An investigation into the minimum number of tissue groups required for 7T in-silico parallel transmit electromagnetic safety simulations in the human head

Purpose: Safety limits for the permitted Specific Absorption Rate (SAR) place restrictions on pulse sequence design, especially at ultra-high fields ($\geq 7$ tesla). Due to inter-subject variability, the SAR is usually conservatively estimated based on standard human models that include an applied safety margin to ensure safe operation. One approach to reducing the restrictions is to create more accurate subject-specific models from their segmented MR images. This study uses electromagnetic simulations to investigate the minimum number of tissue groups required to accurately determine SAR in the human head. Methods: Tissue types from a fully characterized electromagnetic human model with 47 tissue types in the head and neck region were grouped into different tissue clusters based on the conductivities, permittivities, and mass densities of the tissues. Electromagnetic simulations of the head model inside a parallel transmit (pTx) head coil at 7T were used to determine the minimum number of required tissue clusters to accurately determine the subject-specific SAR. The identified tissue clusters were then evaluated using two additional well-characterized electromagnetic human models. Results: A minimum of 4 clusters plus air was found to be required for accurate SAR estimation. These tissue clusters are centered around gray matter, fat, cortical bone, and cerebrospinal fluid. For all three simulated models the pTx maximum 10gSAR was consistently determined to within an error of<12% relative to the full 47-tissue model. Conclusion: A minimum of 4 clusters plus air are required to produce accurate personalized SAR simulations of the human head when using pTx at 7T.


Introduction
The energy deposition in human tissue permitted from radiofrequency pulses in MRI is limited by the energy deposition per unit mass, also known as the specific absorption rate or SAR. There are limits on both local SAR (averaged using 10g averaging volumes) and global SAR (1). Since SAR scales roughly with the square of the external magnetic field strength, accurate assessment of SAR is especially important for ultra-high field (≥7T) MRI-scanners. Due to individual differences in composition and morphometry of human anatomy, SAR varies across individuals for any given pulse. Additionally, local SAR cannot directly be measured in a clinical setting because of various technical and practical complexities.
Using computational human body models, it is possible to estimate SAR using electromagnetic simulations, such as with a finite-difference time-domain (FDTD) method (2,3). Those simulations determine, amongst other parameters, the electric field E(r) at each spatial location r in the model. This can be used to calculate the SAR at each location through: where V is the size of the averaging volume and ( ) and ( ) are the conductivity and density of the tissue at location r, respectively.
These computational human body models are typically not available on an individual subject basis.
Therefore, generic computational models have been used to determine the expected SAR, and then a further safety margin has been applied. Even when only considering a homogeneous group of adults, a safety margin of 1.5 is needed to correct for inter-subject variability with a chance of less than 1% of exceeding the calculated SAR in a pTx mode (4).
Parallel transmit MRI (5,6) has been shown to overcome B1-inhomogeneities that are present at high field, and to reduce SAR (7). Although pTx enables spatial field manipulation, it can also create SAR 'hot-spots' by focusing the electric fields in an undesirable manner (6). This means that accurate subject-specific SAR is of particular interest when operating with pTx.
Previous work (8) studied the improvement of SAR simulations for ultra-high field pTx by non-linearly warping a standard electromagnetic model to match the anatomy of other individual subjects. That work demonstrated that morphometry alone is insufficient in determining accurate personalized SAR, but rather that personalized tissue composition must additionally be addressed.
Segmenting subject-specific MRI data has also been used to generate simplified anatomical models.
Two examples which have been studied previously are models consisting of fat, lung, and water for whole-body models at 3T (9), and fat, muscle, and skin images for prostate at 7T (10). In the head, previous work studied simulation results for several combinations of clusters based on anatomical proximity and similarity, but did not find cluster combinations that resulted in a stable estimation of SAR hotspots (11). More recent work has segmented several combinations of tissues with similar tissue properties in multiple subjects, and evaluated the resulting SAR for only a single B1-shim at 3T (12).
In this work we use a well-defined human electromagnetic model to evaluate the minimum number of tissue clusters needed to accurately estimate SAR for pTx in the head at 7T (13). The accuracy of SAR estimation using the identified tissue groups is then evaluated on two additional models with different ages and gender.

