Assessment of the effect of tissue motion in diffusion MRI: Derivation of new apparent diffusion coefficient formula

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February 2018, 12(1): 125-152. doi: 10.3934/ipi.2018005

Assessment of the effect of tissue motion in diffusion MRI: Derivation of new apparent diffusion coefficient formula

Elie Bretin , Imen Mekkaoui ^, and Jérôme Pousin

ICJ UMR5208, INSA-Lyon, 20 Av. A. Einstein, 69100 Villeurbanne, France

* Corresponding author: imen.mekkaoui@insa-lyon.fr

Received April 2017 Revised September 2017 Published December 2017

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 When dealing with deformable organs, the Bloch-Torrey equation is no longer valid.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.

Keywords: Diffusion magnetic resonance imaging, Bloch-Torrey equation, cardiac diffusion imaging, diffusion encoding gradient, apparent diffusion coefficient, asymptotic analysis.

Mathematics Subject Classification: Primary: 35K57, 35E10; Secondary: 65M60.

Citation: Elie Bretin, Imen Mekkaoui, Jérôme Pousin. Assessment of the effect of tissue motion in diffusion MRI: Derivation of new apparent diffusion coefficient formula. Inverse Problems & Imaging, 2018, 12 (1) : 125-152. doi: 10.3934/ipi.2018005

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	H. C. Torrey, Bloch equation with diffusion terms, Physical Review, 104 (1956), 563-565. doi: 10.1103/PhysRev.104.563. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	H. C. Torrey, Bloch equation with diffusion terms, Physical Review, 104 (1956), 563-565. doi: 10.1103/PhysRev.104.563. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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

Figure 1. Spin echo diffusion encoding sequence. Two identical gradients are applied around the $180^o$ RF pulse. $G$ is the gradient intensity, $\delta$ the gradient duration and $\Delta$ the gradient spacing

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2]. The region of interest (the left ventricle zone) is shown inside the yellow squares. (Right) A domain $\Omega(0)$ in the form of a ring is chosen for representing the left ventricle zone">Figure 2. (Left) Cardiac MRI images generated by the simulator introduced in [2]. The region of interest (the left ventricle zone) is shown inside the yellow squares. (Right) A domain $\Omega(0)$ in the form of a ring is chosen for representing the left ventricle zone

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Figure 3. Behavior of the function $S$ over one cardiac cycle. $T_s = 333$ms, $T_d = 667$ms

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Figure 4. STEAM diffusion encoding sequence

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Figure 5. $\|D \mathbf{u}\|_2$ calculated during the application of the diffusion encoding gradients for different values of

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Figure 6. (Top) Diffusion MRI images at different moments of cardiac cycle. (Bottom) Exact diffusion coefficient

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Figure 7. (a) Relative error in diffusion coefficient. (b) Localization of the sweet spots when the cardiac deformation is approximately equal to its temporal mean during the cardiac cycle

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Figure 8. The squared norm of $\nabla \Phi(\mathbf{x},t)$ calculated at different moments of the cardiac cycle: (a) TD = 50ms, (b) TD = 200ms, (c) TD = 350ms, (d) TD = 600ms, (e) TD = 900ms

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Figure 9. Diffusion images reconstructed in systole. $1^\text{st}$ column: Before correction at: TD = 0ms, TD = 100ms, TD = 350ms. $2^\text{nd}$ column: After correction. $3^\text{rd}$ column: Absolute error between the exact diffusion and the corrected diffusion images

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Figure 10. Diffusion images reconstructed in diastole. $1^\text{st}$ column: Before correction at: TD = 750ms, TD = 900ms. $2^\text{nd}$ column: After correction. $3^\text{rd}$ column: Absolute error between the exact diffusion and the corrected diffusion images

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Figure 11. Exact diffusion

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Figure 12. Images constructed at TD = 250ms. (a) Diffusion after correction for a noisy motion with SNR = 40dB. (b) Error in diffusion. (c) Diffusion after correction for a noisy motion with SNR = 30dB. (d) Error in diffusion

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Figure 13. Images constructed at TD = 850ms. $1^{st}$ row: Diffusion encoding gradient applied in $x$-direction: (a) Diffusion before correction. (b) Diffusion after correction. (c) Absolute error between the exact diffusion and the corrected diffusion images. $2^{nd}$ row: Diffusion encoding gradient applied in $y$-direction: (d) Diffusion before correction. (e) Diffusion after correction. (f) Absolute error between the exact diffusion and the corrected diffusion images

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Figure 14. Images constructed at TD = 250ms. (a) Diffusion after correction for a noisy motion with SNR = 40dB. (b) Error in diffusion. (c) Diffusion after correction for a noisy motion with SNR = 30dB. (d) Error in diffusion

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Figure 15. Images constructed at TD = 250ms. (a) Diffusion after correction with variability of 10% on $T_s$ and $T_d$. (b) Error in diffusion. (c) Diffusion after correction with variability of 20% on $T_s$ and $T_d$. (d) Error in diffusion

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Figure 16. Diffusion images reconstructed with different values of $\varepsilon$. $1^{\text{st}}$ row: $\varepsilon\approx$5e-4. $2^{\text{nd}}$ row: $\varepsilon\approx$1e-3. $3^{\text{rd}}$ row: $\varepsilon\approx$ 5e-3

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Figure 17. The exact diffusion presented on an irregular ring

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Figure 18. Diffusion images reconstructed at: $1^{st}$ row: TD = 250ms. $2^{nd}$ row: TD = 350ms. Diffusion before correction (first column). Diffusion after correction (second column). Error in diffusion (third column)

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