Target-field method for MRI biplanar gradient coil design

Based on the target-field method of solving Fredholm integral equations of the first kind, a new approach is presented in the paper for designing gradient coils that can be used in a permanent-magnet magnetic resonance imaging (MRI) system with biplanar poles. To restrict the current distribution on the coil plane within a finite radius, the current density is pre-expanded into Fourier series by orthogonal basis functions. By setting the target-field points and Bz values over the imaging region of interest, corresponding integral equations are derived from the Biot-Savart law to calculate the current densities. They form a matrix equation, in which the unknown elements of the column vector are the Fourier coefficients for the unknown current density. As long as these target-field points are well chosen, the Fourier coefficients can be solved by inverse matrix calculation instead of the regularization method for Fredholm integral equations of the first kind. Then the current density is discretized using the stream-function method to generate the winding patterns. To verify the feasibility of this approach, the gradient magnetic field generated by the current density is calculated via the Biot-Savart law. Optimized parameters are obtained through computer simulations for some shielded and unshielded transverse gradient coils. The performance of this approach has been demonstrated as well.