ArticleMagnetic resonance in medicine2025
Exact analytical solutions of the Bloch equation for the hyperbolic-secant and chirp pulses.
Article in Magnetic resonance in medicine, 2025. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 3 papers.
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3 citing papers in PubMed.
- Magnetic resonance imaging.Science advances · 2026Review
- Altering MRI rotating frame relaxations by changing the truncation level of Hyperbolic Secant pulse.Journal of magnetic resonance (San Diego, Calif. : 1997) · 2026Article
- Exact analytical solutions of the Bloch equation for the hyperbolic-secant and chirp pulses.Magnetic resonance in medicine · 2025Article
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Abstract
purposeTo improve the accuracy and generality of analytical solutions of the Bloch equation for the hyperbolic-secant (HS1) and chirp pulses in order to facilitate application to truncated and composite pulses and use in quantitative methods. THEORY AND
methodsPrevious analytical solutions of the Bloch equation during an HS1 pulse driving function are refined and extended in this exact solution for arbitrary initial magnetization and pulse parameters including asymmetrical truncation. An unapproximated general solution during the chirp pulse is derived in a non-spinor formulation for the first time. The solution on the extended complex plane for the square pulse is included for completeness.
resultsThe exact solutions for the HS1, chirp, and square pulses demonstrate high consistency with Runge-Kutta simulations for all included pulse and isochromat parameters. The HS1 solution is strictly more accurate than the most complete prior general solution. The analytical solution of the BIR-4 composite pulse constructed using asymmetrically truncated HS1 component pulses likewise agrees with simulation results.
conclusionThe derived analytical solutions for the Bloch equation during an HS1 or chirp pulse are exact regardless of pulse parameters and initial magnetization and precisely conform with simulations enabling their use in quantitative MRI applications and setting a foundation for the analytical consideration of relaxation and pulses in multiply rotating frames.
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