Matthew J. Middione^{1}, Michael Loecher^{1}, Kévin Moulin^{1}, and Daniel B. Ennis^{1}

^{1}Radiological 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 M_{0}≤10^{-2}mT/m•s, M_{1}≤10^{-4}mT/m•s^{2}, and M_{2}≤10^{-5}mT/m•s^{3} 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.

### Introduction

Diffusion-weighted imaging (DWI) measurements provide clinical
value, but sensitivity to bulk motion frequently contributes to signal losses
that can confound diffusion measurements, especially in moving tissues like the
heart. Bulk motion sensitivity can be mitigated by using
velocity (M_{1}=0) and/or acceleration-compensated (M_{1}=M_{2}=0)
gradient waveforms to null the intravoxel phase dispersion from the bulk (coherent)
motion of spins while preserving the intravoxel phase dispersion from incoherent
diffusing spins. 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. The intrinsic
symmetry of conventional diffusion encoding strategies ensure exactly zero
residual gradient moments as the waveforms are equal and opposite on either
side of the refocusing pulse. However, optimization methods, which generate
asymmetric gradient waveforms^{1-5}, are increasingly used for time
optimal gradient waveform design and/or to add additional constraints that
mitigate bulk motion, eddy currents, and concomitant field terms. In designing
an optimization scheme, it is important to identify optimization criteria that
enable fast solutions to be obtained. If convergence criteria are unnecessarily
strict, then convergence times can be unacceptably long (i.e. minutes), whereas
real-time (i.e. less than 10-50ms) solve times are preferred. The purpose of
this work was to define acceptable limits for residual gradient moments that
confer ≤5% measurement
bias for ADC in the presence of bulk motion.### 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=1000mm^{2}/s, ADC=3x10^{-3}mm^{2}/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
n^{th} 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 n^{th} spin, and M_{0}, M_{1} and M_{2} as the zero, first, and
second order residual gradient moments. To analyze the impact of residual M_{0}, ADC'
was computed for different pixel sizes (1-10mm) with residual
values of M_{0} and M_{1}=M_{2}=0. To analyze the
impact of residual M_{1}, ADC’ was computed for simulated intravoxel
velocity gradients, of varying maximum velocities (0-0.2m/s), with residual values of M_{1}
and M_{0}=M_{2}=0. To analyze the impact of residual M_{2},
ADC’ was computed for simulated intravoxel acceleration gradients, of varying
maximum accelerations (0-1m/s^{2}), with residual values of M_{2} and M_{0}=M_{1}=0.
The following residual gradient moments were used in the simulations: 0, 10^{-6},
10^{-5}, 10^{-4}, 10^{-3}, 10^{-2}, 10^{-1},
10^{0}, and 10 with units of mT/m•s for M_{0},
mT/m•s^{2}
for M_{1} and mT/m•s^{3}
for M_{2}.### Results

The impact of residual gradient moments on the measured ADC is shown
in Figure 1 as a function of residual M_{0} and
pixel size (Figure 1A), residual M_{1} and
intravoxel velocity gradients (Figure 1B) and residual M_{2}
and intravoxel acceleration gradients (Figure 1C).### Discussion

_{}
Achieving precise gradient moment nulling is difficult. The simulations in
this work show that knowledge of the pixel size and expected tissue motion can
help define acceptable residual moment nulling thresholds while maintaining ADC
measurements within 5%. The degree of intravoxel signal dephasing depends
on the product of intravoxel velocity and acceleration gradients and any
residual gradient moment(s). Residual M_{0} will always lead to an
increase in the measured ADC due to intravoxel signal dephasing. A negligible (≤5%) increase in the measured ADC is observed
when the residual M_{0} is on the order of 10^{-2}mT/m•s for all simulated pixel sizes
(0-10mm) and 10^{-1}mT/m•s for simulated pixel sizes ≤4.5mm. Hence, higher resolution imaging is less
prone to bulk motion artifacts. Residual M_{1} and M_{2} similarly
lead to an increase in the measured ADC. A negligible (≤5%) increase in the measured ADC is observed
when the residual M_{1} is on the order of 10^{-4}mT/m•s^{2} or when the residual M_{2} is
on the order of 10^{-5}mT/m•s^{3}. Defining acceptable thresholds
for residual gradient moments is important when using convex optimized
diffusion encoding gradient design approaches, where the residual moments need
to be specified as a constraint in the optimization. Setting these values to a
specific non-zero, but nearly zero value ensures minimal signal dephasing while
also providing shorter optimization convergence times. ### 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

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