achieving an S_nn value > −1.2 dB, each reflecting >75% of the incident power, indicate the utility of the GS algorithm by achieving a ≈ 10% CoV(∣∣${\mathrm{B}}_1^{+}$∣∣) in the slice with coil 8 nulled in each respective case. With these same two patients, CP_ul and CP_l achieve >20% with comparable peak Local SAR_10g. GS also had a lower average SAR_Head with 0.22 W kg⁻¹ vs the CP_ul and CP_l cases 0.26 and 0.24 W kg⁻¹, respectively. Local SAR_10g on average was lowest for CP_ul with 3.3 versus 3.79 and 5.3 W kg⁻¹ for the GS and CP₁ cases. The 10W kg⁻¹ IEC limit was violated by 3, 0, and 4 subjects for GS, CP_ul, and CP_l respectively. CP_l has a CoV(∣∣${\mathrm{B}}_1^{+}$∣∣) spanning a range of 56%, wherease GS and CP_ul have ranges 14 and 19.7% indicating that for slice-based shimming of an axial slice near the center of the coil, traditional circularly polarized shimming is an excellent first order approach to achieve higher transmit efficiency with lower SAR_Head and Local SAR_10g. When RF shimming over the entire brain volume (Fig. 5), the GS algorithm outperforms in transmit homogeneity at the cost of higher SAR_Head and Local SAR_10g. GS’s average CoV(∣∣${\mathrm{B}}_1^{+}$∣∣) across the population is 28.2% whereas the CP_ul has 37.5%. The 51-tissue compartment Duke model is shown as the gold diamond in Fig. 3 and Fig. 4 and was plotted as a reference model for each metric.

Fig. 3.

Three RF shimming methods were applied to the pTx array’s field maps across all 72 head models. The field of excitation consisted of a 4 mm axial slice in the center of the RF coil. The non-linear GS algorithm outperformed each CP shim condition (unloaded and loaded circularly polarized) in field homogeneity, the algorithm was run without regard for SAR limits. The GS algorithm had the lowest mean SAR_Head compared to the CP shims. Local SAR_10g increased for 46 of the 72 models when GS shims were applied versus the traditionally circularly polarized weights. 3 subjects exceeded the 10 W kg⁻¹ IEC limit with the GS shims compared to 0 and 4 subjects for the CP cases. Transmit efficiency for the GS algorithm was lower on average. 51 of the 72 subjects had lower transmit efficiency with the GS shim versus traditionally circularly polarized CP_ul condition.

Fig. 4.

On the brain volume, GS achieves more optimal transmit fields but suffers higher local and global SAR due to the size of the FOV. More elaborate shimming algorithms may be needed for optimal homogeneity across the entire brain volume.

3.3. Global SAR variability

Applying eqs. 6 and 7, Average SAR values over a 6-min sequence can be estimated across the population. The SAR_seq will depend on the number of pulses, flip angles, pulse lengths, and pulse types. These can vary for a given sequence and certain subjects may reach tissue heating limits before others. Fig. 5 shows the results of applying eq. 7 to 3D FSE and 3D GRE. Each line in Fig. 5 represents a subject from the 72 models. The x-axis in each plot represents the number of hard RF pulses, which increases with the number of TRs. Each point on the line represents some number of pulses within a 6-min scan. The final number of pulses represents the ≈ 2 × 2 × 2 mm 3D resolution. Higher flip angle pulse sequences like the FSE sequence are at risk of exceeding the 3.2 W kg⁻¹ limit for SAR_Head especially when longer excitation and more elaborate RF pulses like Shinnar-Le Roux (SLR) or Hyperbolic Secant (HS) shapes are used. These results could also be applied to SAR_global, but the SAR per pulse is lower since the global average includes the shoulder region with potentially lower RMS electric field values.

3.4. Local SAR_10g variability

Local SAR_10g has been reported to be the limiting safety metric in many studies [41-43], thus this study focuses most on local SAR_10g spatial and tissue variability. Fig. 6 shows the relationship between peak spatial SAR_10g and the nearest coil to this point’s position. Nearest-coil distance is defined as the Euclidean distance between the local SAR_10g hotspot (maximum), and the nearest voxel labelled as the RF coil. Most of the higher value hot spots occur within 20 mm of the coil. Transmission coils may need to be kept farther than 20 mm from the subject. In Fig. 7, the coil can be seen with the Duke model as a reference along with all 216 peak SAR_10g spatial locations (72 models * 3 shim scenarios). Most of the peaks are at the points of the head that are closest to the coil at the front or back of the head, save some points within the head. Also, notable are the peak SAR_10g locations near the capacitive decoupling circuits between the coils at the back of the head or coil numbers 3, 4, 5 and 6. Fig. 8 demonstrates agreement with previous work [41] that skin is most likely to be the highest in local SAR, given different shim scenarios with this specific coil. In Fig. 9, the highest one thousand local SAR_10g values are identified by tissue class and plotted for every model and slice shim scenario. The results indicate there may be changes in tissue heating distribution due to different shim algorithms. The spatial distributions of local SAR due to varying the shim methods is shown in Fig. 10 and demonstrates clearly that shim conditions can influence the distribution of local SAR. The tune and match (T&M), Duke, and Worst Case SAR_10g are plotted over all 3 shimming conditions for the center slice. The Duke model demonstrates that higher local SAR values tend to congregate on the periphery of the head. Supplementary Fig. 2 provides insight into peak local SAR_10g correlation with overall coil performance metrics. The 8ch. total power was calculated by taking the average of each coils percent power transmission calculated from the S_nn values.

Fig. 6.

The peak local SAR plotted against the minimum distance from this local SAR_10g point and the nearest point on the 8ch. coil. The nearest coil distance is defined as the Euclidean distance between the hotspot and the nearest coil labelled voxel in the cartesian simulation space. Tissue within 20 mm of a coil’s trace have a higher risk of exceeding SAR constraints.

Fig. 7.

With the Duke model as a reference, peak spatial local SAR_10g points for each head subject are plotted for each shim condition. Most of the points occur nearest the RF coil decoupling circuits at the anterior and posterior of the head. The coil is also included as a reference. The copper RF shield is not included in the plot. Special care must be taken in coil design to ensure extremities of the face such as the nose and ears are not exposed to higher local SAR.

Fig. 8.

For each shim condition, most of the peak spatial local SAR_10g points occur in the skin tissue compartment. It is therefore important in model segmentation to represent the skin accurately as these tissues are more likely to experience higher local SAR values.

Fig. 9.

For each subject and shim condition, the hottest 1000 voxels’ tissue and local SAR_10g values were recorded. Across tissue distributions, the skin tissue voxels are the hottest SAR_10g voxels for each shim condition.

Fig. 10.

Three specific subjects’ local SAR_10g spatial distributions are plotted with respect to the 10 W kg⁻¹ IEC limit. The T&M case is the head model with tuned, matched, and decoupled circuit configurations for every coil. The Duke model is the most anatomically accurate model with 51 tissue compartments.

4. Discussion

In this study, a decoupled pTx array’s (Fig. 1) electric and magnetic fields were simulated using 72 head models to demonstrate RF shim and SAR variations due to intersubject variability. The S-parameters for each coil demonstrate the loading variation due to each subject’s tissue composition and position within the coil (Fig. 2). Notably, the diagonal of the S-matrix changed the most due to varying loading conditions (Fig. 2c). The capacitive decoupling circuits were more robust to changes in the load than the matching circuit. The work demonstrates the effect of designing surface loops with only 1 loading condition in mind. RF coils robust to varying loading conditions or self-matched coils [25] will ensure ideal power transfer no matter the subject. Simulation work focusing on uniform models and optimum distance for efficiency has already been done [17] but the human head is far from uniform and population heterogeneity may lead to very different scattering parameters. Arrays should consider as many human models as possible as has been done with gradient coils [44,45]. Hybridized approaches for coil design including transmit homogeneity or SAR variation across many models could make RF coils more universal. Similar to this idea, Li and coworkers implemented a hybrid circuit and ${\mathrm{B}}_1^{+}$ spatial field cost function optimizing the circuit using co-simulation while accounting for the magnetic field homogeneity across 4 models [46]. Hybridizing the co-simulation method to include other domains such as SAR variation and intersubject differences will further improve UHF coil design. It is unknown whether other decoupling strategies, such overlap or self-decoupled transmit arrays will be as robust to changes in the load as the capacitive decoupling circuits used here.

