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Assessment of the effect of tissue motion in diffusion MRI: Derivation of new apparent diffusion coefficient formula
ICJ UMR5208, INSA-Lyon, 20 Av. A. Einstein, 69100 Villeurbanne, France |
We investigate in this paper the diffusion magnetic resonance imaging (MRI) in deformable organs such as the living heart. The difficulty comes from the hight sensitivity of diffusion measurement to tissue motion. Commonly in literature, the diffusion MRI signal is given by the complex magnetization of water molecules described by the Bloch-Torrey equation. When dealing with deformable organs, the Bloch-Torrey equation is no longer valid. Our main contribution is then to introduce a new mathematical description of the Bloch-Torrey equation in deforming media. In particular, some numerical simulations are presented to quantify the influence of cardiac motion on the estimation of diffusion. Moreover, based on a scaling argument and on an asymptotic model for the complex magnetization, we derive a new apparent diffusion coefficient formula. Finally, some numerical experiments illustrate the potential of this new version which gives a better reconstruction of the diffusion than using the classical one.
References:
[1] |
P. T. Callaghan,
A simple matrix formalism for spin echo analysis of restricted diffusion under generalized gradient waveforms, J. Magn. Reson., 129 (1997), 74-84.
doi: 10.1006/jmre.1997.1233. |
[2] |
P. Clarysse, C. Basset, L. Khouas, P. Croisille, D. Friboulet, C. Odet and I. E. Magnin, Two-dimensional spatial and temporal displacement and deformation field fitting from cardiac magnetic resonance tagging, Medical Image Analysis, 4 (2000), 253-268. Google Scholar |
[3] |
J. Dou, T. G. Reese, W. Y. Tseng and V. J. Wedeen,
Cardiac diffusion MRI without motion effects, Magn Reson Med, 48 (2002), 105-114.
doi: 10.1002/mrm.10188. |
[4] |
G. Duvaut, Mécanique des Milieux Continus, Masson, 1990. Google Scholar |
[5] |
U. Gamper, P. Boesiger and S. Kozerke, Diffusion imaging of the in vivo heart using spin echoes-considerations on bulk motion sensitivity, Magn. Reson. Med., 57 (2007), 331-337. Google Scholar |
[6] |
M. A. Horsfield and D. K. Jones, Applications of diffusion-weighted and diffusion tensor MRI to white matter diseases, NMR Biomed., 15 (2002), 570-577. Google Scholar |
[7] |
M. Lazar,
Mapping brain anatomical connectivity using white matter tractography, NMR Biomed., 23 (2010), 821-835.
doi: 10.1002/nbm.1579. |
[8] |
D. Le Bihan and E. Breton, Imagerie de diffusion in vivo par résonance magnétique nucléaire, CR Académie des Sciences, 301 (1985), 1109-1112. Google Scholar |
[9] |
D. Le Bihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis and M. Laval-Jeantet, MR imaging of intra-voxel incoherent motions: Application to diffusion and perfusion in neurologic disorders, Radiology, 161 (1986), 401-407. Google Scholar |
[10] |
J. -L. Lions and E. Magenes,
Probèlmes aux Limites non Homogènes et Applications,
(French) Travaux et Recherches Mathématiques, No. 20. Dunod, Paris, 1970. |
[11] |
Mattiello, P. J. Basser and D. Lebihan, Analytical expressions for the b matrix in NMR diffusion imaging and spectroscopy, J. Magn. Reson. Series A, 108 (1994), 131-141. Google Scholar |
[12] |
I. Mekkaoui, K. Moulin, P. Croisille, J. Pousin and M. Viallon,
Quantifying the Effect of Tissue Deformation on Diffusion-Weighted MRI: A Mathematical Model and an Efficient Simulation Framework applied to Cardiac Diffusion Imaging, Physics in Medicine and Biology, 61 (2016), 5662-5686.
doi: 10.1088/0031-9155/61/15/5662. |
[13] |
B. F. Moroney, T. Stait-Gardner, B. Ghadirian, N. N. Yadav and W. S. Price,
Numerical analysis of NMR diffusion measurements in the short gradient pulse limit, J. Magn. Reson., 234 (2013), 165-175.
doi: 10.1016/j.jmr.2013.06.019. |
[14] |
D. V. Nguyen, J. R. Li, D. Grebenkov and D. Le Bihan,
A finite element methods to solve the Boch-Torrey equation applied to diffusion magnetic resonance imaging, Journal of Computational Physics, 263 (2014), 283-302.
