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Exact algebraization of the signal equation of spoiled gradient echo MRI

Dathe, Henning and Helms, Gunther LU (2010) In Physics in Medicine and Biology 55. p.4231-4245
Abstract
The Ernst equation for Fourier transform nuclear magnetic resonance (MR) describes the spoiled steady-state signal created by periodic partial excitation. In MR imaging (MRI), it is commonly applied to spoiled gradient-echo 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 low-order 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 half-angle tangent substitution and its hyperbolic analogue. They are... (More)
The Ernst equation for Fourier transform nuclear magnetic resonance (MR) describes the spoiled steady-state signal created by periodic partial excitation. In MR imaging (MRI), it is commonly applied to spoiled gradient-echo 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 low-order 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 half-angle tangent substitution and its hyperbolic analogue. They are monotonic and approach identity for small α and small TR/T1 with a third-order 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 T1-weighting and the influence of the local radiofrequency field. (Less)
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author
organization
publishing date
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
1361-6560
DOI
10.1088/0031-9155/55/15/003
language
English
LU publication?
yes
id
4adff3ce-e5cf-4278-9764-faa002988086 (old id 8773596)
alternative location
http://stacks.iop.org/PMB/55/4231
date added to LUP
2016-02-27 00:55:59
date last changed
2018-05-29 12:16:57
@article{4adff3ce-e5cf-4278-9764-faa002988086,
  abstract     = {The Ernst equation for Fourier transform nuclear magnetic resonance (MR) describes the spoiled steady-state signal created by periodic partial excitation. In MR imaging (MRI), it is commonly applied to spoiled gradient-echo 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 low-order 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 half-angle tangent substitution and its hyperbolic analogue. They are monotonic and approach identity for small α and small TR/T1 with a third-order 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 T1-weighting and the influence of the local radiofrequency field.},
  author       = {Dathe, Henning and Helms, Gunther},
  issn         = {1361-6560},
  language     = {eng},
  pages        = {4231--4245},
  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/10.1088/0031-9155/55/15/003},
  volume       = {55},
  year         = {2010},
}