deploy: 2a57023

MRI Signal Equation.html

@@ -495,7 +495,7 @@ <h2>Idealized Signal Equation<a class="headerlink" href="#idealized-signal-equat
 \[s(t) = \int m(\vec{r}) \  \exp\left( -i \gamma \int_0^t \vec{G}(\tau) \cdot \vec{r} \ d\tau \right) \  d\vec{r}\]</div>
 <p>using the notation <span class="math notranslate nohighlight">\(m(\vec{r}) = M_{XY}(\vec{r}, t=0)\)</span> for the initial transverse magnetization for reasons that will become clear soon.</p>
 <p>In many situations, an idealized signal equation is useful, particularly for understanding image formation.  The idealized signal equation neglects relaxation, RF coil profile, and off-resonance, but we will add them in later.</p>
-<p>This is simllified using the concept of k-space:</p>
+<p>This is simplified using the concept of k-space:</p>
 <div class="math notranslate nohighlight">
 \[s(t) = \int m(\vec{r})\  e^{-i 2 \pi \vec{k}(t) \cdot \vec{r}} \  d\vec{r}\]</div>
 <p>From this idealized formulation, it is easiest to see that the signal is the Fourier Transform of the transverse magnetization in the subject:</p>

Spin Physics.html

@@ -608,7 +608,7 @@ <h3>Relaxation Equations<a class="headerlink" href="#relaxation-equations" title
 <p>This equation means that the transverse magnetization decays from its original value (when <span class="math notranslate nohighlight">\(t=0\)</span>) down to 0, with an exponential time constant <span class="math notranslate nohighlight">\(T_2(\vec{r})\)</span>.</p>
 <p>For the longitudinal magnetization:</p>
 <div class="math notranslate nohighlight">
-\[M_Z(\vec{r},t) = M_Z(\vec{r},0)e^{-t/T_1} + M_0(\vec{r},)(1- e^{-t/T_1(\vec{r})})\]</div>
+\[M_Z(\vec{r},t) = M_Z(\vec{r},0)e^{-t/T_1} + M_0(\vec{r})(1- e^{-t/T_1(\vec{r})})\]</div>
 <p>This equation means that the longitudinal magnetization returns or recovers to its equilibrium value, <span class="math notranslate nohighlight">\(M_0(\vec{r})\)</span>, with an exponential time constant <span class="math notranslate nohighlight">\(T_1(\vec{r})\)</span>.</p>
 <p>Both are illustrated below.</p>
 <div class="cell docutils container">

_sources/Key MRI Concepts.md

@@ -0,0 +1,77 @@
+# Key MRI Concepts and Equations
+
+SEE ALSO MRI Math Concepts
+
+[MRI Math Concepts](./MRI%20Math%20Concepts.ipynb)
+
+## Spin Physics
+
+Larmor Frequency
+
+M0
+
+Polarization
+
+$$M_{XY}(\vec{r},t) = M_{XY}(\vec{r},0) e^{-t/T_2(\vec{r})}$$
+
+$$M_Z(\vec{r},t) = M_Z(\vec{r},0)e^{-t/T_1} + M_0(\vec{r})(1- e^{-t/T_1(\vec{r})})$$
+
+## Contrast
+
+T2/T2*
+
+T1
+
+spoiled GRE contrast
+
+$$S \propto M_0 \sin(\theta) \exp(-TE/T_2) \frac{1- \exp(-TR/T_1)}{1- \cos(\theta) \exp(-TR/T_1)}$$
+
+
+Inversion Recovery
+$$S_{IR} \propto M_0  \exp(-TE/T_2) (1 - 2\exp(-TI/T_1) + \exp(-TR/T_1) )$$
+
+## RF Pulses
+
+flip angle
+
+SAR
+
+TBW = BW_RF T_{RF}
+
+Slice thickness
+
+slice shifting
+
+
+## Spatial Encoding
+
+k-space
+
+$$\vec{k}(t) = \frac{\gamma}{2\pi} \int_0^t \vec{G}(\tau) d\tau$$
+
+$$M_{XY}(\vec{r}, t) = M_{XY}(\vec{r}, 0) e^{ -i 2 \pi \vec{k}(t) \cdot \vec{r} }$$ 
+
+$$\begin{align}
+s(t) & = \int_\mathrm{Volume} M_{XY}(\vec{r},t) \ d\vec{r} \\
+ & =  \int_{\textrm{Volume}} M_{XY}(\vec{r},0) \exp(-i2\pi \vec{k}(t) \cdot \vec{r}) \ d\vec{r}
+ \end{align}$$ 
+
+$$s(t) = \int m(\vec{r})\  e^{-i 2 \pi \vec{k}(t) \cdot \vec{r}} \  d\vec{r}$$
+
+
+$$s(t) = \mathcal{F}\{m(\vec{r}) \} |_{\vec{k} = \vec{k}(t)}$ = M(\vec{k}(t))$$
+
+## Image Characeristics
+
+SNR
+
+FOV/resolution - in general, in Cartesian sequence
+
+
+## MRI Signal Equation
+
+## Fast Imaging
+
+Scan times
+
+effective TE

_sources/MRI Signal Equation.ipynb

@@ -55,7 +55,7 @@
     "\n",
     "In many situations, an idealized signal equation is useful, particularly for understanding image formation.  The idealized signal equation neglects relaxation, RF coil profile, and off-resonance, but we will add them in later. \n",
     "\n",
-    "This is simllified using the concept of k-space:\n",
+    "This is simplified using the concept of k-space:\n",
     "\n",
     "$$s(t) = \\int m(\\vec{r})\\  e^{-i 2 \\pi \\vec{k}(t) \\cdot \\vec{r}} \\  d\\vec{r}$$\n",
     "\n",

_sources/Spin Physics.ipynb

@@ -220,7 +220,7 @@
     "\n",
     "For the longitudinal magnetization:\n",
     "\n",
-    "$$M_Z(\\vec{r},t) = M_Z(\\vec{r},0)e^{-t/T_1} + M_0(\\vec{r},)(1- e^{-t/T_1(\\vec{r})})$$\n",
+    "$$M_Z(\\vec{r},t) = M_Z(\\vec{r},0)e^{-t/T_1} + M_0(\\vec{r})(1- e^{-t/T_1(\\vec{r})})$$\n",
     "\n",
     "This equation means that the longitudinal magnetization returns or recovers to its equilibrium value, $M_0(\\vec{r})$, with an exponential time constant $T_1(\\vec{r})$.\n",
     "\n",