Niklas Wehkamp^{1}, Philipp Rovedo^{2}, and Maxim Zaitsev^{3}

^{1}Department of Radiology, Medical Physics, Medical Center – University of Freiburg, Freiburg, Germany, ^{2}Neurozentrum, Medical Center – University of Freiburg, Freiburg, Germany, ^{3}High Field MR Center, Center for Medical Physics and Biomedical Engineering, Medical University of Vienna, Vienna, Austria

We present numerical optimization of variable winding pitch coil designs for miniature coils with respect to target field homogeneity. The optimization is based on a continuous coil winding layout that was used to simulate field homogeneity in the target volume via the Biot-Savart law. The example coil designs expand the range of applications for frequency adjustable field probes, as they increase the B_{0} modification field’s homogeneity by factor of 7. Due to the specific requirements of miniature B_{0} modification coils, the dimensions of the coil are restricted in length, width and wire diameter.

The coil designs discovered in this work can for example be used to improve the B

The coil description considers the coil in an unwrapped state. The unwrapping allows to describe the variable winding pitch as a function that can be modeled by polynomials. The coil in its unwrapped and wrapped state is illustrated in Figure 1.

The regular winding pitch in its unwrapped state corresponds to a straight line with a steady slope c described by $$$f_1$$$:

$$\mathbf {f_1} (\mathbf {z} )={c} {\mathbf {z} } ,$$

$$\mathbf {f_1'} (\mathbf {z} )={c} .$$

Additional, position dependent polynomial terms allow to modify the slope of the function and therefore the winding pitch of the coil. An additional cubed term stretches the center windings and compresses the windings towards the end of the coil. $$$f_2$$$ is an example of this modulated winding pitch with the additional term $$$a · z^{3}$$$ and the coefficient $$$b$$$:

$$\mathbf {f_2} (\mathbf {z} )={a} {\mathbf {z} ^{3}} + {b} {\mathbf {z}} ,$$

$$\mathbf {f_2'} (\mathbf {z} )=3{a} {\mathbf {z} ^{2}} + {b} .$$

The manufacturing of a coil with a 3D printed dissolvable former as shown in

$$c_{max} = {\frac {2\pi {R_{coil}}}{d_{wire}} } .$$

Furthermore, the minimum slope c min was constrained to allow for only one winding direction:

$$c_{min} = 0 .$$

Combining function $$$f_1'$$$ and $$$f_2'$$$ results in the non linear constraint:

$$c_{min} \leq \ 3a\mathbf {z}^{2} + {b} \leq\ c_{max} .$$

The resulting nonlinear optimization problem was initialized at multiple randomly initialized values to improve the global optimization behavior. The corresponding Matlab program for the presented variable winding pitch coil optimization can be found on https://github.com/Nikbert/solenoidPitchOptimization.

The presented examples were calculated for a spherical target volume with diameter of 1.4 mm and a coil length of 4 mm. The diameter was varied from 2 mm to 4 mm in 0.2 mm steps. The coil elements were divided into 3600 segments. The grid resolution for the simulation was set to 60×60×60 voxel for a 4×4×4 mm

1. Kelz J., 3D-printed dissolvable inserts for efficient and customizable fabrication of NMR transceiver coils, Journal of Magnetic Resonance, p. 4, 2019.

2. Idziak S., Haeberlen U., Design and construction of a high homogeneity rf coil for solid-state multiple-pulse NMR, Journal of Magnetic Resonance, vol. 50, pp. 281–288, Nov. 1982.

3. Wehkamp N., Rovedo P., Fischer E., et al, Frequency-adjustable magnetic field probes, Magnetic Resonance in Medicine, vol. 85, p1123-1133, 2021

Illustration of a) unwrapped regular (blue) and variable winding (red). b)Wrapped regular winding pitch. c) Wrapped variable winding pitch

Comparison of simulated B_{0} field optimized regular and variable winding pitchcoil design. With a coil diameter of 3 mm and length 4 mm.

Left side: Spread per B_{0} field shift for the regular and optimized variable winding pitch coildesign. The coil length was constant at 4 mm and the coil radius was varied from 1 to 2 mm.

Right side: Factor of field spread improvement of the optimized variable winding pitch coil design in relation to the optimized regular winding pitch coil design.

DOI: https://doi.org/10.58530/2022/3951