From 214628db89c9c6c39f0fb095877b383ef03675ea Mon Sep 17 00:00:00 2001 From: agentmess Date: Fri, 20 Sep 2024 21:49:39 +0000 Subject: [PATCH] deploy: 8c1472438ffcc01529feb97e653b503733b0b064 --- MR Physics - Bloch Equation.html | 65 +++++-- MRI Signal Equation.html | 6 +- MRI System.html | 69 +------ Magnetic Fields and RF Coils.html | 181 +++++++++++------- Spatial Encoding.html | 26 +-- Spectral-Spatial RF Pulses.html | 2 +- ...9a82bf131b57435e60c4f790118c0a4a6efcc2.png | Bin 0 -> 14091 bytes ...ef88ad632cfcb6fe0c0a689a6a0a017029c46a.png | Bin 13908 -> 0 bytes ...ed9ba81a70eda400b453b6c991e53889fa98d1.png | Bin 35797 -> 0 bytes ...143d755a699dccc217a7c557d6babcba687ae2.png | Bin 16380 -> 0 bytes ...13e0fcce878287f8d0fa8686052838cd27405c.png | Bin 0 -> 32103 bytes ...76f7c8dbdda62b0ea0a5a26ea75a74fb0075eb.png | Bin 16760 -> 0 bytes _images/MRI_experiment_diagram.png | Bin 2315542 -> 1557651 bytes ...0c6b92a06eeec76424914253aa6cf0de62db76.png | Bin 0 -> 15393 bytes ...4b1cfcbe4a08e35d5bf233c72cc86b98a01833.png | Bin 0 -> 11493 bytes _sources/MR Physics - Bloch Equation.ipynb | 79 ++++++-- _sources/MRI System.ipynb | 34 +--- _sources/Magnetic Fields and RF Coils.ipynb | 39 ++-- reports/Magnetic Fields and RF Coils.err.log | 29 +++ searchindex.js | 2 +- 20 files changed, 314 insertions(+), 218 deletions(-) create mode 100644 _images/28286d12c196037a83e7e4db0d9a82bf131b57435e60c4f790118c0a4a6efcc2.png delete mode 100644 _images/37ced63ecf5238c86476a8762def88ad632cfcb6fe0c0a689a6a0a017029c46a.png delete mode 100644 _images/4b2090a487a6d8711253946380ed9ba81a70eda400b453b6c991e53889fa98d1.png delete mode 100644 _images/5e169185decf7509f83b3aa4a0143d755a699dccc217a7c557d6babcba687ae2.png create mode 100644 _images/668c45a8294cc6945c627894d613e0fcce878287f8d0fa8686052838cd27405c.png delete mode 100644 _images/81df12c24547b093a68a5fe30376f7c8dbdda62b0ea0a5a26ea75a74fb0075eb.png create mode 100644 _images/b20086ac8db05aa4be274eaf240c6b92a06eeec76424914253aa6cf0de62db76.png create mode 100644 _images/cd5358b8578e3f0017622a95b14b1cfcbe4a08e35d5bf233c72cc86b98a01833.png create mode 100644 reports/Magnetic Fields and RF Coils.err.log diff --git a/MR Physics - Bloch Equation.html b/MR Physics - Bloch Equation.html index b188892..3bf34ae 100644 --- a/MR Physics - Bloch Equation.html +++ b/MR Physics - Bloch Equation.html @@ -417,15 +417,21 @@

Contents

@@ -511,6 +517,8 @@

THE Bloch Equation\(T_2(\vec{r})\) - the transverse (\(M_{XY}\)) or spin-spin relaxation time constant

All of which can vary across our subject (and all a valuable source of contrast!).

+

There is a very full-featured, interactive Bloch Equation Simulator available online, that is valuable to understand the behavior of the net magnetization:
+Bloch Equation Simulator

Precession#

@@ -523,6 +531,9 @@

Precession

+
+

Simulation of Precession#

+

Open up the Bloch Equation Simulator. You will see a visualization of a net magnetization vector that is precessing around the magnetic field (thin line).

B0 = 1.5e3; % 1.5 T = 1500 mT
@@ -567,6 +578,7 @@ 
Precession
 
RF Excitation#

 
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

@@ -596,11 +608,12 @@ 
Lab versus Rotating Frame

-
For another illustration of the stationary/lab versus rotating frames, try the “Change Frame” option in this Bloch simulator:

-
http://drcmr.dk/BlochSimulator/

Common flip angle RF excitations#

+

For a constant amplitude RF pulse, the flip angle depends on the duration of the RF pulse, \(T_{rf}\) and the strength of the RF magnetic field, \(b_{1,0}\):

+
+\[\theta = \gamma b_{1,0} T_{rf} \]
@@ -616,14 +629,16 @@

Common flip angle RF excitations -

Simulations#

-

The following Bloch equation simulations show

+
+

Simulations of RF Excitation#

+

Again, open up the Bloch Equation Simulator.

    -

  1. First, when a non-resonant magnetic field is applied. An additional magnetic field is applied orthogonal to the main magnetic field, but not applied at the Larmor frequency, and there is no creation of transverse magnetization.

