no. 5
Introduction to magnetic resonance imaging for mathematicians
[Une introduction à l'imagerie par résonance magnétique pour les mathématiciens]
Epstein, Charles L.1
1 University of Pennsylvania, Department of Mathematics, Philadelphia (USA)
Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1697-1716.

Nous introduisons les concepts et modèles de base en résonance magnétique nucléaire (RMN). Nous décrivons une expérience d'imagerie simple ainsi que la réduction du problème d'excitation sélective à un problème de scattering inverse.

The basic concepts and models used in the study of nuclear magnetic resonance are introduced. A simple imaging experiment is described, as well as, the reduction of the problem of selective excitation to a classical problem in inverse scattering.

DOI : 10.5802/aif.2063
Classification : 78A46, 81V35, 65R10, 65R32
Epstein, Charles L. 1

1 University of Pennsylvania, Department of Mathematics, Philadelphia (USA) 
@article{AIF_2004__54_5_1697_0,
     author = {Epstein, Charles L.},
     title = {Introduction to magnetic resonance imaging for mathematicians},
     journal = {Annales de l'Institut Fourier},
     pages = {1697--1716},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     doi = {10.5802/aif.2063},
     mrnumber = {2127862},
     zbl = {02162438},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2063/}
}
TY  - JOUR
AU  - Epstein, Charles L.
TI  - Introduction to magnetic resonance imaging for mathematicians
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 1697
EP  - 1716
VL  - 54
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2063/
DO  - 10.5802/aif.2063
LA  - en
ID  - AIF_2004__54_5_1697_0
ER  - 
%0 Journal Article
%A Epstein, Charles L.
%T Introduction to magnetic resonance imaging for mathematicians
%J Annales de l'Institut Fourier
%D 2004
%P 1697-1716
%V 54
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2063/
%R 10.5802/aif.2063
%G en
%F AIF_2004__54_5_1697_0
Epstein, Charles L. Introduction to magnetic resonance imaging for mathematicians. Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1697-1716. doi : 10.5802/aif.2063. http://www.numdam.org/articles/10.5802/aif.2063/

[AKNS] M. Ablowitz, D. Kaup, A. Newell & H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies Appl. Math. 53 (1974) p. 249-315 | MR | Zbl

[Ab] A. Abragam, Principles of Nuclear Magnetism, Clarendon Press, 1983

[BC] R. Beals & R. Coifman, Scattering and inverse scattering for first order systems, CPAM 37 (1984) p. 39-90 | MR | Zbl

[Bl] F. Bloch, Nuclear induction, Phys. Review 70 (1946) p. 460-474

[Cal] P.T. Callaghan, Principles of nuclear magnetic resonance microscopy, Clarendon Press, 1993

[Ca1] J. Carlson, Exact solutions for selective-excitation pulses, J. Magn. Res. 94 (1991) p. 376-386

[Ca2] J. Carlson, Exact solutions for selective-excitation pulses. II. Excitation pulses with phase control, J. Magn. Res. 97 (1992) p. 65-78

[Ep] C.L. Epstein, Minimum power pulse synthesis via the inverse scattering transform, J. Magn. Res. 167 (2004) p. 185-210

[EBW] R. Ernst, G. Bodenhausen & A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions, Clarendon, 1987

[FT] L. Faddeev & L. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer Verlag, 1987 | MR | Zbl

[Gr1] F. Grünbaum, Trying to beat Heisenberg, Lecture Notes in Pure and Applied Math. vol. 122, Marcel Dekker, 1989, p. 657-666 | Zbl

[Gr2] F. Grünbaum, Concentrating a potential and its scattering transform for a discrete version of the Schrödinger and Zakharov-Shabat operators, Physica D 44 (1990) p. 92-98 | MR | Zbl

[GH] F. Grünbaum & A. Hasenfeld, An exploration of the invertibility of the Bloch transform, Inverse Problems 2 (1986) p. 75-81 | MR | Zbl

[Ha] E.M. Haacke, R.W. Brown, M.R. Thompson & R. Venkatesan, Magnetic Resonance Imaging, Wiley-Liss, 1999

[Ho1] D. Hoult, The principle of reciprocity in signal strength calculations - A mathematical guide, Concepts Magn. Res. 12 (2000) p. 173-187

[Ho2] D. Hoult, Sensitivity and power deposition in a high field imaging experiment, JMRI 12 (2000) p. 46-67

[Ma] J. Magland, Discrete Inverse Scattering Theory and NMR pulse design, PhD. Thesis, University of Pennsylvania, 2004

[Me] E. Merzbacher, Quantum Mechanics, 2nd ed., John Wiler \& Sons, 1970 | MR | Zbl

[PRNM] J. Pauly, P. Le Roux, D. Nishimura & A. Macovski, Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm, IEEE Trans. Med. Imaging 10 (1991) p. 53-65

[MR] D.E. Rourke & P.G. Morris, The inverse scattering transform and its use in the exact inversion of the Bloch equation for noninteracting spins, J. Magn. Res. 99 (1992) p. 118-138

[SL1] M. Shinnar & J. Leigh, The application of spinors to pulse synthesis and analysis, Magn. Res. in Med. 12 (1989) p. 93-98

[SL2] M. Shinnar & J. Leigh, Inversion of the Bloch equation, J. Chem. Phys. 98 (1993) p. 6121-6128

[To] H.C. Torrey, Bloch equations with diffusion terms, Phys. Review 104 (1956) p. 563-565

Cité par Sources :