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 givesThe 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.^[ref]

Choose the convention $\Delta \varphi =\pi v/v_{\mathrm{enc}}$. If $v_{\mathrm{enc}}=100cm/s$, then $v=50cm/s$ produces $\Delta \varphi =\pi /2$. But $v=120cm/s$ 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 $-80cm/s$. The limit describes an encoding ambiguity, not a physical maximum speed of the fluid.

For a fixed phase uncertainty ${\sigma }_{\varphi }$ in this simple model, propagation gives ${\sigma }_v=(v_{\mathrm{enc}}/\pi ){\sigma }_{\varphi }$. Reducing $v_{\mathrm{enc}}$ improves velocity sensitivity under that assumption while narrowing the unambiguous range. For example, ${\sigma }_{\varphi }=0.02$ radians and $v_{\mathrm{enc}}=100cm/s$ give ${\sigma }_v\simeq 0.64cm/s$. 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.^[ref]