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Magnetic resonance velocity encoding makes a component of motion measurable through the phase of a nuclear magnetic resonance signal. Controlled magnetic field gradients give different phase shifts to spins with different trajectories. Phase-contrast magnetic resonance imaging compares encodings to extract a velocity component, rather than interpreting ordinary image brightness as speed.^[ref]

In an original idealized model, assume constant velocity $v$ along one encoding axis and effective gradients with equal gradient zeroth moments. Then the encoding difference obeyswhere $\gamma$ is the angular gyromagnetic ratio and $\Delta m_1$ is the gradient first moment difference. For $\gamma =2\pi \cdot 42.58\times 10^6\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$, $\Delta m_1=2\times 10^{-8}\mathrm{T}{\mathrm{s}}^2/m$ and $v=0.10m/s$, this gives $\Delta \varphi \simeq 0.535\mathrm{rad}$. These stipulated values illustrate the conversion, rather than a particular scanner experiment.

Reversing the moment difference reverses the phase sign under this convention. Reducing its magnitude decreases sensitivity to a given velocity while increasing the velocity encoding limit. A measured voxel can contain a range of velocities, and background phase or phase aliasing can complicate reconstruction.

The encoded component also differs from volumetric flow rate: estimating transported volume requires combining the appropriate normal velocity with area and time, not merely recording one peak speed.

Phase-contrast magnetic resonance imaging estimates motion from differences in the phase of nuclear magnetic resonance signals acquired with controlled magnetic field gradients. The encoding measures a component of velocity, with its sign set by the gradient and reconstruction convention. A 1982 paper described an early flow-sensitive phase-encoding approach.^[ref]

For a constant-velocity teaching model, write position along the encoding direction as $x(t)=x_0+vt$. If $G(t)$ is the effective gradient, including refocusing signs, thenHere $\gamma$ is the gyromagnetic ratio in radians per second per tesla, and $m_1$ is the gradient first moment. With balanced zeroth moments, two encodings giveThese equations are an explicit idealized derivation, not a claim that every measured voxel contains one uniform velocity.

Define the velocity encoding limit by $v_{\mathrm{enc}}=\pi /(\gamma |\Delta m_1|)$. If it is $10cm/s$, a corrected phase difference of $+\pi /2$ corresponds to $+5cm/s$ under the chosen sign convention. Phase wrapping modulo $2\pi$ causes an ambiguity beyond the encoding range, related to aliasing.

A 1991 gated study measured differently timed cerebrospinal fluid motion in ventricular and subarachnoid compartments over the cardiac cycle.^[ref] Opposite velocities during equal parts of a cycle can average to zero despite substantial oscillatory motion. A volumetric flow rate additionally requires integrating the normal velocity across a defined cross-section; peak speed alone does not supply transported volume.

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]