Methods
In this section we describe how the electromagnetic simulations were carried out and evaluated, how tissue clustering was achieved, and how an identified set of clusters was evaluated on different virtual human models. Firstly, electromagnetic simulations were carried out for 7T (298 MHz) using models from the Virtual Population (IT'IS Foundation, Switzerland) (14) provided as part of the electromagnetic simulation package Sim4Life (Zurich MedTech, Switzerland). Simulations were performed initially for the original Duke model before additional simulations with altered tissue properties were done. Below, the general simulation and analysis pipeline is described, followed by a discussion of the modifications which were made to the model in later simulations. For all models, a portion of the shoulders was included to reduce the simulation time while avoiding problems at the boundary of the model (11,15). An optimized 8-channel pTx-coil (outer radius 146 mm, 5 tuning capacitors and 1 matching capacitor per channel, maximum coupling -11.3dB, no RF-shield) was also modelled. Multi-channel simulations were run by simulating each channel separately, with all other channels loaded with a 50 Ω load. The simulation setup with Duke in the pTx-coil is shown in Figures   1a-b. A non-uniform grid with an average grid resolution of (2.135 mm) 3 and a maximum grid resolution of (3 mm) 3 was used. All simulations were terminated after a convergence level of -30 dB was reached (as in (11)), which typically resulted in a simulated time of approximately 160 ns (or nearly 50 periods at 298 MHz). The simulations were run on a system using an Intel Xeon CPU E5-2680 (v4), running at 2.40 GHz with 14 cores and 28 logical processors. Running a single simulation using the voxelization and convergence described above typically took about six hours. The 10g averaged SAR for 64 different pTx configurations was calculated in Sim4Life using the IEEE/IEC 62704-1 standard (16). These 10gSAR maps were used to calculate an 8×8-element Q-matrix (17) for every voxel in the model using the approach described by Beqiri et al. (18). Using the Q-matrices, each voxel's maximum 10gSAR (for 1 W total input power) was determined through eigen-decomposition of the Q-matrices (19,20). Using this, the worst-case 10gSAR can be defined as the highest maximum 10gSAR for all voxels in the model for all possible B1-shims. The maximum 10gSAR for 500 random B1 shims was also calculated, normalized to 1 W total input power per shim. Those 500 shims were generated by setting a random value for the power and phase for each channel, after which the powers were multiplied by a normalization factor such that the total input power was 1 W. Circular polarization (CP)-mode was included as an additional B1-shim. For a given B1-shim, i, the SAR corresponding to the Q-matrix of a voxel at location r is given by where the dagger ( †) denotes the complex conjugate.
A list of 47 tissues from the head region was extracted from the Virtual Population v3.0 (ViP, IT'IS Foundation, Switzerland) (14). Amongst these 47 tissue types there were several with identical dielectric properties leading to only 41 unique tissue types. Tissues were grouped using k-means (21) clustering, implemented using the kMeans function in the scikit-learn Python package (22). The kmeans algorithm groups n vectors, using Euclidian distance, into k (with k ≤ n) clusters. For k = 1 to 6, clusters were identified using tissue conductivity and permittivity alone, setting the mass density to a fixed value of 1 g/cm 3 (as used in previous studies (9)), and then again with the additional inclusion of density (which is important when calculating SAR using Eqn 1). During clustering the n tissue types were weighted by their volumes in the original model. An example of k-means clustering with k=5 is shown in Supporting Information Figure S1. For comparison, a 41-cluster model was also evaluated, representing the full model. An overview of all the n original tissue types can be found in Supporting Information Table S1.
Using the Duke model, simulations were set up for each tissue clustering method. Tissues within each cluster were assigned identical conductivity, permittivity, and density. The values were set to the corresponding value of the centroid of that cluster (i.e. the volume-weighted mean of the tissue properties in the cluster) in accordance with the results of the k-means clustering.
Using the identified minimum number of tissue clusters, the properties of 'real' tissues close to the cluster centroids were then identified and used to define tissue properties for that cluster. These tissue

