Antoine Vallatos^{1}, Haitham F. I. Al-Mubarak^{1}, James M. Mullin^{1}, and William M. Holmes^{1}

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.

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 identified^{1}, 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.

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 In such conditions, the linear relation between measured phase and velocity (equation 1) is no longer valid, leading to important velocimetry errors.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:

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

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.

Experiments with a single tube at 1.2 mm^{3} 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 cm^{3} s^{-1}: (d) Velocity map and map of the R^{2 }value of the fit to the plot of phase against gradient. (e) Propagators for voxels A and B indicated in (d).