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

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)