Filip Szczepankiewicz^{1,2,3}, Irvin Teh^{4}, Erica Dall'Armellina^{4}, Sven Plein^{4}, Jurgen E. Schneider^{4}, and Carl-Fredrik Westin^{2,3}

^{1}Clinical Sciences Lund, Lund University, Lund, Sweden, ^{2}Radiology, Brigham and Women's Hospital, Boston, MA, United States, ^{3}Harvard Medical School, Boston, MA, United States, ^{4}Leeds Institute of Cardiovascular and Metabolic Medicine, University of Leeds, Leeds, United Kingdom

Motion compensation is vital for cardiac diffusion MRI. In this paper we propose an optimized gradient waveform design that allows tensor-valued diffusion encoding with motion compensation. We demonstrate that it works for in vivo cardiac imaging and we show that it is more efficient than previous designs.

$$$~~~~~$$$In this work we propose a design that uses asymmetric numerically optimized gradient waveforms with nulling of arbitrary moments, and we show that this design improves the encoding efficiency and removes several technical limitations of previous designs.

$$\mathbf{m}_n=\gamma \int_0^{\mathrm{TE}}\mathbf{g}(t)\cdot t^n\mathrm{d}t,$$

to a limit $$$||\mathbf{m}_n||<L_n$$$, where $$$\gamma$$$ is the gyromagnetic ratio and $$$t$$$ is the time. We note that

$$$~~~~~$$$To demonstrate the feasibility of this approach, motion compensated waveforms $$$(\mathbf{m}_0,~\mathbf{m}_1,~\mathbf{m}_2)$$$ that yield linear and planar b-tensors [3] were used for diffusion weighted imaging of a healthy heart in vivo. The moment limits were $$$L_0=0,~L_1=10^{-4},~L_2=10^{-4}$$$ in units of s

$$$~~~~~$$$The impact of motion compensation in the heart was evaluated in terms of the diffusion weighted signal averaged over multiple rotations of planar and linear b-tensor encoding, as well as in terms of the calculated mean diffusivity (MD).

$$$~~~~~$$$The encoding efficiency of waveforms with variable limits on motion encoding was investigated in terms of the maximal b-value achievable for a fix encoding duration, and the necessary encoding time to reach

$$$~~~~~$$$As expected, the encoding performance was reduced as higher moment nulling was enforced (Fig. 5). However, the novel design outperformed the previous design, shortening the encoding times by up to 23 ms under the given premise. For a larger timing asymmetry, e.g. when ZOOM-it is not used, the advantage becomes even more pronounced (data not shown).

- improved performance over previous designs (shorter echo-times and/or higher SNR)
- more flexible, allowing for arbitrary distribution of the encoding time
- automatically account for crushing and therefore achieve more accurate compensation
- ensures Maxwell compensation for arbitrary rotations of the waveforms
- ensures Maxwell compensation even for systems with prominent gradient non-linearity [6]

[1] U. Gamper, P. Boesiger, and S. Kozerke, “Diffusion imaging of the in vivo heart using spin echoes-considerations on bulk motion sensitivity.” MRM, 2007.

[2] C. Stoeck, C. von Deuster, M. Genet, D. Atkinson, and S. Kozerke, “Second-order motion-compensated spin echo diffusion tensor imaging of the human heart.” MRM, 2016.

[3] C-F. Westin, H. Knutsson, O. Pasternak, F. Szczepankiewicz, E. Özarslan, D. van Westen, C. Mattisson, M. Bogren, L. O’Donnell, M. Kubicki, D. Topgaard, and M. Nilsson, “Q-space trajectory imaging for multidimensional diffusion MRI of the human brain.” MRM, 2019.

[4] S. Lasič, F. Szczepankiewicz, E. Dall’Armellina, A. Das, C. Kelly, S. Plein, J. E. Schneider, M. Nilsson, and I. Teh, “Motion compensated b-tensorencoding for in vivo cardiac diffusion-weighted imaging.” NMR in Biomed, 2019.

[5] J. Sjölund, F. Szczepankiewicz, M. Nilsson, D. Topgaard, C.-F. Westin, and H. Knutsson, “Constrained optimization of gradient waveforms for generalized diffusion encoding,” JMR, 2015.

[6] F. Szczepankiewicz, C.-F. Westin, and M. Nilsson, “Maxwell-compensated design of asymmetric gradient waveforms for tensor-valued diffusion encod-ing.” MRM, 2019.

[7] F. Szczepankiewicz, C. Eichner, A. Anwander, C.-F. Westin, and M. Paquette, “The effect of concomitant fields caused by gradient non-linearity when using M- and K-nulled maxwell compensation,” Proc. Int. Soc. Magn. Reson. Med. (no. 1547, submitted), 2020.

[8] F. Nery, F. Szczepankiewicz, L. Kerkelä, M. G. Hall, E. Kaden, I. Gordon,D. L. Thomas, and C. A. Clark, “In vivo demonstration of microscopic anisotropy in the human kidney using multidimensional diffusion MRI." MRM, 2019.

Figure 1 - Examples of the proposed asymmetric design (top four rows) and the symmetric design by Lasic et al. [4] (bottom row). Here, we assume that the first encoding period is 3 ms longer than the second to simulate a weak timing asymmetry. The top row is similar to previous designs that yield linear encoding with motion compensation. The total duration to yield *b* = 2 ms/μm^{2} is given on each x-axis. The notation in parenthesis denotes waveforms constrained within a sphere (L2 norm) or a cube (Max norm) [5].

Figure 2 - Linear tensor encoded signal averaged over directions using m_{0}, m_{1} and m_{2}-nulling for b-values up to 0.7 ms/μm^{2} shows that motion compensation is necessary for cardiac imaging. Non-compensated waveforms suffer gross signal loss due to motion, whereas the proposed motion compensated method retains the signal even at relatively high b-values.

Figure 3 - Planar tensor encoded signal averaged over directions using m_{0}, m_{1} and m_{2}-nulling for b-values up to 0.7 ms/μm^{2} shows the same patterns as for the linear encoding in Fig. 2.

Figure 4 - Maps of mean diffusivity clearly show that waveforms that are not compensated for motion cannot be used for in vivo cardiac imaging and quantification of diffusivity. Waveforms that are m_{1}-nulled (velocity) are a marked improvement, however some regions still exhibit elevated MD that are likely to be artifacts. As expected, m_{2}-nulled (velocity and acceleration) waveforms appear the most robust.

Figure 5 - The achievable b-value is reduced as higher moments of the waveform are constrained. Plots show b-values for encoding duration *δ*_{1 }= 33 ms, *δ*_{2 }= 30 ms, separated by 8 ms (for refocusing). Numerical values next to each marker show the minimal encoding duration, in ms, required to achieve *b *= 2 ms/μm^{2}. The proposed optimization is more efficient than Lasic et al. [4], resulting in encoding times that are shorter by 11 to 23 ms for spherical encoding (right plot).