The efficient computation of the RF field is crucial for 3D dielectric pad design.^1,2 Such computations are usually carried out by FDTD-based solvers. This approach offers great generality and allows for the inclusion of the RF coil, shield and heterogeneous body model, but simulations may be very time-intensive. Alternatively, Maxwell’s equations in integral form can be used to model the RF field.³ An advantage of this approach is that the problem size is much smaller than in FDTD, since it is essentially determined by the size of the dielectric pad. However, structures such as the RF coil and shield are hard to include into such a model, and large variations in permittivity are difficult to model accurately. We therefore propose a hybrid method, which exploits the advantages of both approaches, i.e. the flexibility of FDTD is combined with the localized nature of the integral approach. The resulting hybrid method is accurate, fast, and enables us to efficiently design 3D dielectric pads.

Since a pad forms a small perturbation of a given background configuration (coil/shield/body), we use the integral approach to setup a scattering formalism. Specifically, with the help of the so-called Sherman-Morrison-Woodbury (SMW) formula,⁴ the electric field in the total configuration can be written as

$$\textbf{E}^\text{tot}=\textbf{E}^\text{back}+\textbf{Z}\left(\textbf{I}-\textbf{V}^\text{T}\textbf{Z}\right)^{-1}\textbf{V}^\text{T}\textbf{E}^\text{back}$$

where $$$\textbf{E}^\text{back}$$$ represents the field in the background configuration, while the second term represents the scattered field due to the pad. As soon as the electric field is found, the magnetic field follows from Maxwell’s equations.

The library matrix Z contains the Green’s functions and is constructed using an FDTD solver (XFdtd, Remcom Inc.) by computing the field response for point sources at possible pad locations in the background configuration. Matrix Z is background-dependent and does not specify any of the dielectric properties of the pad, nor its location. It has to be constructed only once and can be computed offline. During online pad design, only the pad-matrix V changes and computing the action of the inverse in the above formula can be carried out very rapidly, since its size is equal to the number of voxels occupied by the pad. This number is obviously much smaller than the total number of voxels in the computational domain, and therefore the corresponding RF field can be computed very efficiently.

To evaluate the proposed method, matrix Z was created for a human head model (Duke, IT’IS foundation) placed in a 7T quadrature birdcage coil. The offline construction of this matrix typically takes several hours with conventional FDTD software. Subsequently, a dielectric pad (18x18x1 cm³, ε_r=285, σ=0.25 S/m) was placed on the side of the head and the resulting RF field was computed using the new method (Figure 1) and FDTD. A speed-up factor of ~35 was achieved for this problem (8.5 vs. 300 seconds), while even larger factors of ~50 were obtained for smaller pads. The simulated B₁⁺ maps are shown in Figure 2 along with a measured B₁⁺ map obtained in vivo using a DREAM B₁⁺ mapping sequence (2.5 mm² resolution, 5 mm slice thickness, STEAM/imaging tip angle = 50°/10°).⁵ The measured and simulated B₁⁺ maps are in excellent agreement with each other. The secondary field induced by the dielectric pad is clearly visible as well. The hybrid method proposed in this abstract provides an effective platform for efficient pad design. By exploiting the advantages of FDTD and an integral approach in an off- and online part, respectively, pad parameter sweeps can be carried out very efficiently to determine its appropriate geometry and constitution. In the first part of our method, a background library matrix is constructed, which can be computationally intensive and, depending on the particular background configuration, may take several hours to complete. Fortunately, these computations can be carried out offline and speed up factors of ~35 are achieved in subsequent simulations during the online part of the method. Given that the thickness, width, length, and constitution of the pad need to be optimized (leading to hundreds of combinations), the proposed method is much more appropriate than standard FDTD approaches. Other potential applications include the evaluation of, for example, small implanted medical devices and the modeling of SAR effects. The proposed hybrid method is able to efficiently compute the effect of dielectric shims in configurations where the coil, shield and body are taken into account. The method is fast, accurate, and agrees with experiments, which makes it a key building block for systematic 3D pad design.