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Accurate Flow MRI: the importance of velocity distribution asymmetry
Antoine Vallatos1, Haitham F. I. Al-Mubarak1, James M. Mullin1, and William M. Holmes1

1Glasgow Experimental MRI Centre, Institute of Neuroscience and Psychology, University of Glasgow, Glasgow, United Kingdom

### Synopsis

This work proposes a theoretical and experimental investigation into the unexplored effect of asymmetric distribution of intra-voxel velocities on the accuracy of Flow MRI. Our experimental results show that asymmetric velocity distributions can compromise the linearity of measured phase against applied gradient, leading to important velocimetry errors. A theoretical expression of the observed phase measurement errors is introduced, relating them to velocity distribution properties such as variance, skewness and kurtosis. This enables to explain previously reported velocimetry errors and propose solutions so as to increase the accuracy of velocity measurements.

### Introduction

Flow MRI or phase-shift velocimetry relies on a linear relation between the phase of the signal measured using pulsed filed gradient experiments, $φ$, and the average velocity $V$ within each voxel:

$$φ=\gamma\delta G\Delta V (1)$$

where $\gamma$ is the gyromagnetic ratio, $\delta$ the duration of the motion encoding gradients, G the gradient strength and $\Delta$ the time between two encoding gradients. The technique is widely used for measuring arterial blood velocities, and considerable effort is put in developing it into an accurate diagnostic tool for the assessment of various diseases.

Over the recent years, Flow MRI accuracy issues have been identified1, with reported underestimation of peak and average velocities. Inaccuracy is often attributed to partial volume effects caused by poor resolution, low SNR caused by inappropriate maximal velocity prediction or pulsatile flow effects. In this work we investigated, theoretically and experimentally, the unexplored effect of asymmetric distribution of intra-voxel velocities (caused by stagnating or differential flow) on the accuracy of Flow MRI.

### Methods

All experiments were carried on a 7T Bruker Biospec instrument using a Pulsed Field Gradient sequence. Phase-shift velocimetry and measurements of the probability distribution of molecular displacements Z during $\Delta$ (or propagator, $P(Z,\Delta)$ were performed on pipe flow, flow through a carotid phantom and in vivo measurements of flow in rat carotid. For pipe flow, syringe pumps were used and the maximum velocity varied from 0.1 mm s-1 to 1 mm s-1. For carotid phantom flow, A-mount suction-shoe gear pumps were used and the maximum velocity varied from 0.1 m s-1 to 1.5 m s-1. The effect of different experimental parameters was investigated.

### Results

Our results show when velocity distributions are not symmetric (Fig.1 a) the phase to velocity encoding gradient (q) plots exhibit non linear behaviour (Fig.1 b). Depending on the gradient value (calculated on the basis of the user defined value of maximum velocity) important velocimetry errors (more than 60%) were observed under certain parameter conditions (Fig.1 c). At typical flow rates encountered in vivo, velocimetry errors were related to the phase gradient non-linearity regions, where R2 of the fit deviated from 1 (Fig.1 d), themselves related to the presence of asymmetric velocity distributions (Fig.1 e).

### Discussion

The presence of asymmetric velocity distributions (Fig.1 a) was shown to compromise the phase to velocity encoding gradient (q) linearity (Fig.1 b), that is a condition for accurate phase-shift velocimetry. In such conditions, the linear relation between measured phase and velocity (equation 1) is no longer valid, leading to important velocimetry errors. A correction term was introduced (equation 2), allowing to quantitatively relate the observed velocimetry errors to the asymmetry of molecular displacement distribution (propagator) within the voxels expressed in terms of variance, skewness and kurtosis:

$$φ_{error}=tan^{-1} \left(\frac{\frac{-q^3}{3!} Skew(Z)}{1-\frac{-q^2}{2!} Var(Z)+\frac{q^4}{4!} Kurt(Z)}\right) (2)$$

### Conclusion

Our work demonstrated, both theoretically and experimentally, that asymmetries in the molecular velocity distribution within voxels can lead to important phase-shift velocimetry errors, compromising the accuracy of Flow MRI measurements. These errors were theoretically related to properties of intravoxel velocity distributions such as variance, skewness and kurtosis. This allowed to explain previously reported errors in velocity measurements and propose several ways in order to increase the accuracy of Flow MRI.

### Acknowledgements

No acknowledgement found.

### References

1. Toger J, Bidhult S, Revstedt J, Carlsson M, Arheden H, Heiberg E. Independent validation of four-dimensional flow MR velocities and vortex ring volume using particle imaging velocimetry and planar laser-Induced fluorescence. Magn Reson Med. 2016;75(3):1064-1075.

### Figures

Experiments with a single tube at 1.2 mm3 min-1 (symmetric distribution) and combined with a stationary flow tube (asymmetric distribution): (a) Propagators, (b) measured phase and (c) deviations from expected velocity values. Experiments with a single tube at 6.5 cm3 s-1: (d) Velocity map and map of the R2 value of the fit to the plot of phase against gradient. (e) Propagators for voxels A and B indicated in (d).

Proc. Intl. Soc. Mag. Reson. Med. 25 (2017)
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