Redundant Spatial Harmonic Information in Zeugmatography with Linear Encoding (R-SHIZLE) Theoretically Encodes Intra-Acquisition Decay

Introduction: Much work has been done using time series of T2* or T2 weighted images to consider physiologic processes ([1], [2]). Such weighted images include contributions from other weighting factors including T1, proton spin density, and inflow effects. Such contributions limit the ability to extract quantifiable measurements from a single image acquisition after a single excitation. To obtain absolute measurements, multiple image acquisition techniques have been used. Additionally, to calculate differential measurements, multiple acquisitions from separate excitations during different conditions have been used. Here we provide theoretical work for making quantitative measurements of T2* or T2 from a single image acquisition following a single excitation pulse by utilizing the expected symmetry of k-space observations. Theory: The signal acquired in MRI can be modeled as the complex-valued Fourier transform of the object being imaged. As the object being imaged is physical, its ideal image representation is real-valued. A fundamental property of the Fourier transform is that conjugate symmetry exists about the origin in the k-space representation of a real-valued image. However, the k-space data is acquired over a period of discrete time points, with each k-space observation being acquired at a different time point and thus weighted by a different amount of intra-acquisition (T2 or T2*) decay. Additionally, unmodeled magnetic field inhomogeneity leads to a complex-valued image-space representation of the object being imaged. Both of these complications break the expected conjugate symmetry of k-space observations about the origin. The signal at a time point can be shown to be [ ] [ ] [ ] dxdy y k x k i t y x B i y x T t y x t k k S y x y x