deploy: 8c14724

MR Physics - Bloch Equation.html

@@ -417,15 +417,21 @@ <h2> Contents </h2>
                 <ul class="visible nav section-nav flex-column">
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning Goals</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-bloch-equation">THE Bloch Equation</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#precession">Precession</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#precession">Precession</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simulation-of-precession">Simulation of Precession</a></li>
+</ul>
+</li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rf-excitation">RF Excitation</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example-constant-amplitude-pulse">Example: Constant amplitude pulse</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lab-versus-rotating-frame">Lab versus Rotating Frame</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#common-flip-angle-rf-excitations">Common flip angle RF excitations</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simulations">Simulations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simulations-of-rf-excitation">Simulations of RF Excitation</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#relaxation">Relaxation</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simulation-of-relaxation">Simulation of Relaxation</a></li>
 </ul>
 </li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#relaxation">Relaxation</a></li>
 </ul>
             </nav>
         </div>
@@ -511,6 +517,8 @@ <h2>THE Bloch Equation<a class="headerlink" href="#the-bloch-equation" title="Pe
 <li><p><span class="math notranslate nohighlight">\(T_2(\vec{r})\)</span> - the transverse (<span class="math notranslate nohighlight">\(M_{XY}\)</span>) or spin-spin relaxation time constant</p></li>
 </ul>
 <p>All of which can vary across our subject (and all a valuable source of contrast!).</p>
+<p>There is a very full-featured, interactive Bloch Equation Simulator available online, that is valuable to understand the behavior of the net magnetization:<br />
+<a class="reference external" href="https://www.drcmr.dk/BlochSimulator/">Bloch Equation Simulator</a></p>
 </section>
 <section id="precession">
 <h2>Precession<a class="headerlink" href="#precession" title="Permalink to this heading">#</a></h2>
@@ -523,6 +531,9 @@ <h2>Precession<a class="headerlink" href="#precession" title="Permalink to this
 \gamma \vec{M}(t) \times \vec{B}(t)
 \]</div>
 <p><img alt="RF reception" src="_images/RF_reception.gif" /></p>
+<section id="simulation-of-precession">
+<h3>Simulation of Precession<a class="headerlink" href="#simulation-of-precession" title="Permalink to this heading">#</a></h3>
+<p>Open up the <a class="reference external" href="https://www.drcmr.dk/BlochSimulator/">Bloch Equation Simulator</a>.  You will see a visualization of a net magnetization vector that is precessing around the magnetic field (thin line).</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="n">B0</span> <span class="p">=</span> <span class="mf">1.5e3</span><span class="p">;</span> <span class="c">% 1.5 T = 1500 mT</span>
@@ -567,6 +578,7 @@ <h2>Precession<a class="headerlink" href="#precession" title="Permalink to this
 </div>
 </div>
 </section>
+</section>
 <section id="rf-excitation">
 <h2>RF Excitation<a class="headerlink" href="#rf-excitation" title="Permalink to this heading">#</a></h2>
 <p>RF excitation occurs when an oscillating magnetic field is applied orthogonally to the main magnetic field.  If we apply RF at the Larmor frequency, the magnetic field would be</p>
@@ -596,11 +608,12 @@ <h3>Lab versus Rotating Frame<a class="headerlink" href="#lab-versus-rotating-fr
 \end{bmatrix}\end{split}\]</div>
 <p>Now we can analyze RF excitation as a rotation around RF magnetic field amplitude and do not have to include Larmor frequency in our analysis, shown in the illustration below:</p>
