A Mathematical Analysis of Clustering-Free Local SAR Compression Algorithms for MRI Safety in Parallel Transmission

Parallel transmission (pTX) is a versatile solution to enable UHF MRI of the human body, where radiofrequency (RF) field inhomogeneity appears very challenging. Today, state of the art monitoring of the local SAR in pTX consists in evaluating the RF power deposition on specific SAR matrices called Virtual Observation Points (VOPs). It essentially relies on accurate electromagnetic simulations able to return the local SAR distribution inside the body in response to any applied pTX RF waveform. In order to reduce the number of SAR matrices to a value compatible with real time SAR monitoring
$(\boldsymbol {\ll 1}{\boldsymbol {0}^{\boldsymbol {3}}})$
, a VOP set is obtained by partitioning the SAR model into clusters, and associating a so- called dominant SAR matrix to every cluster. More recently, a clustering-free compression method was proposed, allowing for a significant reduction in the number of SAR matrices. The concept and derivation however assumed static RF shims and their extension to dynamic pTX is not straightforward, thereby casting doubt on the strict validity of the compression approach for these more complicated RF waveforms. In this work, we provide the mathematical framework to tackle this problem and find a rigorous justification of this criterion in the light of convex optimization theory. Our analysis led us to a variant of the clustering-free compression approach exploiting convex optimization. This new compression algorithm offers computational gains for large SAR models and for high-channel count pTX RF coils.