doi: 10.1016/j.jcp.2014.01.009. |
[15] |
D. G. Nishimura, Principles of Magnetic Resonance Imaging, Stanford University, California, 1996. Google Scholar |
[16] |
J. Pfeuffer, U. Flogel, W. Dreher and D. Leibfritz,
Restricted diffusion and exchange of intracellular water: theoretical modelling and diffusion time dependence of 1H NMR measurements on perfused glial cells, NMR in Biomedicine, 11 (1998), 19-31.
doi: 10.1002/(SICI)1099-1492(199802)11:1<19::AID-NBM499>3.0.CO;2-O. |
[17] |
W. S. Price, Pulsed-field gradient nuclear magnetic resonance as a tool for studying translational diffusion: Part 1. Basic theory, Concepts in Magnetic Resonance, 9 (1997), 299-336. Google Scholar |
[18] |
S. Rapacchi, H. Wen, M. Viallon, D. Grenier, P. Kellman and P. Croisille,
Low b-Value Diffusion-Weighted Cardiac Magnetic Resonance Imaging, Invest. Radiol., 46 (2011), 751-758.
doi: 10.1097/RLI.0b013e31822438e8. |
[19] |
T. G. Reese, R. M. Weisskoff, R. N. Smith, B. R. Rosen, R. E. Dinsmore and V. J. Wedeen, Imaging myocardial fiber architecture in vivo with magnetic resonance, Magn. Reson. Med, 34 (1995), 786-791. Google Scholar |
[20] |
T. G. Reese, V. J. Wedeen and R. M. Weisskoff,
Measuring diffusion in the presence of material strain, J. Magn. Reson, 112 (1996), 253-258.
doi: 10.1006/jmrb.1996.0139. |
[21] |
D. Rohmer and G. T. Gullberg,
A Bloch-Torrey equation for diffusion in a deforming media, Technical report, University of California, (2006).
doi: 10.2172/919380. |
[22] |
G. Russell, K. D. Harkins, T. W. Secomb, J. P. Galons and T. P. Trouard,
A finite difference method with periodic boundary conditions for simulations of diffusion-weighted magnetic resonance experiments in tissue, Physics in Medicine and Biology, 57 (2012), 35-46.
doi: 10.1088/0031-9155/57/4/N35. |
[23] |
B. S. Spottiswoode, X. Zhong, A. T. Hess, C. M. Kramer, E. M. Meintjes, B. M. Mayosi and F. H. Epstein,
Tracking myocardial motion from cine DENSE images using spatiotemporal phase unwrapping and temporal fitting, IEEE Trans. Med. Imaging., 26 (2007), 15-30.
doi: 10.1109/TMI.2006.884215. |
[24] |
E. O. Stejskal and J. E. Tanner,
Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient, J. Chem. Phys., 42 (1965), 288-292.
doi: 10.1063/1.1695690. |
[25] |
C. T. Stoeck, A. Kalinowska, C. V. Deuster, J. Harmer, R. W. Chan and M. Niemann et al., Dual-phase cardiac diffusion tensor imaging with strain correction PloS One 9(2014), e107159.
doi: 10.1371/journal.pone.0107159. |
[26] |
J. E. Tanner and E. O. Stejskal,
Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient spin-echo method, J. Chem. Phys., 49 (1968), 1768-1777.
doi: 10.1063/1.1670306. |
[27] |
H. C. Torrey,
Bloch equation with diffusion terms, Physical Review, 104 (1956), 563-565.
doi: 10.1103/PhysRev.104.563. |
[28] |
W. I. Tseng, T. G. Reese, R. M. Weisskoff, T. J. Brady and V. J. Wedeen,
Myocardial fiber shortening in humans: Initial results of MR imaging, Radiology, 216 (2000), 128-139.
doi: 10.1148/radiology.216.1.r00jn39128. |
[29] |
W. Y. Tseng, T. G. Reese, R. M. Weisskoff and V. J. Wedeen,
Cardiac diffusion tensor MRI in vivo without strain correction, Magn. Reson. Med., 42 (1999), 393-403.
doi: 10.1002/(SICI)1522-2594(199908)42:2<393::AID-MRM22>3.0.CO;2-F. |
[30] |
S. Warach, D. Chien, W. Li, M. Ronthal and R. R. Edelman,
Fast magnetic resonance diffusion-weighted imaging of acute human stroke, Neurology, 42 (1992), 1717-1723.