 <p><img alt="RF excitation Rotating Frame" src="_images/RF_hard_rotating_frame_on-resonance.gif" /></p>
-<p>For another illustration of the stationary/lab versus rotating frames, try the “Change Frame” option in this Bloch simulator:</p>
-<p><a class="reference external" href="http://drcmr.dk/BlochSimulator/">http://drcmr.dk/BlochSimulator/</a></p>
 </section>
 <section id="common-flip-angle-rf-excitations">
 <h3>Common flip angle RF excitations<a class="headerlink" href="#common-flip-angle-rf-excitations" title="Permalink to this heading">#</a></h3>
+<p>For a constant amplitude RF pulse, the flip angle depends on the duration of the RF pulse, <span class="math notranslate nohighlight">\(T_{rf}\)</span> and the strength of the RF magnetic field, <span class="math notranslate nohighlight">\(b_{1,0}\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\theta = \gamma  b_{1,0} T_{rf} \]</div>
 <table class="table">
 <thead>
 <tr class="row-odd"><th class="head text-center"><p><img alt="RF 45-degree flip" src="_images/RF_45flip.gif" /></p></th>
@@ -616,14 +629,16 @@ <h3>Common flip angle RF excitations<a class="headerlink" href="#common-flip-ang
 </tbody>
 </table>
 </section>
-<section id="simulations">
-<h3>Simulations<a class="headerlink" href="#simulations" title="Permalink to this heading">#</a></h3>
-<p>The following Bloch equation simulations show</p>
+<section id="simulations-of-rf-excitation">
+<h3>Simulations of RF Excitation<a class="headerlink" href="#simulations-of-rf-excitation" title="Permalink to this heading">#</a></h3>
+<p>Again, open up the <a class="reference external" href="https://www.drcmr.dk/BlochSimulator/">Bloch Equation Simulator</a>.</p>
 <ol class="arabic simple">
-<li><p>First, when a non-resonant magnetic field is applied.   An additional magnetic field is applied orthogonal to the main magnetic field, but <em>not</em> applied at the Larmor frequency, and there is no creation of transverse magnetization.</p></li>
-<li><p>After, this is corrected, and the RF pulse is applied at the Larmor frequency, <span class="math notranslate nohighlight">\(\omega_0 = \gamma B_0\)</span>.  With a resonant RF pulse, we have excitation of the net magnetization away from the direction of the main magnetic field, and creation of transverse magnetization, <span class="math notranslate nohighlight">\(M_X\)</span> and <span class="math notranslate nohighlight">\(M_Y\)</span>.</p></li>
-<li><p>Finally, the simulation is converted into the rotating from.  It is hard to visualize the transverse magnetization in the lab because it is rotating at the Larmor frequency.  The excitation is more clearly visualized in the rotating frame</p></li>
+<li><p>Lab versus rotating frame:  The default view is in the lab (stationary) frame.  If you select the ‘B0’ option from the ‘Frame’ in the top left it will change to the rotating frame.</p></li>
+<li><p>RF Excitation: Change to ‘Equilibrium’ scene in bottom left.  Then, use the ‘90x hard’ button to apply a constant amplitude pulse.</p></li>
+<li><p>Other flip angles: Go back to ‘Equilibrium’ scene, and try the other hard RF pulse flip angles.</p></li>
 </ol>
+<p>Below are additional Bloch equation simulations and associated code for RF excitation show non-resonant magnetic fields, resonant magnetic fields, and excitation in the rotating frame.</p>
+<p>First, when a non-resonant magnetic field is applied.   An additional magnetic field is applied orthogonal to the main magnetic field, but <em>not</em> applied at the Larmor frequency, and there is no creation of transverse magnetization.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="c">% lab frame</span>
@@ -686,6 +701,7 @@ <h3>Simulations<a class="headerlink" href="#simulations" title="Permalink to thi
 <img alt="_images/60e7d3679630f0b0d959bdef8300d6cf3abfda46cfae772509150d26b636c34f.png" src="_images/60e7d3679630f0b0d959bdef8300d6cf3abfda46cfae772509150d26b636c34f.png" />
 </div>
 </div>
