×

Fast approximate solution of Bloch equation for simulation of RF artifacts in magnetic resonance imaging. (English) Zbl 1187.78059

Summary: The technique used to spot information in Magnetic Resonance Imaging (MRI) uses electromagnetic fields. Even minor perturbations of these magnetic fields can disturb the imaging process and may render clinical images inaccurate or useless. Modelling and numerical simulation of the effects of static field inhomogeneities are now well established. Less attention has been paid to mathematical modelling of the effects of radio-frequency (RF) field inhomogeneities in the imaging process. When considering RF field inhomogeneities, the major difficulty is that the mathematical expression of the magnetisation vector is not anymore explicitly known in contrast with the unperturbed case. Indeed, the Bloch equation becomes an ordinary differential equation with nonconstant coefficients that cannot be solved analytically. The use of standard numerical schemes for ordinary differential equations to compute the magnetisation vector appears to be costly and not well suited for MRI image simulation. In this paper, we present an original method for solving the Bloch equation based on a truncated series expansion of the solution. The computational cost of the method reduces to the computation of the eigenelements of a block tridiagonal matrix of a very small size.

MSC:

78A70 Biological applications of optics and electromagnetic theory
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
92C55 Biomedical imaging and signal processing

Software:

SIMRI
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] ()
[2] Callaghan, P.T., Principles of nuclear magnetic resonance microscopy, (1994), Oxford University Press
[3] Abele, M.G.; Rusinek, H.; Bertora, F.; Trequattrini, A., Compensation of field distortion with ferromagnetic material and permanent magnets, Journal of applied physics, 75, 10, (1994), 6990-699
[4] Alexander, A.; Tsuruda, J.; Parker, D., Elimination of eddy current artifacts in diffusion weighted echo planar images: the use of bipolar gradients, Journal of magnetic resonance in medicine, 38, 1016-1021, (1997)
[5] Schenck, J.F., The role of magnetic susceptibility in magnetic resonance imaging, Medical physics, 23, 6, 815-850, (1996)
[6] Balac, S.; Caloz, G., Mathematical modeling and numerical simulation of magnetic susceptibility artifacts in magnetic resonance imaging, Computer methods in biomechanics and biomedical engineering, 3, 335-349, (2001)
[7] Balac, S.; Benoit-Cattin, H.; Lamotte, T.; Odet, C., Analytic solution to boundary integral computation of susceptibility induced magnetic field inhomogeneities, Mathematical and computer modelling, 39, 437-455, (2004) · Zbl 1070.78007
[8] Camacho, C.R.; Pleves, D.B.; Henkelman, R.M., Non-susceptibility artifacts due to metallic objects in MR imaging, Jmri, 5, 75-88, (1995)
[9] Malko, J.A.; Hoffman, J.C.; Jarret, P.J., Eddy-current induced artifacts caused by a MR compatible halo device, Radiology, 173, 563-564, (1989)
[10] Bennett, L.H.; Wang, P.S.; Donahue, M.J., Artifacts in magnetic resonance imaging from metals, Journal on applied physics, 79, 8, 4712-4714, (1996)
[11] Sled, J.G.; Pike, G.B., Standing-wave a RF penetration artifacts caused by elliptic geometry: an electrodynamic analysis of MRI, IEEE transactions on medical imaging, 17, 4, 653-662, (1998)
[12] ()
[13] Jin, J., The finite element method in electromagnetics, (1993), John Wiley & Sons · Zbl 0823.65124
[14] Bloch, F., Nuclear induction, Physical review, 70, 7, 460-474, (1946)
[15] Sebastiani, G.; Barone, P., Mathematical principles of basic magnetic resonance imaging in medicine, Signal processing, 25, 227-250, (1991)
[16] Hsieh, P.F.; Sibuya, Y., Basic theory of ordinary differential equations, (1999), Springer
[17] Walter, W., Ordinary differential equations, (1998), Springer
[18] Levante, T.O.; Baldus, M.; Meier, B.H.; Ernst, R.R., Formalized quantum mechanical Floquet theory and its application to sample spinning in nuclear magnetic resonance, Molecular physics, 86, 5, 1195-1212, (1995)
[19] Bain, A.D.; Dumont, R.S., Introduction to Floquet theory: the calculation of spinning sideband intensities in magic-angle spinning NMR, Concepts in magnetic resonance, 13, 3, 159-170, (2001)
[20] Casas, F.; Oteo, J.A.; Ros, J., Floquet theory: exponential perturbative treatment, Journal on physics A, mathematical and general, 34, 16, 3379-3388, (2001) · Zbl 0984.34046
[21] Hart, W.L., The cauchy – lipschitz method for infinite systems of differential equations, American journal of mathematics, 43, 4, 226-231, (1921) · JFM 48.0476.02
[22] S. Balac, M. Sadkane, On the computation of eigenvectors of a symmetric tridiagonal matrix: comparison of accuracy improvements of Givens and inverse iteration methods. Technical report, Laboratoire de Mathématiques Appliquées de Lyon, France, 2003. http://hal.archives-ouvertes.fr/hal-00137149
[23] Stewart, G.W.; Sun, J.-G., Matrix perturbation theory, (1990), Academic Press New York
[24] Jackson, J.D., Classical electrodynamics, (1999), John Wiley and Sons · Zbl 0114.42903
[25] Benoit-Cattin, H.; Collewet, G.; Belaroussi, B.; Saint-Jalmes, H.; Odet, C., The SIMRI project: A versatile and interactive mri simulator, Journal on magnetic and resonance imaging, 173, 97-115, (2005)
[26] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press · Zbl 0576.15001
[27] Nyenhuis, J.A.; Yee, O.P., Simulation of nuclear magnetic resonance spin echoes using the Bloch equation: influence of magnetic field inhomogeneities, Journal of applied physics, 76, 10, 6909-6911, (1994)
[28] Rourke, D.E., Solutions and linearization of the nonlinear dynamics of radiation damping, Concepts in magnetic resonance, 14, 2, 112-129, (2002)
[29] Balac, S.; Caloz, G.; Cathelineau, G.; Chauvel, B.; De Certaines, J.D., An integral representation method for numerical simulation of MRI artifacts induced by metallic implants, Journal of magnetic resonance in medicine, 45, 724-727, (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.