Linear Current Elements Research Articles

Biot–Savart-Based Design and Workbench Validation at 100 MHz of Transverse Field Surface RF Coils

Radiofrequency (RF) surface coils are extensively used as receivers in magnetic resonance imaging (MRI) and magnetic resonance spectroscopy (MRS) systems thanks to their high signal-to-noise ratio (SNR). For specific magnetic resonance applications, the design of dedicated RF surface coils with a transverse (to the coil’s plane) RF magnetic field pattern can be necessary. Such transverse-field RF coils are constituted by several central linear (parallel or crossing) conductor elements connected by return current paths. Typically, the outer shape of such RF coils is circular or squared, although other geometries can be used. This paper describes the implementation and validation of a transverse-field RF surface coil simulator based on magnetostatic analysis, which permits the design and optimization of square butterfly and figure-of-eight RF coils with adjustable size and mutual distance between the central linear current elements. The simulation results, compared with the ones provided by a standard square loop RF coil, were validated with 100 MHz workbench measurements performed on three home-built prototypes. Finally, two novel quadrature RF coil structures designed by overlapping two orthogonal square butterfly and figure-of-eight RF coils were simulated and theoretically characterized. The RF coils described here should be suitable for a wide range of MRI/MRS preclinical/clinical applications, mainly at fields below 3 T.

Analytical Lorentz Force Model Between 1-D Linear Currents in Arbitrary Relative Positions and Directions

We derive an analytical formula for the Lorentz force and torque between 1-D linear current elements in arbitrary relative positions explicitly without remaining integral. A circuit is represented as the links of linear elements, and the force and torque on an element are represented as equivalent nodal forces on either side of it. We then treat the circuit as a link mechanism. Furthermore, we consider the singularity that appears in connected elements in a single circuit due to zero distance and avoid it using an approximation with a small radius. This singularity treatment makes the formula applicable even for calculation of the partial force from the entire circuit. Since the formula does not require numerical integration, it can reduce computation time. The accuracy of the approximation is also evaluated by comparing our calculations with the existing experimental results in the literature and numerical integrations.

A theoretical and experimental study on transverse field radio frequency surface coils

We have analysed, theoretically and experimentally, transverse field RF surface coils comprising two parallel linear current elements (figure-of-eight, FO8). A quasi-static numerical integration of the Biot–Savart law was used to simulate the B 1 field distribution of FO8 coils comprising a range of coil diameters and linear element separations. RF coil prototypes, suitable for 1H and 31P magnetic resonance imaging/spectroscopy at 1.5 T, that closely match the simulated models, were built and tested using workbench and MRI methods. A comparison with standard loop RF coils was made. Testing with phantoms showed a good agreement between the theoretical and experimental RF field distributions. Our findings should be useful to optimize the sensitivity and spatial selectivity of FO8 coils by a careful choice of the geometrical parameters. A number of applications where the use of FO8 RF coils could be of benefit are discussed, these include in vivo nuclear magnetic resonance imaging/spectroscopy.

Improved 1.5 T Magnetic Resonance Spectroscopy in the Human Calf with a Spatially Selective Radio Frequency Surface Coil

We describe the use of a transverse field RF surface coil that improves 1.5 T proton MR spectroscopy in the human calf. A 2-element figure-of-eight (FO8) transverse field RF surface coil (diameter 2R=10 cm; separation between the two linear current elements 2s=1cm) and a circular loop (CL) coil of equal diameter where built and tested with proton PRESS spectra at 1.5 T. The 1 H PRESS spectra obtained in the resting calf muscle of healthy volunteers showed that the FO8 coil allows a higher PRESS SNR (up to a factor 4.5) within a region of about 20 mm centred at about 12 mm from the coil plane, as compared to a standard CL coil. We found also a faster PRESS SNR decrease in the muscle tissue for anterior/posterior distance >20 mm using the FO8 coil. The measured PRESS SNR in the fat tissues of the calf showed a signal mostly localised within 10 mm from the coils surface and with an improved SNR (up to 5.5 times) observed in the presence of the FO8 coil as compared to the CL coil of equal diameter. The FO8 coil design can be advantageous for MRS applications, since it allows higher SNR from a small VOI positioned centrally within a relatively narrow region at a given depth in the human calf. The reported spatial SNR features of the FO8 coil design should also be useful for 1 H and 31 P MRS metabolites quantification in the human brain.

