= Phase-contrast magnetic resonance imaging
<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.<A flow velocity zeugmatographic interlace for NMR imaging in humans by P. R. Moran>[Primary abstract; flow-sensitive phase modulation and velocity information]{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, then
$$
\phi=\gamma\int G(t)x(t)\,dt=\gamma(x_0m_0+vm_1),\quad m_0=\int G(t)\,dt,\quad m_1=\int tG(t)\,dt.
$$
Here $\gamma$ is the <gyromagnetic ratio> in radians per second per tesla, and $m_1$ is the <gradient first moment>. With balanced zeroth moments, two encodings give
$$
\Delta\phi=\gamma v\Delta m_1.
$$
These 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 $10\,\mathrm{cm/s}$, a corrected phase difference of $+\pi/2$ corresponds to $+5\,\mathrm{cm/s}$ 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>.<Normal flow patterns of intracranial and spinal cerebrospinal fluid defined with phase-contrast cine MR imaging by D. R. Enzmann and others>[Abstract; 16 flow-encoded images per 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.
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