Introduction

Halbach array is popular for providing the transverse main magnetic field ($$$\bf{B_0}$$$) to the bore in a portable MRI scanner. In a Halbach-based system, a solenoid is commonly used as the transmit coil [1]. However, a solenoid surfers by large length-to-radius ratio caused by the required homogeneity in $$$\bf{B_1}$$$ and consequently by high inductance. The large inductance leads to a relative strong conservative electric field, making the solenoid sensitive to the sample and the ambient environments. Moreover, large inductance make the coil difficult to be tuned and matched because large capacitors (over thousands of pF) are needed. There is a need for a short solenoid that provides field homogeneity with relatively low inductance, especially for a system using magnet array where the space for housing a solenoid is limited. Here, we present the design and optimization of a variable-pitch solenoid that shows advantages of high homogeneity with a practically small length-to-radius ratio and a low inductance. It is for a low-field portable MRI scanner that the length of the magnet array is short. Genetic algorithm (GA) was used for the optimization and Biot-Savart law was applied for the forward calculation.

Methods

Fig.$$$\,1$$$ (a) and (b) shows a Halbach magnet array which is the environment the short solenoid is designed for. The Halbach array is 19 cm long along the z-direction. Fig.1 (c) shows the measured magnetic field that is generated by the array. It is pointing in $$$xy$$$-plane with an average field strength of $$$67\,$$$mT and inhomogeneity of $$$42000\,$$$ppm in the central circular region with a diameter of 12 cm. Therefore, the resonant frequency is at about 2.85 MHz which falls into the quasi-static regime of electromagnetic waves [2]. Fig. 2 (a) and (b) show the side view and front view of the solenoid under design. The pitch is labeled as $$$P_i^j$$$ where i = 1 ... N and N is the total number of turns, j = 1...4, where a turn is split into four equal parts carrying different pitches every $$$90^0$$$. Biot-Savart law is applied for the forward calculation of the magnetic field generated by the solenoid for the optimization. Fig. 2 (c) shows the flow chart of GA. The parameter that are under optimization is $$$P_i^j$$$ ($$$i=1,...12$$$, j=1...4) with the total number of turns set to be 12. The rest of the parameters are preset as shown in Fig. 2. The optimization criteria are $$$\bf{B_1}$$$ homogeneities in the FOV (a cylindrical volume with a diameter of 12 cm and a length of 5 cm ).

Results

The result of a 200-iteration optimization loop is shown in Fig. $$$\,3(a)$$$, and the decrease of fitness values (inhomogeneities of $$$\bf{B_1}$$$ field in FOV) with the progress of iteration steps can be observed, which means the integration of Biot-Savart law and GA works properly. The best candidate solution in Fig. $$$\,3$$$(a) is set as an initial candidate solution for the next optimization loop. The final best solution in terms of pitch after several optimization loops is shown in Fig.$$$\,3$$$(b). It is seen that the shorter pitches are near the edge for compensating the inhomogeneity. The best solenoid has a total length of 19.2 cm. It was modeled, built, and shown in Fig.$$$\,4$$$(a)-(b). A reference solenoid of the same length with a uniform pitch is modeled. It is 12-turn and the pitch is $$$16.02 mm$$$. Fig. 4 (c) and (d) shows the B1 field (z-components) of the optimized and that of the reference in the FOV, respectively. The field inhomogeneity in FOV of the optimized solenoid is greatly reduced by about $$$56\%$$$ from $$$88500$$$ ppm to $$$39000$$$ ppm of the reference one, although the average field is slightly weaker (0.53 Gauss compared to 0.59 Gauss of the reference solenoid).

Discussion

Fig. 5 shows the field inhomogeneity versus the number of turns for a constant-pitch solenoid. The horizontal red line shows the inhomogeneity of the optimized variable-pitch solenoid. As shown, in order to have the same inhomogeneity, a solenoid with a constant pitches is $$$50\%$$$ longer in length than the optimized solenoid with variable pitches.

Conclusion

A short solenoid is successfully designed with significantly improved field homogeneity compared to a constant-pitch solenoid of the same length in the same FOV. Genetic algorithm is used for the optimization. It shows advantages of high homogeneity with a practically small length-to-radius ratio compared with a constant-pitch solenoid. It can fit in the Halbach array for a portable MRI scanner with field homogeneity guaranteed.

Acknowledgements

No acknowledgement found.