Questions

1. (15 points) A static magnetic field, $B_{0}$, points along the positive $z$-axis. A classical spinning particle with positive gyromagnetic ratio $\gamma$ and fixed magnetic moment $\mu$ has its spin initially $(t=0)$ in the laboratory $x-z$ plane, at an angle $\theta$ with the $z$-axis. in the direction of the negative $x$-axis. Ignoring any relaxation effects and using your analytical experience, write down expressions in terms of the given quantities for the laboratory components of the magnetic moment vector $\mu(t)$ without any derivation. (Use the magnetic moment equation of motion only, if you need to, or want to check your answer.)

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

2. (20 points) Assume that you are performing a 2D field echo experiment, with a 90 degree excitation pulse, and $T_{R} >> T_{2}$. Given that $T_{1}$ = 1200ms, $T_{2}$ = 80ms, and $T_{2}^{*}$ = 30ms for your target tissue.
   1. Assume that you need the signal for your experiment to be equal to at least 0.5 times the signal for an experiment with $T_{R} = \infty$. What is $T_{R,min}$ (ms) in this case?

      
      
      
      
      
      
      
      
      
      
      
      
      
      

   2. Assume that you are doing an experiment where $T_{R} = \infty$ If you can only afford to lose half of your signal to $T_{2}^{*}$ decay, what is $T_{E,max}$ (ms) in this case?

      
      
      
      
      
      
      
      
      
      
      
      
      
      

3. (15 points) Given $T_{2}$ = 40ms and $T_{2}' = 15\,ms$ please answer the following.
   1. What is $T_{2}^{*}$ in this case?

      
      
      
      
      
      
      
      

   2. Write an expression for $T_{E,SE}$ as a function of $T_{E,FE}$,$T_{2}$, and $T_{2}^{*}$ such that the signal at the echo is the same for the SE and the FE experiments.

      
      
      
      
      
      
      
      

4. (20 points) Given the following sequence diagram and a spin located at $(x,y,z)$ in a field $B_{z}(x,y,z) = B_{0} + \Delta B$. Assume the spins have $\phi = 0$ after the rf pulse, and $G$ is the magnitude of the read gradient pulses.
   

   

   1. In the rotating reference frame, what is the phase of the spin at the end of the dephase gradient ($t_{2}$).

      
      
      
      
      
      
      
      

   2. What is the phase at the echo time?

      
      
      
      
      
      
      
      

   3. If this were a spin echo experiment what would be the phase at the echo?

      
      
      
      
      
      
      
      

   4. What is the phase at the end of all read gradient activity($t_{4}$)?

      
      
      
      
      
      
      
      

5. (20 points) Given the following spin echo sequence diagram. Please label problem section for each diagram.
   
   
   
   
   
   
   
   
   
   
   
   1. Draw the measured k-space for this experiment. Do not draw how you arrive at each k-space line. For the sake of clarity, assume that you only collect 5 k-space lines for this experiment, and number the lines in temporal order of collection.

   2. Underneath the sequence diagram, carefully re-draw the PE and RD gradients such that all gradient activity starts after the $\pi$ pulse, and you measure the same k-space coverage.

   3. In a separate k-space diagram, re-draw the complete k-space coverage for the first repetition of the sequence from part (5a). Include the path taken to get to the start of the k-space line. Use a dotted line to indicate the effect of the $\pi$-pulse if necessary.

   4. In a separate k-space diagram, re-draw the k-space coverage for the first repetition of the sequence from part (5b). Include the path taken to get to the start of the k-space line. Use a dotted line to indicate the effect of the $\pi$-pulse if necessary.

   
   
   
   
   
   
   
   
   
   
   
   
   
   

6. (20 points) Assume that you are doing a 2D field echo experiment. Neglect gradient rise times. Given the following imaging parameters
   

   $FOV_{read}$ = 256mm	$FOV_{PE}$ = 256mm
   $G_{read}$=10mT/m	$\tau_{pe}$ = 2ms
   $N_{x}$ = 256	$N_{y}$ = 256
   $T_{R}$ = 500ms

   What are:
   

   $\Delta t$ ($\mu$ s)	$\Delta G_{pe}$ (mT/m)
   $T_{s}$ (ms)	$T_T$(s)
   $G_{pe,max}$ (mT/m)	$T_T$(s) if $N_{z}$ = 16, i.e., you turn this into a 3D experiment

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

7. (15 pts) Given a uniform, spherical MRI sample centered at the origin of a magnet where $\vec{B}_{0} = B_{0}\hat{z}$. Describe the position and orientation of a single loop of wire that will measure no signal. Please specify the coil center $(x,y,z)$ and direction of the area vector, $\vec{a} = \hat{?}$, for the coil?

   
   
   
   
   
   
   
   
   
   
   
   
   
   

8. (15 pts) Assume that you are using a circular loop as a receive coil, and you are interested in making an image of a point a distance $d$ from your coil, along the axis of the coil. How will the magnitude of the measured signal be different if you use a coil where the radius of the coil is equal to $d$, or if the radius of the coil is equal to $2d$? Express your answer as a fraction $S(r=d)/S(r=2d)$. Which coil would you choose to use if information from the area of this point is of paramount importance in your experiment, and you want to collect your information quickly?

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

9. (15 points) Given a particular gradient coil (case a). What will be the fractional change in minimum rise time to maximum gradient for your gradient system if the inductance of the coil is reduced by 30% (case b), but you use the same gradient amplifier? Express your answer as a fraction rise time (case a)/rise time (case b)?

   
   
   
   
   
   
   
   
   
   
   
   
   
   

1. Assume that you are using a circular loop as a receive coil, and you are interested in making an image of a point a distance $d$ from your coil, along the axis of the coil. How will the magnitude of the measured signal be different if you use a coil where the radius of the coil is equal to $d$, or if the radius of the coil is equal to $2d$? Express your answer as a fraction $S(r=d)/S(r=2d)$. Which coil would you choose to use?
2. Given a particular gradient coil (case a). What will be the fractional change in minimum rise time to maximum gradient for your gradient system if the inductance of the coil is reduced by 30% (case b), but you use the same gradient amplifier? Express your answer as a fraction rise time (case a)/rise time (case b)?
3. Given a 1.5T magnet with a quadrature rf transmit coil. How much more power will you need to apply to provide the same flip angle to spins if you have a 3T system with a linear rf transmit coil.

HWS 15.3 15.4 15.5 15.7 15.10 15.11