Inverse Problems and Imaging
2018, Volume 12, Issue 1: 125-152. Doi: 10.3934/ipi.2018005
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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: March 31, 2017
Revised: August 31, 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 When dealing with deformable organs, the Bloch-Torrey equation is no longer valid.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.
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.
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
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 3. Behavior of the function $S$ over one cardiac cycle. $T_s = 333$ms, $T_d = 667$ms
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 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)
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Elie Bretin
Imen Mekkaoui
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 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)