Description
1) Consider the following MR sequence where two subsequent RF pulses are applied with tip angles . Assume that #(0&) = ) and *+(0&) = 0, where ) is the equilibrium magnetization. Also assume that ≫ /. Do NOT assume that ≫ 0.
a) Find #(01) and *+(01).
b) Find #( &) and *+( &).
c) Find #( 1) and *+( 1).
d) Find #( ) and *+( ) for >.
a a
RF
◦
2) If we repeat the RF pulses in Question 1 for a sufficient number of times (waiting for TR in between subsequent RF pulses), the expression for *+( ) right after an RF pulse will reach what is called a “steady-state” value, which can be expressed as follows:
*+ = #44 :; &</>?
Note that
44 is the steady-state value for
right before each RF pulse. Show that
44 for the
equation
above can be expressed as:
#( )
#
#
0&@ABC/BD
.
#44 = ) 0&EFGH @ABC/BD
Hint: Start at = &, and use the fact that the following should be satisfied in steady-state:
#( &)= # + 1 & = #44
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3) Draw a pulse sequence diagram where the data is acquired on a circular trajectory in k-space, following an RF excitation pulse that is not slice-selective. Right after RF excitation, start from (0,0) in k-space and go up to (0, 1) mm-1. Then, make a full circle as shown on the right, with constant velocity in k-space.
Assume that Gmax = 4 G/cm is the maximum gradient strength for the MRI scanner. Choose the gradient/timing parameters so that the k-space trajectory is completed in the shortest time possible.
Assume that the data acquisition takes place only during the circular part of the trajectory. Mark the data acquisition window in your pulse sequence.
Label your timing diagram including the amplitudes of the gradients.
4) MATLAB Question: Include your MATLAB codes and related images in your solution.
Mapping: If you estimate the 0 for every pixel in an MRI image and display this as an image, it is called a “ 0 map”. So, every pixel in this “ 0 map” image would correspond to the estimated 0 of the corresponding pixel in the MRI image.
Download the files brainT1_mri.mat and roiellipse.m posted on Moodle. The file brainT1_mri.mat contains two sets of simulated brain MRI images:
◦ image1 and image2: Simulated 0-weighted MRI images with = 45° and = 90°, respectively. For both images = 15 and = 600 . No noise is added.
◦ image1_noisy and image2_noisy: Same as above, with a very small amount of noise added to both images.
a) The following imaging equation gives the signal level during echo time for a steady-state pulse sequence as in Question 2. Here, 0 and / are functions on , .
, = ) sin
&>]/>?
1 − &>_/>D
1 − cos &>_/>D
Show that this expression can be re-written as:
,
=
&>_/>D
,
+ )
&>] >?
1 −
&>_ >D
sin
tan
b) Assume that two images 0( , ) and /( , ) are acquired with different tip angles 0 and /, while keeping and the same. Show that an estimate of 0 can be obtained from these two images using the following relation:
0( , ) = ln / / − 0 0 − ln sin/ / − sin0 0
c) Display the noise-free images (i.e., image1 and image2). Calculate and display the 0 map using the equation in part (b) for the noise-free dataset.
d) Estimate 0 for white matter using roiellipse.m. Here is the MATLAB code that you can use to do that:
figure, imshow(T1map,[])
BW = roiellipse; % type “help roiellipse” to see how to use T1_est = mean(T1map(BW))
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This code will create a figure displaying your 0 map. It will then place a draggable and scalable ellipse on it. Drag this ellipse on a part of the image that contains white matter only. Maximize the window containing the figure first, so that you can move/scale the ellipse more carefully. When you are ready, click the “Continue” button on the bottom left of the image. This will create a mask of the selected elliptical region (BW), and then calculate the mean 0 value in that region using the mask. In your solution, include the image with ellipse showing the selected region, together with the estimated 0 value.
e) Estimate 0 for gray matter and CSF (cerebrospinal fluid) using the steps in part (b). Include the images with ellipse in your solution, together with the estimated T1 values.
f) Display the noisy images (i.e., image1_noisy and image2_noisy). Calculate and display the 0 map for the noisy dataset. Because these images are noisy, the computed 0 values may be unreasonably high (or even complex valued). Do the following to restrict the range of 0 values displayed to between 0-3000 ms, and to force 0 to be real-valued:
figure, imshow(abs(T1map),[0 3000])
g) Estimate 0 for white matter, gray matter, and CSF using the noisy 0 map. Because of the noise in the 0 map, it helps to select a reasonably large ellipse (rather than a very small one), so that the estimation of mean 0 is more accurate. Include the images with ellipse in your solution, together with the estimated 0 values.
h) Which noisy 0 estimate showed the biggest deviation from its noise-free version? Why?
Hint: If you are not sure which part of the brain is gray/white matter or CSF, here is a segmented version of the simulated MRI image:
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