+<p>To achieve excitation the RF pulse is applied at the Larmor frequency, <span class="math notranslate nohighlight">\(\omega_0 = \gamma B_0\)</span>.  With a resonant RF pulse, we have excitation of the net magnetization away from the direction of the main magnetic field, and creation of transverse magnetization, <span class="math notranslate nohighlight">\(M_X\)</span> and <span class="math notranslate nohighlight">\(M_Y\)</span>.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="c">% RF pulse at Larmor frequency</span>
@@ -718,6 +734,7 @@ <h3>Simulations<a class="headerlink" href="#simulations" title="Permalink to thi
 <img alt="_images/5a13ff251dea32a7faaac41626649ee77b778d3c79c7932059c146f3ce325276.png" src="_images/5a13ff251dea32a7faaac41626649ee77b778d3c79c7932059c146f3ce325276.png" />
 </div>
 </div>
+<p>Finally, the simulation is converted into the rotating frame.  It is hard to visualize the transverse magnetization in the lab because it is rotating at the Larmor frequency.  The excitation is more clearly visualized in the rotating frame.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="c">% rotating frame</span>
@@ -800,6 +817,17 @@ <h2>Relaxation<a class="headerlink" href="#relaxation" title="Permalink to this
 <div class="math notranslate nohighlight">
 \[M_Z(t) = M_Z(0)e^{-t/T_1} + M_0(1- e^{-t/T_1})\]</div>
 <p>Where here the shorthand complex notation for the transverse magnetization is being used: <span class="math notranslate nohighlight">\(M_{XY}(\vec{r},t) = M_X(\vec{r},t) + i M_Y(\vec{r},t)\)</span></p>
+<p><strong>T1 and T2 relaxation after RF Excitation</strong></p>
+<p><img alt="Relaxation T1 and T2" src="_images/relaxation_t1t2.gif" /></p>
+<section id="simulation-of-relaxation">
+<h3>Simulation of Relaxation<a class="headerlink" href="#simulation-of-relaxation" title="Permalink to this heading">#</a></h3>
+<p>One more time, open up the <a class="reference external" href="https://www.drcmr.dk/BlochSimulator/">Bloch Equation Simulator</a>.  Under the ‘Relaxation’ options in the top left, you can adjust T1 and T2 relaxation rates.</p>
+<ol class="arabic simple">
+<li><p>Experiment with different T1 and T2 relaxation values</p></li>
+<li><p>When the magnetization turns to equilibrium, use a RF pulse and you will see relaxation occuring again.</p></li>
+<li><p>From Equilibrium, try a 180-degree flip angle and try adjusting both T1 and T2.  Which parameter influence the relaxation in this situation?</p></li>
+</ol>
+<p>Below are additional Bloch equation simulations and associated code of relaxation.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-octave notranslate"><div class="highlight"><pre><span></span><span class="n">t</span> <span class="p">=</span> <span class="nb">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="c">% s</span>
@@ -856,8 +884,7 @@ <h2>Relaxation<a class="headerlink" href="#relaxation" title="Permalink to this
 <img alt="_images/02ce669ad0b5dd91283feee8b19371dc6510c2ed6d0eca05a5454529a4eeeab6.png" src="_images/02ce669ad0b5dd91283feee8b19371dc6510c2ed6d0eca05a5454529a4eeeab6.png" />
 </div>
 </div>
-<p><strong>T1 and T2 relaxation after RF Excitation</strong></p>
-<p><img alt="Relaxation T1 and T2" src="_images/relaxation_t1t2.gif" /></p>
+</section>
 </section>
 </section>
 
@@ -926,15 +953,21 @@ <h2>Relaxation<a class="headerlink" href="#relaxation" title="Permalink to this
     <ul class="visible nav section-nav flex-column">
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning Goals</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-bloch-equation">THE Bloch Equation</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#precession">Precession</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#precession">Precession</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simulation-of-precession">Simulation of Precession</a></li>
+</ul>
