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)
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
- 1361-6560
- DOI
- 10.1088/0031-9155/55/15/003
- project
- Algebraization of MRI signal equations
- language
- English
- LU publication?
- yes
- additional info
- 15
- id
- 4adff3ce-e5cf-4278-9764-faa002988086 (old id 8773596)
- alternative location
- http://stacks.iop.org/PMB/55/4231
- date added to LUP
- 2016-04-01 10:41:36
- date last changed
- 2022-02-02 20:05:20
@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}},
doi = {{10.1088/0031-9155/55/15/003}},
volume = {{55}},
year = {{2010}},
}