Understanding the Combined Effect of ${k}$ -Space Undersampling and Transient States Excitation in MR Fingerprinting Reconstructions

Magnetic resonance fingerprinting (MRF) is able to estimate multiple quantitative tissue parameters from a relatively short acquisition. The main characteristic of an MRF sequence is the simultaneous application of 1) transient states excitation and 2) highly undersampled <inline-formula> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula>-space. Despite the promising empirical results obtained with MRF, no work has appeared that formally describes the combined impact of these two aspects on the reconstruction accuracy. In this paper, a mathematical model is derived that directly relates the time-varying RF excitation and the <inline-formula> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula>-space sampling to the spatially dependent reconstruction errors. A subsequent in-depth analysis identifies the mechanisms by which MRF sequence properties affect accuracy, providing a formal explanation of several empirically observed or intuitively understood facts. The new insights are obtained which show how this analytical framework could be used to improve the MRF protocol.