With respect to hardware, open questions remain considering the dependence of other coil designs such as dipole and loop-dipole combinations on intersubject differences [16,47]. Recently, dipole based transceivers have been rising in popularity due to their simplicity and comparable SNR, transmit efficiency, and SAR management capabilities [48,49]. In light of this, the first images of the human brain at 10.5 T were acquired safely with fractionated and bumped dipole elements [50]. Further work considering the interaction between other coil designs and intersubject differences would benefit UHF safety at large. Finally, the RF shield is known to affect SNR at UHF [51] and its interaction between subjects and SAR variation was not investigated here.

Optimizing the transmit field over the 4 mm axial slice, the utility of the GS algorithm is demonstrated across many subjects (Fig. 3). The optimization’s cost function was only regularized by the weights of each shim squared which is proportional to the power delivered to each coil port. GS had the lowest CoV(∣∣${\mathrm{B}}_1^{+}$∣∣) with only 3 models that violated the Local SAR_10g 10 W kg⁻¹ limit. This is a concrete example of RF shimming outperforming conventional CP in transmit homogeneity, with comparable global and local SAR over a population. GS initialization phase was that of the CP_l shim weights and greater optimization may be reached if the phase was initialized to the CP_ul state. Nevertheless, for whole brain RF shimming (Fig. 4), the CP_ul performance is impressive given the size of the ROI and demonstrates the strength and utility of traditional CP shimming for 8ch. systems. Future RF shimming work over multiple head models could use other regularization terms including explicit SAR measurements [37,52].

With CP shim weights and a given pulse sequence, disparity of SAR over the head increases as the number of pulses within a 6-min window increases (Fig. 5). This implies that in pulse-heavy sequences, certain patients will exceed the 3.2 W kg⁻¹ average SAR within a 6-min window before others. However, the slope of the line is close to constant while average SAR metrics can be monitored on the table by pickup coils [53] or equivalent circuit models [54]. There is little to no correlation between transmit efficiency with the reflected power at the port (Supplementary Fig. 2). This is simply due to the electric and magnetic fields being proportional to the square root of the port power. Variation due to the models may be more important or impactful to transmit metrics than the port’s power efficiency within a certain range of S_nn values since a subject with 68.7% total power efficiency has lower CoV(∣∣${\mathrm{B}}_1^{+}$∣∣) than the T&M model with 99% total power when the two slices have mean target field in the FOV of 0.98 and 0.99 μT respectively.

In most cases transmit efficiency may represent the limiting factor of peak spatial Local SAR_10g (Supplementary Fig. 2). Transmit efficiency is calculated with the same Q-matrix given to local SAR algorithms without the time consuming 10 g averaging techniques (order of minutes). If Q-matrices become available within the bore, flagging lower transmit efficiency coil and subject combinations at risk for exceeding local SAR may aid in SAR management. It is also clear that ideal repositioning of the patient may eliminate SAR risk altogether [10]. The tendency of local SAR to be concentrated at the edges of the sample and decrease nearly exponentially with distance from the coil has been discussed in the case of uniform spheres [17]. Given regions most susceptible to higher local SAR_10g an optimal distance for transmit only coils may exist. The range of S_nn values over 72 models raise questions concerning T/R coils near the head (e.g., how to balance SAR with SNR) and decoupling performance given varying head geometries. It also indicates special attention should be paid to the face region of each subject, to ensure array designs are robust to differing head geometries and nose sizes.

Not included in this work is the effect of intersubject differences on temperature changes in the head. The Pennes bioheat equation requires SAR, specific heat capacity of tissue, tissue density, blood perfusion rate, and tissue thermal conductivity [2]. Common body models use fixed estimates of these parameters however they may vary between patients. Including these variables in the head models would affect SAR and temperature estimations. For example, Wang et al. showed as temperature rates exceeded 39° C with SAR above 9 W/Kg, the perfusion rates also increased, thus removing heat from surrounding regions at greater rates [55].

5. Conclusion

In this work, the change in resistive load from differing patients’ composition and position in the coil is shown to be significant enough to affect an array’s S-matrix elements when tuning over a single head model. When designing an RF array, multiple loading conditions should be included to maintain acceptable S-matrix values across a population. For slice shimming, it is shown over a population that non-linear optimization schemes such as the Gerchberg-Saxton algorithm provide better transmit homogeneity with comparable global and local SAR. It is shown that average SAR in a window of time is dependent on the number of pulses and the subject, possibly limiting pulse-heavy sequences. RF coil designs with traces <2 cm away from the subject are at a greater risk of violating local SAR constraints. Similarly, it is also confirmed that local SAR hotspots most often occur on the periphery of the subjects, with skin representing the most and highest-value local SAR hotspots.