Analytic Calculation of the E-Fields induced by Gradient Coil Switching

Introduction: Nerve stimulation induced by rapidly switching magnetic field gradients poses an upper limit on the gradient strengths that can be employed in fast magnetic resonance imaging and diffusion measurements’. The likelihood of stimulation is usually assessed by evaluation of the maximum rate of change of the magnetic field, inside the gradient coi12. Nerve stimulation actually occurs as a result of the electric field, E, generated inside the body by the time varying magnetic fields. Calculation of this field requires evaluation of the electric potential, V, due to charge accumulated at the boundaries between areas of different electrical conductivity, as well as the vector potential, A, resulting from the current flowing through the coil windings. Previously this calculation has been implemented using finite element methods, which have been applied to simple coil geometries and to model systems314. Here we imagine a human subject to be an infinite conducting cylinder placed coaxially in a set of gradient coils. We have obtained analytic expressions for the electric field inside the subject which can be used to evaluate the areas where stimulation by a given gradient coil is likely to occur; they may also be employed at the coil design stage to produce gradient coils giving higher switched gradients at stimulation threshold. Theory: We consider a coil on a cylinder of radius, a, the windings of which can be described by a current distribution, J, with axial, J,, and azimuthal, J+, components. The Fourier transform of Jb with respect to I$ and z is written as in the usual manner as J?(k). The coil surrounds an infinite cylinder of radius, po < a, conductivity c7 and relative permittivity, E,. The electric field inside the conducting cylinder is given by E = -8A/dt VV. For gradient switching frequencies and human body dimensions and conductivity, quasi-static conditions5 obtain and all propagation and skin depth effects can be neglected. In this situation the vector potential can be calculated directly from the coil currents using standard expressions”. The electric potential is set by the boundary condition that the radial component of the electric field, Ep, must be zero at p = ~0. Defining the Fourier transform of Ep at radius p with respect to 4 and 8 as Ey(lc,p) we then find