Electrodynamic constraints on minimum SAR in parallel excitation

(n) (r), the periodic excitation patterns, by a Fourier transform, using Parseval’s theorem, ξ can be expressed as a quadratic form in terms of fp [5]: ξ = 1/P (f p H Φf p ) 1 P . Φ is a positive definite covariance matrix. In accelerated parallel excitation, the aliased single-coil excitation patterns are linearly combined such that the aliasing lobes cancel out and the central lobe generates the desired profile. This condition represents the constraint for the design of pulses that minimize power deposition in the subject. We also know that the electromagnetic field inside the subject can be expressed as a sum of weighted basis functions that are multipole expansion solutions of source-free Maxwell’s equations [4]. Therefore, in order to calculate the ultimate intrinsic SAR, we have to solve a constrained optimization of the weighting factors, minimizing ξ under the design constraint. For this work, a solution was calculated in the case of EPI-trajectory small tip-angle parallel excitations following a method outlined by Zhu [5]. We used the optimal weighting factors to find the excitation patterns f (n) (r) that result in minimum average SAR. This was carried out one set of aliased pixels (x,y) at a time: f p =Φ −1 C x,y H (C x,y Φ −1 C x,y H ) −1 µ x,y ,