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

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 When dealing with deformable organs, the Bloch-Torrey equation is no longer valid.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

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

    Citation:

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  • 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 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)

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