ELECTROMAGNETIC SOURCE EQUIVALENCE AND EXTENSION OF THE COMPLEX IMAGE METHOD FOR GEOPHYSICAL APPLICATIONS

In this work, source equivalence and computation of the re∞ected (induced) electromagnetic fleld in geophysical situations are studied. It is shown that the application of Huygens' principle allows for full generalization of Fukushima's equivalence theorem that applies only for magnetic fleld. The source equivalence is revisited for a vertical line current element, and it is shown that the equivalent charge required to replace the original source by a planar equivalent source together with the surface charge associated with the re∞ected fleld generates a purely vertical total electric fleld on the ground. Consequently, if the magnetic fleld and horizontal components of the total electric fleld on the ground are of interest, only equivalent currents need to be considered. The classical Complex Image Method (CIM) is derived from the exact image theory for planar impedance surfaces. The classical CIM is extended by considering a divergence-free source current that may have components also perpendicular to the ground plane. The extension is seen to generate a complex image charge not present in the classical CIM. Further, a generalized application of the extended CIM to geophysical situations having divergence-free volume source currents is introduced. The application involves decomposition of the source into linear current elements and rotations, translations and re∞ections of the electromagnetic fleld expressions associated with each element. The validity of the new approach is verifled for an

Optimization of multi-element transverse field radio frequency surface coils

In several MRI and MRS applications it is necessary to select a well-defined region of interest (ROI) with optimized signal-to-noise ratio and radio frequency (RF) field homogeneity. We have shown, theoretically and experimentally, that transverse field RF surface coils comprising a multitude of parallel linear elements are characterized by improved B1 spatial homogeneity and sensitivity in the central ROI, with respect to the standard square loop design. Finite-element modelling of RF coils tuned to 64.0 MHz was used to study the effect of the number of linear current elements and relative positioning with respect to B1 field homogeneity and sensitivity in the central ROI, both when empty and in the presence of a homogeneous tissue model. Workbench B1 field calibrations obtained with RF coil prototypes tuned to 64.0 MHz are in good agreement with our theoretical finite-element modelling. We have shown that the number of linear current elements and their relative positioning can be carefully selected to optimize either the B1 field homogeneity or sensitivity in the central ROI. We have shown that a number of multi-element RF coil geometries can be chosen to obtain an optimized RF field distribution and/or field amplitude. For example, we found that a six-element RF coil with 6 mm element separation is a good compromise for optimizing the B1 sensitivity (up to a factor 2) and the spatial homogeneity (about 30 mm with ΔB1/B1 ⩽ 20%) in the central ROI, with respect to the standard square loop or the two-element FO8 design. MRI images obtained at 2.35 T (proton frequency 100.34 MHz) with standard GE pulse sequences showed that the six-element FO8 prototype significantly increases the RF field homogeneity and RF field amplitude, with respect to the SL coil design.

Equivalence between Ampere and Grassmann's forces

We calculate the force on part of a single closed circuit due to the remaining circuit in four different geometries according to the forces of Ampere and Grassmann. All analytical calculations are performed using surface or volume current elements in order to avoid the divergences which appear with linear current elements of zero diameter. We conclude that when we consider the action of a closed circuit as a whole and utilize only circuits with closed lines of current, there will be an equivalence between the expressions of Grassmann and Ampere. This means that both of them are compatible with the experimental findings related to Ampere's bridge, contrary to the opinion of some authors.