    2. -

  2. After, this is corrected, and the RF pulse is applied at the Larmor frequency, \(\omega_0 = \gamma B_0\). 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, \(M_X\) and \(M_Y\).

    3. -

  3. 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

    4. +

  4. 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.

    5. +

  5. RF Excitation: Change to ‘Equilibrium’ scene in bottom left. Then, use the ‘90x hard’ button to apply a constant amplitude pulse.

    6. +

  6. Other flip angles: Go back to ‘Equilibrium’ scene, and try the other hard RF pulse flip angles.

+

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.

+

First, when a non-resonant magnetic field is applied. An additional magnetic field is applied orthogonal to the main magnetic field, but not applied at the Larmor frequency, and there is no creation of transverse magnetization.

% lab frame
@@ -686,6 +701,7 @@ 
Simulations
+

To achieve excitation the RF pulse is applied at the Larmor frequency, \(\omega_0 = \gamma B_0\). 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, \(M_X\) and \(M_Y\).

% RF pulse at Larmor frequency
@@ -718,6 +734,7 @@ 
Simulations
+

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.

% rotating frame
@@ -800,6 +817,17 @@ 
Relaxation
 \[M_Z(t) = M_Z(0)e^{-t/T_1} + M_0(1- e^{-t/T_1})\]

Where here the shorthand complex notation for the transverse magnetization is being used: \(M_{XY}(\vec{r},t) = M_X(\vec{r},t) + i M_Y(\vec{r},t)\)

+

T1 and T2 relaxation after RF Excitation

+

Relaxation T1 and T2

+
+

Simulation of Relaxation#

+

One more time, open up the Bloch Equation Simulator. Under the ‘Relaxation’ options in the top left, you can adjust T1 and T2 relaxation rates.

+
    +

  1. Experiment with different T1 and T2 relaxation values

    2. +

  2. When the magnetization turns to equilibrium, use a RF pulse and you will see relaxation occuring again.

    3. +

  3. From Equilibrium, try a 180-degree flip angle and try adjusting both T1 and T2. Which parameter influence the relaxation in this situation?

    4. +
+

Below are additional Bloch equation simulations and associated code of relaxation.

t = linspace(0,1); % s
@@ -856,8 +884,7 @@ 
Relaxation
-

T1 and T2 relaxation after RF Excitation

-

Relaxation T1 and T2

+
@@ -926,15 +953,21 @@

Relaxation
  • Learning Goals
  • THE Bloch Equation
    • -
  • Precession
    • +
  • Precession +
  • RF Excitation +
    • +
  • Relaxation
    • -
  • Relaxation
    • diff --git a/MRI Signal Equation.html b/MRI Signal Equation.html index 8ca39d5..911837e 100644 --- a/MRI Signal Equation.html +++ b/MRI Signal Equation.html @@ -579,8 +579,8 @@

    Relaxation during signal acquisition -_images/04e32389def1da9556efd712d119655989abaa4ab970ce8cc166a8125b4a22c0.png _images/92ac396d1094116faca36e071635f22ff14da490df79aa5b1e645af612d2ec53.png +_images/04e32389def1da9556efd712d119655989abaa4ab970ce8cc166a8125b4a22c0.png

    Thus the reconstructed image will corrupted by a convolution (denoted by \(*\)) based on the k-space amplitude weighting

    @@ -626,8 +626,8 @@

    Relaxation during signal acquisition -_images/c2bc6f3fee2a679d68cb3956f1dd71771107c64c4330c26801b306718311c22e.png _images/2ac77f1e3c485944eb84049be4e604109e3623a5e7fef06e080c1bd6acd6f1e0.png +_images/c2bc6f3fee2a679d68cb3956f1dd71771107c64c4330c26801b306718311c22e.png

    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 \(T_2^*\) gets worse as the relaxation time is shorter, the blurring it is much worse for EPI (in phase encoding direction) versus Cartesian trajectories.

    @@ -698,8 +698,8 @@

    Off-resonance and Chemical Shift -_images/7116ed83d8359b6883a12abf711538b5c0ffc32e612333da3dc4050edb5f88f6.png _images/8f6a3cbbd80b72ee00d3a30c540d68b3fd29709b58d9cb727045d05a021c1418.png +_images/7116ed83d8359b6883a12abf711538b5c0ffc32e612333da3dc4050edb5f88f6.png

    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.)

    diff --git a/MRI System.html b/MRI System.html index e9a8999..2363484 100644 --- a/MRI System.html +++ b/MRI System.html @@ -423,7 +423,6 @@

    Contents

  • Fundamental MRI Experiment Procedure
    • -
  • Magnetic Fields
    • @@ -486,19 +485,19 @@

    Main magnet - \(B_0\)The purpose of the main magnetic field is “Polarization”. This refers to the preferential alignment of nuclear spins with a magnetic field. The more alignment there is of spins, the greater the MRI signal. The stronger the magnetic field, the more alignment and therefore the stronger signal that will be received.