doi: 10.1212/WNL.42.9.1717. |
[31] |
H. Wen, K. A. Marsolo, E. E. Bennett, K. S. Kutten and R. P. Lewis, Adaptive post-processing techniques for myocardial tissue tracking with displacement-encoded MR imaging, Radiology., 246 (2008), 229-240. Google Scholar |
[32] |
J. Xu, M. D. Does and J. C. Gore, Numerical study of water diffusion in biological tissues using an improved finite difference method, Physics in Medicine and Biology, 52 (2007), 111-126. Google Scholar |
show all references
References:
[1] |
P. T. Callaghan,
A simple matrix formalism for spin echo analysis of restricted diffusion under generalized gradient waveforms, J. Magn. Reson., 129 (1997), 74-84.
doi: 10.1006/jmre.1997.1233. |
[2] |
P. Clarysse, C. Basset, L. Khouas, P. Croisille, D. Friboulet, C. Odet and I. E. Magnin, Two-dimensional spatial and temporal displacement and deformation field fitting from cardiac magnetic resonance tagging, Medical Image Analysis, 4 (2000), 253-268. Google Scholar |
[3] |
J. Dou, T. G. Reese, W. Y. Tseng and V. J. Wedeen,
Cardiac diffusion MRI without motion effects, Magn Reson Med, 48 (2002), 105-114.
doi: 10.1002/mrm.10188. |
[4] |
G. Duvaut, Mécanique des Milieux Continus, Masson, 1990. Google Scholar |
[5] |
U. Gamper, P. Boesiger and S. Kozerke, Diffusion imaging of the in vivo heart using spin echoes-considerations on bulk motion sensitivity, Magn. Reson. Med., 57 (2007), 331-337. Google Scholar |
[6] |
M. A. Horsfield and D. K. Jones, Applications of diffusion-weighted and diffusion tensor MRI to white matter diseases, NMR Biomed., 15 (2002), 570-577. Google Scholar |
[7] |
M. Lazar,
Mapping brain anatomical connectivity using white matter tractography, NMR Biomed., 23 (2010), 821-835.
doi: 10.1002/nbm.1579. |
[8] |
D. Le Bihan and E. Breton, Imagerie de diffusion in vivo par résonance magnétique nucléaire, CR Académie des Sciences, 301 (1985), 1109-1112. Google Scholar |
[9] |
D. Le Bihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis and M. Laval-Jeantet, MR imaging of intra-voxel incoherent motions: Application to diffusion and perfusion in neurologic disorders, Radiology, 161 (1986), 401-407. Google Scholar |
[10] |
J. -L. Lions and E. Magenes,
Probèlmes aux Limites non Homogènes et Applications,
(French) Travaux et Recherches Mathématiques, No. 20. Dunod, Paris, 1970. |
[11] |
Mattiello, P. J. Basser and D. Lebihan, Analytical expressions for the b matrix in NMR diffusion imaging and spectroscopy, J. Magn. Reson. Series A, 108 (1994), 131-141. Google Scholar |
[12] |
I. Mekkaoui, K. Moulin, P. Croisille, J. Pousin and M. Viallon,
Quantifying the Effect of Tissue Deformation on Diffusion-Weighted MRI: A Mathematical Model and an Efficient Simulation Framework applied to Cardiac Diffusion Imaging, Physics in Medicine and Biology, 61 (2016), 5662-5686.
doi: 10.1088/0031-9155/61/15/5662. |
[13] |
B. F. Moroney, T. Stait-Gardner, B. Ghadirian, N. N. Yadav and W. S. Price,
Numerical analysis of NMR diffusion measurements in the short gradient pulse limit, J. Magn. Reson., 234 (2013), 165-175.
doi: 10.1016/j.jmr.2013.06.019. |
[14] |
D. V. Nguyen, J. R. Li, D. Grebenkov and D. Le Bihan,
A finite element methods to solve the Boch-Torrey equation applied to diffusion magnetic resonance imaging, Journal of Computational Physics, 263 (2014), 283-302.
doi: 10.1016/j.jcp.2014.01.009. |
[15] |
D. G. Nishimura, Principles of Magnetic Resonance Imaging, Stanford University, California, 1996. Google Scholar |
[16] |
J. Pfeuffer, U. Flogel, W. Dreher and D. Leibfritz,
Restricted diffusion and exchange of intracellular water: theoretical modelling and diffusion time dependence of 1H NMR measurements on perfused glial cells, NMR in Biomedicine, 11 (1998), 19-31.
