True energy-minimal and finite-size biplanar gradient coil design for MRI

Finite-sized high-performance planar magnetic field gradient coils in today's open configuration magnetic resonance imaging (MRI) systems have always been desirable for ever demanding imaging applications. We present a Lagrange multiplier technique for designing a minimum-energy gradient coil under a finite-size planar geometry constraint in addition to a set of magnetic field constraints. In this new design methodology, the surface current density on a finite size plane is represented by a two-dimensional (2-D) Fourier series expansion. Following the standard approach, we construct a functional F in terms of the stored magnetic energy and a set of field constraint points which are chosen over the desired imaging volume. Minimizing F, we obtain the continuous current density distribution for the finite-size planar gradient coil. Applying the stream function technique to the resulting continuous current distribution, the discrete current pattern can be generated. Employing the Biot-Savart law to the discrete current loops, the gradient magnetic field has been re-evaluated in order to validate the theory. Using this approach, we have been able to design a finite-size biplanar z-gradient coil which is capable of generating a gradient field of 40 mT/m @ 266 A. The excellent agreement between the analytical and numerical results has been achieved.