Brain MRI phase imaging assumes a linear spatial mapping of the internal fieldmap that continues to lack theoretical proof. We herein present one proof by replacing the arithmetic mean (in MRI signal formation from the intravoxel spin precession dephasing mechanism) with the geometric mean. Methods: By replacing the complex arithmetic mean of intravoxel dephasing isochromats with a complex geometric mean, we readily derive a linear spatial mapping of MRI phase imaging from an internal fieldmap without any restriction in phase angles. To justify the replacement of the complex arithmetic mean with the complex geometric mean for realistic brain MRI, we provide numerical T2*MRI simulations to observe the similarity and difference between arithmetic-and geometric-mean phase images in diverse settings with respect to spatial resolution and echo time, with or without proton density weighting.Results: Theoretically, the complex geometric mean model offers a theoretical proof of linear spatial mapping for MRI phase imaging. Numerical simulations of T2*MRI phase imaging show that the geometric mean conforms to the arithmetic mean at a high similarity in the small phase condition (e.g., corr > 0.90 in phase pre-wrapping status at T E < 10 ms) and the similarity falls at large phase angles (e.g., corr ≈ 0.80 in phase-wrapped status at T E = 30 ms). Conclusion: By replacing the arithmetic mean of intravoxel spin precession dephasingwith the geometric mean, we find a theoretical proof for linear MRI phase imaging beyond the small phase condition on spin precession angles.complex-valued MRI, intravoxel dephasing mechanism, isochromats, quantitative susceptibility mapping (QSM), spin precession angle, T 2 * magnitude, T 2 * phase | INTRODUCTIONThe output of T 2 *-weighted MRI (T2*MRI) yields a complex-valued image consisting of a pair of real-valued magnitude and phase images, of which the phase image is used for reconstructing the underlying magnetic susceptibility source (denoted by χ) by computationally solving an inverse MRI problem (CIMRI). 1,2 The inverse solution of T2*MRI has been described by quantitative susceptibility mapping (QSM), [3][4][5] which presumes that the T2*MRI phase imag7e takes on a linear spatial mapping of the internal field map. There remains a lack of theoretical proof for this assumption so far, except for a strict linear approximation in the small phase angle regime. 1,2 Herein we provide one proof through a geometric mean model on the intravoxel spin procession dephasing formula beyond the small phase angle regime.

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Upon checking with clinical multi-GRE complex-valued brain images, we observed zigzag lines of multi-echo phase signals, to which we provided numerical multi-GRE simulations for looking into the MRI phase signal artifacts. Based on intravoxel dephasing mechanism, we calculated a train of multi-GRE complex-valued voxel signals by simulating gradient field reversals under perturbations in either gradient strength (G±δG) or gradient duration (Δ±δΔ), as well as the bi-variable gradient perturbations (δG&δΔ). On linear stepwise gradient variations (e.g. δG ∝ n and δΔ ∝ n, with respect to echo index n), we observed the multi-GRE phase zigzags in cases of various gradient variations. We consider the effect of eddy current on multi-echo GRE as an equivalent gradient variation, which is a practical cause for multi-echo phase zigzag artifacts. Both clinical multi-echo T2*MRI phase data and numerical simulations exhibit similar zigzag lines, as caused by gradient variations over the gradient reversal repetition in a multi-GRE sequence, which could be practically attributed to gradient-reversal-induced eddy current. However, the multi-echo magnitude signals are invariant to gradient field reversal.

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