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Impact of Residual Gradient Moments on Diffusion Weighted Imaging
Matthew J. Middione1, Michael Loecher1, Kévin Moulin1, and Daniel B. Ennis1

1Radiological Sciences Lab, Department of Radiology, Stanford University, Stanford, CA, United States

### Synopsis

Bulk motion corrupts diffusion measurements in the heart but can be mitigated by nulling first and second gradient moments. When using optimization methods to design diffusion gradient waveforms with moment nulling, imperfect gradient waveforms arise from discrete convergence criteria, which impart residual (non-zero) gradient moments. This leads to intravoxel dephasing, signal loss, and inaccurate ADC measurements. Herein, simulations show that residual M0≤10-2mT/m•s, M1≤10-4mT/m•s2, and M2≤10-5mT/m•s3 leads to ≤5% increase in ADC. This work defines convergence criteria requirements for residual gradient moments that enable faster optimizations and more accurate measurements of ADC when using optimization methods for cardiac DWI sequence design.

### Methods

Numerical simulations were performed in Matlab to analyze the impact of residual gradient moments on intravoxel phase dispersion. The measured ADC was calculated for each of the 10,000 simulated spins per voxel as ADC’ $=\frac{1}{-b}ln\bigg[abs\bigg(\sum_{n=1}^{10,000}\frac{S_{DWI}^{n}}{S_{0}}\bigg)\bigg]$, for b=1000mm2/s, ADC=3x10-3mm2/s, and $S_{0}$=10,000, representing the cumulative signal within the voxel in the absence of diffusion encoding gradients and intravoxel phase dispersion. The diffusion weighted signal for the nth spin was calculated as $S_{DWI}^{n}=e^{-bADC}e^{-i\phi_{n}}$, with $\phi_{n}=\gamma r_{n}M_{0}+\gamma v_{n}M_{1}+\gamma a_{n}M_{2}$, $\gamma$ as the gyromagnetic ratio, $r_{n}$, $v_{n}$, and $a_{n}$ as the simulated position, velocity and acceleration of the nth spin, and M0, M1 and M2 as the zero, first, and second order residual gradient moments. To analyze the impact of residual M0, ADC' was computed for different pixel sizes (1-10mm) with residual values of M0 and M1=M2=0. To analyze the impact of residual M1, ADC’ was computed for simulated intravoxel velocity gradients, of varying maximum velocities (0-0.2m/s), with residual values of M1 and M0=M2=0. To analyze the impact of residual M2, ADC’ was computed for simulated intravoxel acceleration gradients, of varying maximum accelerations (0-1m/s2), with residual values of M2 and M0=M1=0. The following residual gradient moments were used in the simulations: 0, 10-6, 10-5, 10-4, 10-3, 10-2, 10-1, 100, and 10 with units of mT/m•s for M0, mT/m•s2 for M1 and mT/m•s3 for M2.

### Results

The impact of residual gradient moments on the measured ADC is shown in Figure 1 as a function of residual M0 and pixel size (Figure 1A), residual M1 and intravoxel velocity gradients (Figure 1B) and residual M2 and intravoxel acceleration gradients (Figure 1C).

### Conclusion

Residual gradient moments can lead to an increase in the measured ADC in DWI due to intravoxel signal dephasing. This work helps to define acceptable thresholds for residual gradient moments, which can be used to enable fast and more accurate measurements of ADC when using optimization methods for the pulse sequence design of diffusion sequences.

### Acknowledgements

Funding NIH R01 HL131975 and HL131823 to DBE.

### References

1. Aliotta et al. Convex optimized diffusion encoding (CODE) gradient waveforms for minimum echo time and bulk motion compensated diffusion weighted MRI. Magn Reson Med 2017;77:717–729.
2. Aliotta et al. Eddy current–nulled Convex Optimized Diffusion Encoding (EN-CODE) for distortion-free diffusion tensor imaging with short echo times. Magn Reson Med 2018;79:663–672.
3. Yang et al. Eddy current nulled constrained optimization of isotropic diffusion encoding gradient waveforms. Magn Reson Med 2018;00:1–15.
4. Sjölund et al. Constrained optimization of gradient waveforms for generalized diffusion encoding. J Magn Reson 2015;261:157-168.
5. Loecher et al. Accelerating 4D-Flow Acquisitions by Reducing TE and TR with Optimized RF and Gradient Waveforms. ISMRM 2018

### Figures

Figure 1: Numerical simulations showing the impact on the measured ADC (ADC’) arising from intravoxel phase dispersion (signal loss) as a function of residual M0 and pixel size (A), residual M1 and intravoxel velocity gradients (B), and residual M2 and intravoxel acceleration gradients (C). Measured ADC’ values that vary ≤5% compared to the simulated ADC (3x10-3mm2/s) and b-value (1000mm2/s), are highlighted by the black borders (lower-left area within plots).

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
3496