doi: 10.1002/(SICI)1099-1492(199802)11:1<19::AID-NBM499>3.0.CO;2-O. |
[17] |
W. S. Price, Pulsed-field gradient nuclear magnetic resonance as a tool for studying translational diffusion: Part 1. Basic theory, Concepts in Magnetic Resonance, 9 (1997), 299-336. Google Scholar |
[18] |
S. Rapacchi, H. Wen, M. Viallon, D. Grenier, P. Kellman and P. Croisille,
Low b-Value Diffusion-Weighted Cardiac Magnetic Resonance Imaging, Invest. Radiol., 46 (2011), 751-758.
doi: 10.1097/RLI.0b013e31822438e8. |
[19] |
T. G. Reese, R. M. Weisskoff, R. N. Smith, B. R. Rosen, R. E. Dinsmore and V. J. Wedeen, Imaging myocardial fiber architecture in vivo with magnetic resonance, Magn. Reson. Med, 34 (1995), 786-791. Google Scholar |
[20] |
T. G. Reese, V. J. Wedeen and R. M. Weisskoff,
Measuring diffusion in the presence of material strain, J. Magn. Reson, 112 (1996), 253-258.
doi: 10.1006/jmrb.1996.0139. |
[21] |
D. Rohmer and G. T. Gullberg,
A Bloch-Torrey equation for diffusion in a deforming media, Technical report, University of California, (2006).
doi: 10.2172/919380. |
[22] |
G. Russell, K. D. Harkins, T. W. Secomb, J. P. Galons and T. P. Trouard,
A finite difference method with periodic boundary conditions for simulations of diffusion-weighted magnetic resonance experiments in tissue, Physics in Medicine and Biology, 57 (2012), 35-46.
doi: 10.1088/0031-9155/57/4/N35. |
[23] |
B. S. Spottiswoode, X. Zhong, A. T. Hess, C. M. Kramer, E. M. Meintjes, B. M. Mayosi and F. H. Epstein,
Tracking myocardial motion from cine DENSE images using spatiotemporal phase unwrapping and temporal fitting, IEEE Trans. Med. Imaging., 26 (2007), 15-30.
doi: 10.1109/TMI.2006.884215. |
[24] |
E. O. Stejskal and J. E. Tanner,
Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient, J. Chem. Phys., 42 (1965), 288-292.
doi: 10.1063/1.1695690. |
[25] |
C. T. Stoeck, A. Kalinowska, C. V. Deuster, J. Harmer, R. W. Chan and M. Niemann et al., Dual-phase cardiac diffusion tensor imaging with strain correction PloS One 9(2014), e107159.
doi: 10.1371/journal.pone.0107159. |
[26] |
J. E. Tanner and E. O. Stejskal,
Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient spin-echo method, J. Chem. Phys., 49 (1968), 1768-1777.
doi: 10.1063/1.1670306. |
[27] |
H. C. Torrey,
Bloch equation with diffusion terms, Physical Review, 104 (1956), 563-565.
doi: 10.1103/PhysRev.104.563. |
[28] |
W. I. Tseng, T. G. Reese, R. M. Weisskoff, T. J. Brady and V. J. Wedeen,
Myocardial fiber shortening in humans: Initial results of MR imaging, Radiology, 216 (2000), 128-139.
doi: 10.1148/radiology.216.1.r00jn39128. |
[29] |
W. Y. Tseng, T. G. Reese, R. M. Weisskoff and V. J. Wedeen,
Cardiac diffusion tensor MRI in vivo without strain correction, Magn. Reson. Med., 42 (1999), 393-403.
doi: 10.1002/(SICI)1522-2594(199908)42:2<393::AID-MRM22>3.0.CO;2-F. |
[30] |
S. Warach, D. Chien, W. Li, M. Ronthal and R. R. Edelman,
Fast magnetic resonance diffusion-weighted imaging of acute human stroke, Neurology, 42 (1992), 1717-1723.
doi: 10.1212/WNL.42.9.1717. |
[31] |
H. Wen, K. A. Marsolo, E. E. Bennett, K. S. Kutten and R. P. Lewis, Adaptive post-processing techniques for myocardial tissue tracking with displacement-encoded MR imaging, Radiology., 246 (2008), 229-240. Google Scholar |
[32] |
J. Xu, M. D. Does and J. C. Gore, Numerical study of water diffusion in biological tissues using an improved finite difference method, Physics in Medicine and Biology, 52 (2007), 111-126. Google Scholar |
















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