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 When dealing with deformable organs, the Bloch-Torrey equation is no longer valid.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.
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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
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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Spin echo diffusion encoding sequence. Two identical gradients are applied around the
(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
Behavior of the function
STEAM diffusion encoding sequence
(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
Diffusion images reconstructed in systole.
Diffusion images reconstructed in diastole.
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.
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
Diffusion images reconstructed with different values of
The exact diffusion presented on an irregular ring
Diffusion images reconstructed at: