An analytical solution for the SSFP signal in MRI

Area of Science:

• Magnetic Resonance Imaging (MRI)
• Signal Processing
• Mathematical Physics

Background:

• Previous analyses of steady-state free-precession (SSFP) signals in rapid MRI involved complex equations with infinite binomial series.
• A need exists for a more direct and unified mathematical treatment of SSFP signals.

Purpose of the Study:

• To derive a novel analytical solution for SSFP signals in rapid MRI.
• To demonstrate the equivalence of previously disparate SSFP signal equations.
• To extend the analysis to higher-order signals and explore applications.

Main Methods:

• The study employed a mathematical transformation to convert infinite binomial series into the power series expansion of the derivative of the inverse sine function.
• This analytical approach was extended to incorporate higher-order signal components.
• The derived solution was applied to specific MRI scenarios, including echo-shifted MRI and TrueFISP sequences.

Main Results:

• A closed-form analytical solution for SSFP signals was successfully derived, avoiding infinite series.
• The results confirmed the identity of previously reported, seemingly different, SSFP signal equations.
• New closed expressions for echo-shifted MRI signals were obtained, alongside an analysis of TrueFISP sequences.

Conclusions:

• The developed analytical method provides a unified and simplified approach to understanding SSFP signals in rapid MRI.
• This work resolves inconsistencies in prior SSFP signal analyses and offers practical applications.
• The findings contribute to a deeper theoretical understanding and potential optimization of advanced MRI techniques.