Homework-6: Basic Imaging Techniques

  1. [25 pts] How many slices could you image with a gradient echo sequence that uses a 20 ms slice selection gradient, 10 ms phase encoding gradient, 10 ms dephasing frequency gradient, 100 ms frequency encoding gradient, and TR = 1 s?

    For a gradient echo sequence, the total sequence takes an amount of time equal to the sum of the slice selection gradient, the longer of the phase encoding gradient or dephasing gradient, and frequency encoding gradient on times. In this case 130 ms. A total of seven 130 ms periods will fit into a 1 s TR peroid. [(1000 ms) / (130 ms) = 7.69 , but since we can not have fractions of a sequence the answer is 7.]

  2. [25 pts] Draw a timing diagram for an inversion recovery imaging sequence which uses a gradient echo sequence rather than a spin-echo sequence to detect the signal present TI after the inversion (180 degree) pulse.

  3. [25 pts] There are two adjacent tissues in an image with T1a = 300 ms, T2a* = 20 ms, ra = 50 ASDU, and T1b = 150 ms, T2b* = 20 ms, rb = 50 ASDU. (ASDU = arbitrary spin density units.) You wish to produce a 90 o gradient echo image of the tissues such that the contrast between the tissues is maximized. What TR should be used?

    C = S1 - S2

    where:

      C = contrast
      S1 = signal from tissue 1
      S1 = signal from tissue 2

    We need to find the maximum C, or dC/dTR = 0.
      dC/dTR = d(S1 - S2)/dTR= 0
      S = k r (1-exp(-TR/T1)) Sinq exp(-TE/T2*) / (1 -Cosq exp(-TR/T1))
      d(1-e-TR/T1(2))/dTR - d(1-e-TR/T1(1))/dTR = 0
      (e-TR/T1(2))/T1(2) = (e-TR/T1(1))/T1(1)
      T1(2)/T1(1) = (e-TR/T1(2))/(e-TR/T1(1))
      T1(2)/T1(1) = exp(-TR/T1(2)+ TR/T1(1))
      ln(T1(2)/T1(1)) = TR/T1(1) + TR/T1(2)
      ln(T1(2)/T1(1)) = TR (T1(2) - T1(1))/ (T1(2) T1(1))
      TR = T1(2) T1(1) ln[T1(2)/T1(1)]/(T1(2) - T1(1))
      TR = 208 ms

  4. [25 pts] You wish to image an oblique slice located at an angle of 45o to the +Z-axis and +y-axis, and 45o between the +Z-axis and +x-axis. Since gradients can only be produced with the three gradient coil systems located around the x, y, and z-axes, what combination of the three gradients should be applied to produce a slice selection, phase encoding, and frequency encoding gradient?

    There are several ways to solve this problem. Those of you that can see in 3D can calculate the projections onto the x, y, and z-axes of the needed gradients. Those that can not see in 3D may wish to use the coordinate transformation or rotation matrices. Start with an XZ image plane with Gslice = -Gy, Gphase = Gz, and Gfreq = -Gx. Rotate the plane and gradient vectors by the desired angles. The two rotation matrices, Rz(a) and Rx(b), perform rotations about z and x by angle a and b, respectively. G is the needed slice, frequency, or phase encoding gradient.

    The image plane is rotted by: q = tan-1((2)1/2/2), and f = 45o, as defined in the above figure.

    The slice selection gradient (Gs) should be created by a combination of y, x, and z gradients.

    Gx = -Gs Sinf Cosq = -0.57735 Gs
    Gy = -Gs Cosf Cosq = -0.57735 Gs
    Gz = Gs Sinq = 0.57735 Gs

    The phase encoding gradient (Gf) should be created by a combination of x, y, and z gradients also.

    Gx = Gf Sinf Sinq = 0.40825 Gf
    Gy = Gf Cosf Sinq = 0.40825 Gf
    Gz = Gf Cosq = 0.8165 Gf

    The frequency encoding gradient (Gf) will be composed of x and y gradients.

    Gx = -Gf Cosf = -0.70711 Gf
    Gy = Gf Sinf = 0.70711 Gf

    (Please note that the direction of the phase and frequency encoding gradients could be interchanged.)