Exact algebraization of the signal equation of spoiled gradient echo MRI
(2010) In Physics in Medicine and Biology 55. p.42314245 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... (More)  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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/8773596
 author
 Dathe, Henning and Helms, Gunther ^{LU}
 organization
 publishing date
 2010
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Physics in Medicine and Biology
 volume
 55
 pages
 4231  4245
 publisher
 IOP Publishing
 external identifiers

 scopus:78049425927
 ISSN
 13616560
 DOI
 10.1088/00319155/55/15/003
 project
 Algebraization of MRI signal equations
 language
 English
 LU publication?
 yes
 additional info
 15
 id
 4adff3cee5cf42789764faa002988086 (old id 8773596)
 alternative location
 http://stacks.iop.org/PMB/55/4231
 date added to LUP
 20160401 10:41:36
 date last changed
 20220202 20:05:20
@article{4adff3cee5cf42789764faa002988086, 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<br/><br> 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<br/><br> 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.}}, author = {{Dathe, Henning and Helms, Gunther}}, issn = {{13616560}}, language = {{eng}}, pages = {{42314245}}, publisher = {{IOP Publishing}}, series = {{Physics in Medicine and Biology}}, title = {{Exact algebraization of the signal equation of spoiled gradient echo MRI}}, url = {{http://dx.doi.org/10.1088/00319155/55/15/003}}, doi = {{10.1088/00319155/55/15/003}}, volume = {{55}}, year = {{2010}}, }