Phase contrast magnetic resonance imaging (PC-MRI) is a specific type of magnetic resonance imaging used primarily to determine flow velocities. PC-MRI can be considered a method of Magnetic Resonance Velocimetry. It also provides a method of magnetic resonance angiography. Since modern PC-MRI is typically time-resolved, it provides a means of 4D imaging (three spatial dimensions plus time).
How it Works Atoms with an odd number of protons or neutrons have a randomly aligned angular spin momentum. When placed in a strong magnetic field, some of these spins align with the axis of the external field, which causes a net 'longitudinal' magnetization. These spins precess about the axis of the external field at a frequency proportional to the strength of that field. Then, energy is added to the system through a Radio frequency (RF) pulse to 'excite' the spins, changing the axis that the spins precess about. These spins can then be observed by receiver coils (Radiofrequency coils) using Faraday's law of induction. Different tissues respond to the added energy in different ways, and imaging parameters can be adjusted to highlight desired tissues. All of these spins have a phase that is dependent on the atom's velocity. Phase shift ( ϕ ) {\displaystyle (\phi )} of a spin is a function of the gradient field G ( t ) {\displaystyle \mathbf {G} (t)} :
ϕ = γ ∫ 0 t B 0 + G ( τ ) ⋅ r ( τ ) d τ {\displaystyle \phi =\gamma \int _{0}^{t}B_{0}+\mathbf {G} (\tau )\cdot \mathbf {r} (\tau )d\tau }
where γ {\displaystyle \gamma } is the Gyromagnetic ratio and r {\displaystyle \mathbf {r} } is defined as:
r ( τ ) = r 0 + v r τ + 1 2 a r τ 2 + … {\displaystyle \mathbf {r} (\tau )=\mathbf {r} _{0}+\mathbf {v} _{r}\tau +{\frac {1}{2}}\mathbf {a} _{r}\tau ^{2}+\ldots } ,
r 0 {\displaystyle \mathbf {r} _{0}} is the initial position of the spin, v r {\displaystyle \mathbf {v} _{r}} is the spin velocity, and a r {\displaystyle \mathbf {a} _{r}} is the spin acceleration. If we only consider static spins and spins in the x-direction, we can rewrite equation for phase shift as:
ϕ = γ x 0 ∫ 0 t G x ( τ ) d τ + γ v x ∫ 0 t G x ( τ ) τ d τ + γ a x 2 ∫ 0 t G x ( τ ) τ 2 d τ + … {\displaystyle \phi =\gamma x_{0}\int _{0}^{t}G_{x}(\tau )d\tau +\gamma v_{x}\int _{0}^{t}G_{x}(\tau )\tau d\tau +\gamma {\frac {a_{x}}{2}}\int _{0}^{t}G_{x}(\tau )\tau ^{2}d\tau +\ldots }
We then assume that acceleration and higher order terms are negligible to simplify the expression for phase to:
ϕ = γ ( x 0 M 0 + v x M 1 ) {\displaystyle \phi =\gamma (x_{0}M_{0}+v_{x}M_{1})}
where M 0 {\displaystyle M_{0}} is the zeroth moment of the x-gradient and M 1 {\displaystyle M_{1}} is the first moment of the x gradient. If we take two different acquisitions with applied magnetic gradients that are the opposite of each other (bipolar gradients), we can add the results of the two acquisitions together to calculate a change in phase that is dependent on gradient:
Δ ϕ = v ( γ Δ M 1 ) {\displaystyle \Delta \phi =v(\gamma \Delta M_{1})}
where Δ M 1 = 2 M 1 {\displaystyle \Delta M_{1}=2M_{1}} .
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