    -

    This magnetic field is typically 1.5 Tesla (1.5 T) or 3 T. It is always on, and designed to be homogeneous. To achieve this large of a mangetic field typically requires using a superconducting magnet, which consists of a superconducting alloy bathed in liquid helium.

    +

    This magnetic field is typically 1.5 Tesla (1.5 T) or 3 T. It is designed to be homogeneous. To achieve this large of a mangetic field typically requires using a superconducting magnet, which consists of a superconducting alloy bathed in liquid helium, and this type of magnet is always on.

    Radiofrequency (RF) coils - \(B_1^+(\vec{r},t), B_1^-(\vec{r},t)\)#

    -

    The purpose of the RF coils is “Excitation” and signal reception.

    +

    The purpose of the RF coils is “Excitation” and “Acquisition”. RF coils are antennas designed to detect changes in electromagnetic fields in the radiofrequency range.

    Excitation refers to depositing energy at a specific frequency that perturbs the spins, causing them to resonate and release energy that is captured as the MRI signal. This is performed using a transmit RF coil, described by the magnetic field \(B_1^+(\vec{r},t)\). -This energy is deposited in the radio-frequency (RF) range, typically around 100 MHz for MRI, which is the resonance frequency.

    -

    After Excitaiton, the spins release energy, also in the RF range, as they return to thermal equilibrium. This is performed by the receive RF coil, described by the magnetic field \(B_1^-(\vec{r},t)\).

    -

    The RF coils are typically separated in transmit and receive coils because of different requirements: the transmit coils for excitation are designed to be homogeneous in depositing energy across the field-of-view (FOV), while receive coils are designed for maximum maximum sensitivity to the relatively small signals detectable from the spins.

    +This energy is deposited in the radio-frequency (RF) range, typically in the 100 MHz range for MRI, which is the resonance frequency.

    +

    After Excitaiton, the spins release energy, also in the RF range, as they return to thermal equilibrium. This is measured in Acquisition where the signal is detected by the receive RF coil, described by the magnetic field \(B_1^-(\vec{r},t)\).

    +

    The RF coils are typically separated into separate transmit and receive coils because of different requirements: the transmit coils for excitation are designed to be homogeneous in depositing energy across the field-of-view (FOV), while receive coils are designed for maximum maximum sensitivity to the relatively small signals detectable from the spins.

    Magnetic field gradient coils - \(\vec{G}(t)\)#

    -

    The purpose of the magnetic field gradient coils is for imaging. Thes coils, commonly refered to simply as “gradients”, create linearly varying magnetic fields. These are designed to add and subtract from the main magnetic field. Since the magnetic field determines the resonance frequency (\(f = \bar{\gamma} \|\vec{B}\|\)), this will create a spatial variations in the magnetic field that can be used to separate signals from different locations and ultimately create images.

    +

    The purpose of the magnetic field gradient coils is for “Spatial Encoding”, which enables creation of images. These coils, commonly refered to simply as “gradients”, are designed to create linearly varying magnetic fields. These varying magnetic fields add and subtract from the main magnetic field. Since the magnetic field determines the resonance frequency (\(f = \bar{\gamma} \|\vec{B}\|\)), this will create a spatial variations in the magnetic field that can be used to separate signals from different locations and ultimately create images.

    @@ -519,7 +518,7 @@

    Fundamental MRI Experiment Procedure\(B_1^-(\vec{r},t)\)

    -

  • Encoding: Create spatial variations in the magnetic field to distinguish spins at different locations

    +

  • Spatial Encoding: Create spatial variations in the magnetic field to distinguish spins at different locations

    • Components required: Magnetic field gradient coils, \(\vec{G}(t)\)

    @@ -528,59 +527,6 @@

    Fundamental MRI Experiment Procedure

    • -
    -

    Magnetic Fields#

    -

    Each of the main components of a MRI system creates magnetic fields, but with different orientations, magnitudes, and frequencies, which are summarized in the following table

    -

    RF 45-degree flip

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    Component

    Notation

    Direction

    Frequency

    Strength

    Purpose

    Main Field

    \(B_0\)

    \(z\)

    0 Hz

    \(\approx 1\) T

    Polarization

    Magnetic Field Gradients

    \(\vec{G}(t)\)

    \(z\)

    \(\approx 1\) kHz

    \(\approx 10\) mT

    Spatial Encoding

    Transmit RF Coils

    \(B_1^+(\vec{r},t)\)

    \(x,y\)

    \(\approx 100\) MHz

    \(\approx 10 \mu T\)

    Excitation

    Receive RF Coils

    \(B_1^-(\vec{r},t)\)

    \(x,y\)

    \(\approx 100\) MHz

    \(\approx 1 \mu T\)

    Reception

    Net Magnetization

    \(\vec{M}(\vec{r},t)\)

    \(z,y,z\)

    \(\approx 100\) MHz

    \(\approx 1 \mu \mathrm{T}\)

    Signal Source

    -

    Also included is the Net Magnetization, \(\vec{M}(\vec{r},t)\), which captures the magnetic fields resulting from nuclear spins that we measure in MRI.

    -

    - + @@ -773,6 +825,7 @@

    Sensitivity Simulation