Discussion
When forming simplified tissue models, it was found that including mass density was important for the resultant model and for use as a clustering parameter. A tissue model simplified using a weighted 3D k-means clustering algorithm produces accurate 10gSAR estimates for k=5 clusters in the head region at 7T. While using fewer clusters may make segmentation easier in a clinical setting, it results in increased errors in both the simulated 10gSAR for specific shims and the simulated worst-case 10gSAR.
Using k=5 clusters is both tractable to segment from in-vivo data and produces only small errors in 10gSAR estimation. The resulting 10gSAR-calculations, shown in Error! Reference source not found. and 5, exhibit a high degree of agreement with the ground truth. This was found to be consistent for three human body models despite strongly different anatomies due to their size, weight, age, and gender. When expressed as a percentage of the shim-wise peak-values, the absolute errors are below 12% for over 99% of the shims for all models. Based on the results in Error! Reference source not found., the remaining 10gSAR-errors generally seem to be over-estimations, which correspond to conservative SAR-estimations. The simulated errors are consistently much lower than the 50% uncertainty margin which is required for a probability of less than 1% of exceeding the actual value when using generic models. Also, the 50% margin is only sufficient when determining SAR for subjects of the same ethnicity as the generic model used for the simulation (e.g. if both model and subject are from the adult Caucasian population) (4). SAR simulation approaches using clustered segmentation, however, seem to offer consistent results for subjects of different genders and for both adults and children and may also be suitable for subjects with non-standard anatomies. The results also appear to be consistent when simulations are run with the model in a different position in the coil and when the resolution of the voxelization is changed. Note that all results in this study are based on static B1-shims -further analysis is required for more complex dynamic pTx-pulses.
The simulations were all carried out using the same 8-channel pTx coil. This coil was tuned in silico and constructed to minimize the influence of its design on the simulations, for example by using physically separated elements to minimize coupling between neighboring elements. The segmentation method has not yet been tested using coils with different designs.
All errors in the presented SAR simulations are based on models with perfect clustered segmentation.
Generating practical clustered models for individual subjects would require experimental determination of the proposed subject-specific tissue clusters. Based on that segmentation, SAR simulations could then be conducted for individual subjects. The exact duration of SAR simulations depends on simulation parameters and computational capacity, but the computation time to date is not shorter than the duration of a typical MRI scan session. Therefore, in practice, the scan or scans which determine the clustered segmentation would likely have to be performed in a separate session, and therefore not necessarily at 7T, provided the resulting model can be correctly positioned in the pTx coil.
Recent work (12) used an automated computer-vision based approach that segmented individual subjects into models consisting of air, bone, fat, and soft tissues for SAR simulation. An extension of this approach, with CSF segmented as an additional cluster, would be interesting for the '4 clusters plus air' segmentation proposed here. However, the general lack of ground-truth information makes experimental validation challenging. Alternative automated segmentation approaches could make use of quantitative mapping of tissue properties such as the dielectric properties or the relaxation times (24,25). In practice this would require high-accuracy quantitative maps of the whole head, for which the separation of bone and air is likely to prove difficult. Therefore, the identification of bone may have to be performed using a separate method, such as using ultra-short echo time scans (26) or a combination of T1-and T2-weighted images (27). Bone-air segmentation can be improved using additional post-processing steps, such as knowledge-based approaches (28) and artificial intelligencebased methods trained from CT (29), although such a network is not currently available for segmentation of the (upper part of the) skull at 3 tesla.

Conclusion
We found that a minimum number of 4 clusters plus air is required to generate personalized SAR models for the human head-region at 7T. A specific clustering approach is proposed whereby clusters are segmented to gray matter, fat, cortical bone, CSF, and air. This clustering resulted in errors in the simulated SAR-distributions and peak-10gSAR values which are much smaller than the errors due to inter-subject variability when using generic models. The peak-10gSAR could be determined with an error of less than 12% for models with different genders, age, and positioning in the scanner.
In order to be able to use this new approach in a clinical setting, an approach for the automated segmentation of the clusters in individual subjects, is still required. With that in place the newly proposed segmentation method could improve the estimation of subject-specific SAR, making it possible to operate 7T MR scanners closer to the true SAR-limits in a clinical setting.