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Ideal current patterns for optimal SNR in realistic heterogeneous head models1Center for Computational and Data-Intensive Science and Engineering (CDISE), Skolkovo Institute of Science and Technology, Moscow, Russian Federation, 2Q Bio, Redwood City, CA, United States, 3Center for Advanced Imaging Innovation and Research (CAI2R), Department of Radiology, New York University School of Medicine, New York, NY, United States, 4Bernard and Irene Schwartz Center for Biomedical Imaging (CBI), Department of Radiology, New York University School of Medicine, New York, NY, United States, 5Sackler Institute of Graduate Biomedical Sciences, New York University School of Medicine, New York, NY, United States
Synopsis
We describe a method to calculate, for the first time, ideal current patterns (ICP) associated with optimal signal-to-noise ratio (SNR) in heterogeneous head models for arbitrary current-bearing substrates. We show ICP for different voxel positions in the brain and neck, magnetic field strengths, and realistic head coil substrates. Even though the optimal SNR distribution is similar for different substrates, we show that the ICP depend instead on the topology of the substrate on which they are restricted. ICP can inform non-convex radiofrequency (RF) coil optimization problems by providing an intuitive initial guess and could lead to task-optimal RF arrays designs.
Introduction
Ideal current patterns (ICP) were initially proposed as the surface current distributions that result in the highest possible signal-to-noise ratio (SNR) or the lowest possible radiofrequency (RF) power deposition consistent with electrodynamic principles1. More recently, ICP have been defined for the ultimate transmit efficiency2. ICP can provide physical insight into the design of optimal coils that approach the ultimate performance3. Theoretical coil performance bounds have been described for simplified sample geometries2,4,5 (uniform spheres or cylinders) and for heterogeneous head models6-8. Until now, the ICP associated with ultimate SNR in the head have been calculated only for simple substrate geometries9. In this work, we show how to calculate ICP associated with optimal performance inside a heterogeneous head model for arbitrarily shaped current-bearing surfaces (or coil substrates).Theory and Methods
The first step to calculate ICP is to generate a numerical basis-set of electromagnetic (EM) fields inside the sample. Our approach is similar to that used for ultimate SNR and SAR amplification factor in a realistic head model7,8. The key difference is that we employ ideal Hertzian dipoles as electric current sources, instead of random, voxelized electric and magnetic dipoles. In fact, it has been shown that electric current sources alone (or only magnetic currents), distributed over a closed surface, are sufficient to represent all possible EM fields inside the surface10. We start by projecting the Hertzian dipoles onto a surface of interest. The surface can be meshed with a triangular grid, and as the dipole cloud becomes denser, the elementary sources approach a continuous current distribution. Then, we generate the coupling matrix from dipoles to EM fields in the head voxels, from the dyadic free-space Green’s functions and apply singular value decomposition (SVD) to generate an incident EM fields basis. For each basis element we compute the corresponding EM field inside the head model using the MARIE ultra-fast volume integral solver11. The next step is to calculate the optimal SNR by treating each basis EM field as if generated by a coil of a hypothetical array. The corresponding weighting coefficients1 allow to appropriately combine the Hertzian dipoles to obtain ICP resulting in optimal SNR. Since the operator that maps dipole currents to incident fields is compact12, SNR converges to the optimal value as the number of basis elements increases. To validate our numerical approach, we calculated ICP yielding ultimate SNR at the center of homogeneous ($$$\epsilon_r=41.2$$$ and $$$\sigma=0.54S/m$$$ at 7T) spherical (radius=10cm) and cylindrical (radius=9cm, length=40cm) samples, placing the current-bearing surface 3cm away from the sample, and compared the results with those obtained from analytical calculations1. Then, we calculated ICP for voxels at the center of the brain, near the brain cortex (surface) and in the neck region, using three realistic coil substrates (Fig.1), loaded with the Duke6 head model at 1.5T, 7T, and 10.5T.Results
Fig.1 shows that the proposed numerical framework (SVD) is in agreement with the analytical approach (DGF). Fig.2a shows the three modeled coil substrates, whereas Fig.2b shows the voxels of interest. Fig.3 and Fig.4 show ICP yielding optimal SNR near the center and surface of the brain, respectively, for a cylindrical and a helmet substrate, at 1.5T, 7T, and 10.5T. In both cases, ICP depend on the topology of the substrate and tend to concentrate around the target voxel. As the field strength increases, ICP become more complex, especially for the cylindrical substrate, where electric dipole patterns begin to emerge. A similar behavior is shown in Fig.5, which illustrates ICP yielding optimal SNR at a voxel in the neck, for the head-and-neck and helmet substrates.Discussion and Conclusions
Even though the ultimate SNR is similar in a uniform sphere and a realistic head model7, and only slightly varies when calculated for the same sample using different substrates (it would be identical for completely enclosing substrates), the corresponding ICP are different. In fact, they depend mostly on the topology of the surface on which they are constrained13. For example, using the same cylindrical substrate, ICP for a voxel at the center of a realistic head resemble those for a voxel at the center of a uniform cylinder. This work confirms the emergence of electric dipole behaviors at ultra-high field strength, especially for cylindrical substrates, and the dominance of loops at lower field and in the case of substrates with spherical symmetry. In addition to helping to visually understand the origin of optimal RF coil performance, ICP could also help designing the next-generation MR coils, by providing an intuitive initial guess for non-convex coil optimization algorithms, leading to truly task-optimal design of RF coils.Acknowledgements
This work was supported in part by NIH R01 EB024536, NIH U01 EB025144 and NSF 1453675, and it was performed under the rubric of the Center for Advanced Imaging Innovation and Research (CAI2R, www.cai2r.net), a NIBIB Biomedical Technology Resource Center (NIH P41 EB017183).References
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Figures
Figure 1: Validation of the proposed framework by comparison of the numerical (SVD) and analytical (DGF) ICP yielding ultimate SNR at the center of a homogeneous ($$$\epsilon_r=41.2$$$ and $$$\sigma=0.54S/m$$$ at 7T) spherical (10cm radius) and cylindrical (9cm radius, 40cm length) sample. The real part of the ICP is plotted at $$$\omega t=\pi/2$$$ at the current-bearing spherical and cylindrical shell placed at 3cm distance from the sample for both cases.
