# American Institute of Mathematical Sciences

• Previous Article
Superconductive and insulating inclusions for linear and non-linear conductivity equations
• IPI Home
• This Issue
• Next Article
Recovery of block sparse signals under the conditions on block RIC and ROC by BOMP and BOMMP
January  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

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 When dealing with deformable organs, the Bloch-Torrey equation is no longer valid.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

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:

show all references

##### References:
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
(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
Behavior of the function $S$ over one cardiac cycle. $T_s = 333$ms, $T_d = 667$ms
STEAM diffusion encoding sequence
$\|D \mathbf{u}\|_2$ calculated during the application of the diffusion encoding gradients for different values of
(Top) Diffusion MRI images at different moments of cardiac cycle. (Bottom) Exact diffusion coefficient
(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
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
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
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
Exact diffusion
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
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
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
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
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
The exact diffusion presented on an irregular ring
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 & 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 & Imaging, 2008, 2 (2) : 205-224. doi: 10.3934/ipi.2008.2.205 [3] Simon Hubmer, Andreas Neubauer, Ronny Ramlau, Henning U. Voss. On the parameter estimation problem of magnetic resonance advection imaging. Inverse Problems & Imaging, 2018, 12 (1) : 175-204. doi: 10.3934/ipi.2018007 [4] Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041 [5] Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems & Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285 [6] 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 & Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311 [7] Mostafa Bendahmane, Kenneth H. Karlsen. Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue. Networks & Heterogeneous Media, 2006, 1 (1) : 185-218. doi: 10.3934/nhm.2006.1.185 [8] Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391 [9] Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 [10] Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493 [11] Bernard Bonnard, Olivier Cots, Jérémy Rouot, Thibaut Verron. Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019029 [12] Mikaela Iacobelli. Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4929-4943. doi: 10.3934/dcds.2019201 [13] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [14] Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735 [15] Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019 [16] Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663 [17] Bastian Harrach. Simultaneous determination of the diffusion and absorption coefficient from boundary data. Inverse Problems & Imaging, 2012, 6 (4) : 663-679. doi: 10.3934/ipi.2012.6.663 [18] Bernard Bonnard, Monique Chyba, Alain Jacquemard, John Marriott. Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Mathematical Control & Related Fields, 2013, 3 (4) : 397-432. doi: 10.3934/mcrf.2013.3.397 [19] M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079 [20] Kin Ming Hui. Collasping behaviour of a singular diffusion equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2165-2185. doi: 10.3934/dcds.2012.32.2165

2018 Impact Factor: 1.469