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ELG7173 - Final Exam Winter 2005

You have 3 hours to complete this exam. The exam has four questions; you are required to answer any two of them. Each question is worth equal marks. This is a closed book exam; however, you are permitted to bring one 8.5" × 14" sheet of notes into the exam. You are permitted to use a calculator. You may not communicate with anyone during the exam except the instructor.

You may make assumptions to simplify the problems as long as they don't change the calculations by more than 10%. You may use the following conditions and equations for your calculations:

  • E = mc², c= 3×108 m/s
  • E = , h = 6.63×10−34 m²kg/s
  • eV = 1.6×10−19 J
  • F = ma
  • electron mass = 9.11×10−31 kg
  • Hydrogen Gyromagnetic ratio, γ = 42.58 MHz/T
  • V = IR
  • Accoustic impedance Z = ρc

1. X-ray Computed Tomography

A Computed Tomography system can be represented in terms of a matrix inverse. In order to simplify the problem, the body is divided into four square regions, R1 .. R4, with 2 cm sides, as shown in figure 1A. Projections P1 .. P6 are defined such that each projection goes through two regions. Assume that the width of each region is 2 cm, independent of the direction of travel of the X-rays.

Figure 1A: Left: Diagram of simple X-ray CT system. The target area is divided into four square regions, R1 .. R4. Each region has 2 cm sides. Six projections, P1 .. P6, are defined, such that P1 travels through R1 and R2. Right: Matrix formulation of the projection data as a function of region attenuations.

    1A. Each projection has 108 X-ray photons per second, with all photons having 1 keV energy. We define each projection, P1 .. P6, as the natural logarithm of the received beam power. Each region, R1 .. R4 is the value of the attenuation, μ, in cm−1. Given R1=0.5 cm−1, and R2=R3=R4=0. Calculate P1 and P2.

    1B. Calculate the matrix H and vector b.

    1C. The matrix inverse to calculate the vector of regions, R, from the vector of projections, P, is

    R = (HtH)−1 Ht (P-b)
    The expression R = Ht (P-b) represents simple (unfiltered) backprojection. What does the term (HtH) represent?

    1D. Given the following projection data: P2=P3=P4=−18.95, and P1=P5=P6=−17.95. Calculate the reconstructed region values using the matrix equation in the previous question. You may wish to use the following matrix equality

    1E. Human Visual System: Many radiologists were reluctant to move away from using film to computer monitors. This resistance has been overcome somewhat with newer high contrast high intensity monitors. How do such monitors help radiologists?

2. Positron Emission Tomography

    2A. Collimation: Describe the difference between electronic collimation and physical collimation in terms of mechanical design and efficiency. Categorize PET and SPECT collimation strategies.

    2B. Efficiency (PET): In class we defined the efficiency of a nuclear medical imaging system as the fraction of the radiation emitted during an interval that is captured and processed by the imaging system. Consider a PET system with 3D collimation imaging the heart (which we can consider to be a sphere of 10 cm diameter at the centre of the field of view.) The detector ring is 1 m in diameter with a z-direction width of 30 cm. Assume the individual PET detectors are 100% efficient, and γ-rays are not attenuated by the body. What is the efficiency of this system?

    2C. Efficiency (Pinhole camera): Instead of a PET imager, a pinhole camera based nuclear medical imaging system is now used. The pinhole is 2 mm in diameter placed on the body surface, 10 cm from the centre of the heart. Assume all photons which enter the pinhole are captured. What is the efficiency of this camera? Hint: You can make simplifying assumptions, since the required accuracy is 10%.

    2D. Coincidence: A new detector technology is invented for PET which allows the difference in timing of coincidence events to be measured to an accuracy of 1 ns. Describe how such information can be used to improve PET image reconstruction. Specifically, describe in general terms how backprojection reconstruction could be modified to use this information.

    2E. Attenuation: Why is attenuation correction important for PET image reconstruction? In class we discussed three methods to estimate the data necessary for attenuation correction. Describe one method to estimate attenuation.

3. Magnetic Resonance Imaging

Consider a MRI system imaging the head (assumed to be a cube of 10 cm × 10 cm × 10 cm) with the following parameters: B0= 1 Tesla, Gx= Gy= Gz= 4 Gauss/cm. The RF pulse envelope is rectangular. To simplify, assume a rectangular pulse of width T has a width of 1/T in the frequency domain.

    3A. What is the bandwidth of the MRI output signal from the field of view that includes the head? What is the length of the phase encoding pulse, T, in order for the maximum phase difference across the head to be 360°?

    3B. What is the length of the RF pulse TRF, in order for the selected slice width to be 2 mm? Assume a rectangular RF pulse envelope. Given the RF pulse time, what is the RF pulse magnetic field strength to induce an inversion (ie. 180°) of the nuclear magnetization?

    3C. Consider that T2* is 20 ms. Using the 90°−FID pulse sequence, each MRI RF signal is recorded for 3 times T2* before a new pulse sequence can be started (in order to prevent interference between the signals from two pulse sequences). If TR is 1.2 s, how many volume imaging pulse sequences can be done in each TR interval.

    3D. Sketch the inversion-recovery pulse sequence for very large TR. The time between the 180° and 90° RF pulses is TI. Show how the signal from the inversion-recovery pulse sequence is ρ*( 1−2×exp(−TI/T1) ) for large TR. Note: This question does not require a proof; illustrate how this value is calculated.

    3E. If two tissue regions differ in T1, show how (theoretically) the inversion-recovery pulse sequence can generate a larger contrast than the spin-echo pulse sequence. Given signals s1 and s2, define the contrast, c, as c=(s1−s2)/s1.

4. Ultrasound

We want to make ultrasound images of the eye (which we consider to be a sphere of 2 cm diameter). Assume a uniform velocity of sound in the eye of 1480 m/s. A 10 MHz ultrasound transducer is used. At this frequency the attenuation is 7 cm−1.

    4A. What is the smallest diameter of ultrasound transducer in order for the eye to be entirely in the near field? What is the advantage of a smaller diameter ultrasound transducer?

    4B. What problem does time gain correction, g(t), solve? In this case, the gain correction uses c= 1500 m/s and attenuation 5dB/cm. What is the ratio between the gain for sound reflected from 1 cm deep and that from sound reflected from 2 cm deep?

    4C. One common feature in ultrasound images is speckle noise. Explain briefly the origin of speckle noise. Why is there no speckle noise in X-ray images?

    4D. In this question we are interested in the combined effect of speckle noise, Nspeckle, and electronic noise, Nelectronic. Nelectronic is constant, while the signal decreases with depth (assume the near field goes to 4 cm). Assume these noise sources are independent, and thus add as (Ntotal)² = (Nspeckle)² + (Nelectronic)². The SNR due to speckle is 1.91. The SNR due to electronic noise is 1.0 at a depth of 4 cm. At what depth into the tissue does the overall SNR reach 1.0? Assume that the reflective tissue is uniformly distributed in the tissue. This means that you don't need to calculate reflectivity in the solution. Hint: You may want to plug in values in the calculator. Recall that the required accuracy is 10%.

    4E. List two challenges in interpreting mode B ultrasound images. Discuss one technique for feature identification that can overcome a challenge described in the previous question. You may choose discuss a technique described in Richard Youmaran's presentation in class.

Last Updated: $Date: 2007-11-24 01:39:40 -0500 (Sat, 24 Nov 2007) $