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
-
Previous Article
Recovery of block sparse signals under the conditions on block RIC and ROC by BOMP and BOMMP
- IPI Home
- This Issue
-
Next Article
Superconductive and insulating inclusions for linear and non-linear conductivity equations
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
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 When dealing with deformable organs, the Bloch-Torrey equation is no longer valid.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.
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 and 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. |
| [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.
|
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
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.
|
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
Figure 4.
STEAM diffusion encoding sequence
Figure 5.
$\|D \mathbf{u}\|_2$ calculated during the application of the diffusion encoding gradients for different values of
Figure 6.
(Top) Diffusion MRI images at different moments of cardiac cycle. (Bottom) Exact diffusion coefficient
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
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
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
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
Figure 11.
Exact diffusion
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
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
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
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
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
Figure 17.
The exact diffusion presented on an irregular ring
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)
| [1] |
Shenglong Hu, Zheng-Hai Huang, Hong-Yan Ni, Liqun Qi. Positive definiteness of Diffusion Kurtosis Imaging. Inverse Problems and Imaging, 2012, 6 (1) : 57-75. doi: 10.3934/ipi.2012.6.57 |
| [2] |
Yunmei Chen, Weihong Guo, Qingguo Zeng, Yijun Liu. A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images. Inverse Problems and Imaging, 2008, 2 (2) : 205-224. doi: 10.3934/ipi.2008.2.205 |
| [3] |
Shanshan Wang, Yanxia Chen, Taohui Xiao, Lei Zhang, Xin Liu, Hairong Zheng. LANTERN: Learn analysis transform network for dynamic magnetic resonance imaging. Inverse Problems and Imaging, 2021, 15 (6) : 1363-1379. doi: 10.3934/ipi.2020051 |
| [4] |
Simon Hubmer, Andreas Neubauer, Ronny Ramlau, Henning U. Voss. On the parameter estimation problem of magnetic resonance advection imaging. Inverse Problems and Imaging, 2018, 12 (1) : 175-204. doi: 10.3934/ipi.2018007 |
| [5] |
Mostafa Bendahmane, Kenneth H. Karlsen. Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue. Networks and Heterogeneous Media, 2006, 1 (1) : 185-218. doi: 10.3934/nhm.2006.1.185 |
| [6] |
Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems and Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041 |
| [7] |
Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311 |
| [8] |
Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems and Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285 |
| [9] |
Qinghua Luo. Damped Klein-Gordon equation with variable diffusion coefficient. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3959-3974. doi: 10.3934/cpaa.2021139 |
| [10] |
Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391 |
| [11] |
Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 |
| [12] |
Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493 |
| [13] |
Bernard Bonnard, Olivier Cots, Jérémy Rouot, Thibaut Verron. Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging. Mathematical Control and Related Fields, 2020, 10 (1) : 47-88. doi: 10.3934/mcrf.2019029 |
| [14] |
Mikaela Iacobelli. Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 4929-4943. doi: 10.3934/dcds.2019201 |
| [15] |
Hong Lu, Ji Li, Mingji Zhang. Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3357-3371. doi: 10.3934/dcdsb.2020065 |
| [16] |
Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735 |
| [17] |
Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 |
| [18] |
Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019 |
| [19] |
Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663 |
| [20] |
Kin Ming Hui, Jinwan Park. Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5473-5508. doi: 10.3934/dcds.2021085 |
2021 Impact Factor: 1.483
Tools
Metrics
Other articles
by authors
[Back to Top]