= Velocity encoding limit
The <velocity encoding limit>, commonly written $v_{\mathrm{enc}}$, specifies the velocity scale represented by a phase difference of magnitude $\pi$ in a chosen <phase-contrast magnetic resonance imaging> encoding. With equal <gradient zeroth moments> and first-moment difference $\Delta m_1$, an ideal constant-velocity model gives
$$
\Delta\phi=\gamma v\Delta m_1,\qquad
v_{\mathrm{enc}}=\frac{\pi}{|\gamma\Delta m_1|}.
$$
The <gyromagnetic ratio> $\gamma$ is angular, in radians per second per tesla, and $\Delta m_1$ is the difference of <gradient first moments>. A sign convention must be specified separately. A 1982 original paper described obtaining motion information through phase-sensitive <nuclear magnetic resonance> encoding.<A flow velocity zeugmatographic interlace for NMR imaging in humans by P. R. Moran>[Abstract; phase encoding of flow information; displayed relations are the stated model]{ref}
Choose the convention $\Delta\phi=\pi v/v_{\mathrm{enc}}$. If $v_{\mathrm{enc}}=100\,\mathrm{cm/s}$, then $v=50\,\mathrm{cm/s}$ produces $\Delta\phi=\pi/2$. But $v=120\,\mathrm{cm/s}$ produces $1.2\pi$, which wraps to $-0.8\pi$ when phase is represented in $[-\pi,\pi)$. A reconstruction that ignores this <aliasing> would report $-80\,\mathrm{cm/s}$. The limit describes an encoding ambiguity, not a physical maximum speed of the fluid.
For a fixed phase uncertainty $\sigma_\phi$ in this simple model, propagation gives $\sigma_v=(v_{\mathrm{enc}}/\pi)\sigma_\phi$. Reducing $v_{\mathrm{enc}}$ improves velocity sensitivity under that assumption while narrowing the unambiguous range. For example, $\sigma_\phi=0.02$ radians and $v_{\mathrm{enc}}=100\,\mathrm{cm/s}$ give $\sigma_v\simeq0.64\,\mathrm{cm/s}$. Real measurements must also account for background phase, intravoxel variation and timing. Gated measurements of <cerebrospinal fluid> show why timing within the <cardiac cycle> matters even when net displacement is small.<Normal flow patterns of intracranial and spinal cerebrospinal fluid defined with phase-contrast cine MR imaging by D. R. Enzmann and others>[Abstract; 16 quantitative flow-encoded images per cardiac cycle]{ref}
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