+</li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rf-excitation">RF Excitation</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example-constant-amplitude-pulse">Example: Constant amplitude pulse</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#lab-versus-rotating-frame">Lab versus Rotating Frame</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#common-flip-angle-rf-excitations">Common flip angle RF excitations</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simulations">Simulations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simulations-of-rf-excitation">Simulations of RF Excitation</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#relaxation">Relaxation</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simulation-of-relaxation">Simulation of Relaxation</a></li>
 </ul>
 </li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#relaxation">Relaxation</a></li>
 </ul>
   </nav></div>
 

MRI Signal Equation.html

@@ -579,8 +579,8 @@ <h2>Relaxation during signal acquisition<a class="headerlink" href="#relaxation-
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="_images/04e32389def1da9556efd712d119655989abaa4ab970ce8cc166a8125b4a22c0.png" src="_images/04e32389def1da9556efd712d119655989abaa4ab970ce8cc166a8125b4a22c0.png" />
 <img alt="_images/92ac396d1094116faca36e071635f22ff14da490df79aa5b1e645af612d2ec53.png" src="_images/92ac396d1094116faca36e071635f22ff14da490df79aa5b1e645af612d2ec53.png" />
+<img alt="_images/04e32389def1da9556efd712d119655989abaa4ab970ce8cc166a8125b4a22c0.png" src="_images/04e32389def1da9556efd712d119655989abaa4ab970ce8cc166a8125b4a22c0.png" />
 </div>
 </div>
 <p>Thus the reconstructed image will corrupted by a convolution (denoted by <span class="math notranslate nohighlight">\(*\)</span>) based on the k-space amplitude weighting</p>
@@ -626,8 +626,8 @@ <h2>Relaxation during signal acquisition<a class="headerlink" href="#relaxation-
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="_images/c2bc6f3fee2a679d68cb3956f1dd71771107c64c4330c26801b306718311c22e.png" src="_images/c2bc6f3fee2a679d68cb3956f1dd71771107c64c4330c26801b306718311c22e.png" />
 <img alt="_images/2ac77f1e3c485944eb84049be4e604109e3623a5e7fef06e080c1bd6acd6f1e0.png" src="_images/2ac77f1e3c485944eb84049be4e604109e3623a5e7fef06e080c1bd6acd6f1e0.png" />
+<img alt="_images/c2bc6f3fee2a679d68cb3956f1dd71771107c64c4330c26801b306718311c22e.png" src="_images/c2bc6f3fee2a679d68cb3956f1dd71771107c64c4330c26801b306718311c22e.png" />
 </div>
 </div>
 <p>In the above plots, the height of the main peak in the center represents the expected SNR, including losses due to blurring, while the signal amplitude outside of the main peak represents blurring that will occur.   These show that the blurring and signal loss from <span class="math notranslate nohighlight">\(T_2^*\)</span> gets worse as the relaxation time is shorter, the blurring it is much worse for EPI (in phase encoding direction) versus Cartesian trajectories.</p>
@@ -698,8 +698,8 @@ <h2>Off-resonance and Chemical Shift<a class="headerlink" href="#off-resonance-a
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="_images/7116ed83d8359b6883a12abf711538b5c0ffc32e612333da3dc4050edb5f88f6.png" src="_images/7116ed83d8359b6883a12abf711538b5c0ffc32e612333da3dc4050edb5f88f6.png" />
 <img alt="_images/8f6a3cbbd80b72ee00d3a30c540d68b3fd29709b58d9cb727045d05a021c1418.png" src="_images/8f6a3cbbd80b72ee00d3a30c540d68b3fd29709b58d9cb727045d05a021c1418.png" />
+<img alt="_images/7116ed83d8359b6883a12abf711538b5c0ffc32e612333da3dc4050edb5f88f6.png" src="_images/7116ed83d8359b6883a12abf711538b5c0ffc32e612333da3dc4050edb5f88f6.png" />
 </div>
 </div>
 <p>For frequency shift, the main peak of the convolution kernels is shifted frfom the origin.  This will result in a shift in the reconstructed image.  The shift is much larger for EPI and is in the phase encoding instead of the frequency encoding direction.  (The residual side lobes are due to sinc interpolation effects, similar to Gibbs ringing.)</p>