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Abstract
The Ernst equation for Fourier transform nuclear magnetic resonance (MR) describes the spoiled steadystate signal created by periodic partial excitation. In MR imaging (MRI), it is commonly applied to spoiled gradientecho acquisition in the steady state, created by a small flip angle α at a repetition time TR much shorter than the longitudinal relaxation time T1. We describe two parameter transformations of α and TR/T1, which render the Ernst equation
as a loworder rational function. Computer algebra can be readily applied for analytically solving protocol optimization, as shown for the dual flip angle experiment. These transformations are based on the halfangle tangent substitution and its hyperbolic analogue. They are monotonic and approach identity for small α and small TR/T1 with a thirdorder error. Thus, the exact algebraization can be readily applied to fast gradient echo MRI to yield a rational approximation in α and TR/T1. This reveals a fundamental relationship
between the square of the flip angle and TR/T1 which characterizes the Ernst angle, constant degree of T1weighting and the influence of the local radiofrequency field.
as a loworder rational function. Computer algebra can be readily applied for analytically solving protocol optimization, as shown for the dual flip angle experiment. These transformations are based on the halfangle tangent substitution and its hyperbolic analogue. They are monotonic and approach identity for small α and small TR/T1 with a thirdorder error. Thus, the exact algebraization can be readily applied to fast gradient echo MRI to yield a rational approximation in α and TR/T1. This reveals a fundamental relationship
between the square of the flip angle and TR/T1 which characterizes the Ernst angle, constant degree of T1weighting and the influence of the local radiofrequency field.
Original language  English 

Pages (fromto)  42314245 
Journal  Physics in Medicine and Biology 
Volume  55 
DOIs  
Publication status  Published  2010 
Bibliographical note
15Subject classification (UKÄ)
 Other Physics Topics
 Medical Image Processing
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Algebraization of MRI signal equations
Helms, G. (PI) & Dathe, H. (CoI)
2007/08/07 → …
Project: Research