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Key MRI Concepts and Equations#

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SEE ALSO MRI Math Concepts

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MRI Math Concepts

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Spin Physics#

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Larmor Frequency

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M0

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Polarization

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+\[M_{XY}(\vec{r},t) = M_{XY}(\vec{r},0) e^{-t/T_2(\vec{r})}\]
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+\[M_Z(\vec{r},t) = M_Z(\vec{r},0)e^{-t/T_1} + M_0(\vec{r})(1- e^{-t/T_1(\vec{r})})\]
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Contrast#

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T2/T2*

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T1

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spoiled GRE contrast

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+\[S \propto M_0 \sin(\theta) \exp(-TE/T_2) \frac{1- \exp(-TR/T_1)}{1- \cos(\theta) \exp(-TR/T_1)}\]
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Inversion Recovery +$\(S_{IR} \propto M_0 \exp(-TE/T_2) (1 - 2\exp(-TI/T_1) + \exp(-TR/T_1) )\)$

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RF Pulses#

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flip angle

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SAR

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TBW = BW_RF T_{RF}

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Slice thickness

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slice shifting

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Spatial Encoding#

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k-space

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+\[\vec{k}(t) = \frac{\gamma}{2\pi} \int_0^t \vec{G}(\tau) d\tau\]
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+\[M_{XY}(\vec{r}, t) = M_{XY}(\vec{r}, 0) e^{ -i 2 \pi \vec{k}(t) \cdot \vec{r} }\]
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+\[\begin{split}\begin{align} +s(t) & = \int_\mathrm{Volume} M_{XY}(\vec{r},t) \ d\vec{r} \\ + & = \int_{\textrm{Volume}} M_{XY}(\vec{r},0) \exp(-i2\pi \vec{k}(t) \cdot \vec{r}) \ d\vec{r} + \end{align}\end{split}\]
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+\[s(t) = \int m(\vec{r})\ e^{-i 2 \pi \vec{k}(t) \cdot \vec{r}} \ d\vec{r}\]
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+\[s(t) = \mathcal{F}\{m(\vec{r}) \} |_{\vec{k} = \vec{k}(t)}$ = M(\vec{k}(t))\]
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Image Characeristics#

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SNR

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FOV/resolution - in general, in Cartesian sequence

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MRI Signal Equation#

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Fast Imaging#

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Scan times

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effective TE

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