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Updated: Sep 27, 2025
High-resolution Functional Magnetic Resonance Imaging Methods for Human Midbrain
Published on: May 10, 2012
Proof of linear MRI phase imaging from an internal fieldmap
Zikuan Chen1,2, Xiulan Zhai2, Zeyuan Chen3
1Diagnostic Radiology, City of Hope National Medical Center, Duarte, CA, USA.
Purpose:
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 TE < 10 ms) and the similarity falls at large phase angles (e.g., corr ≈ 0.80 in phase-wrapped status at TE = 30 ms).
Conclusion:
By replacing the arithmetic mean of intravoxel spin precession dephasing with the geometric mean, we find a theoretical proof for linear MRI phase imaging beyond the small phase condition on spin precession angles.
