Category Archives: Engineering

Explain why attenuation is not a big problem in PET.

Consider a 2-D object consisting of two triangle compartments, as shown in Figure P9.4. Suppose a solution containing a 511 KeV gamma ray emitting radionuclide with concentration f = 0.5 mCi/cm3 fills the lower triangle. The linear attenuation coefficients in the two regions are μ1 = 0.1 cm−1 and μ2 = 0.2 cm−1. Assume perfect detection in all cases and ignore the inverse square law effect.

(a) We image the radioactivity using a 2D SPECT scanner. Compute the projected radioactivities gSPECT(, 0◦) and gSPECT(, 180◦). The camera is located on the +y-axis looking down when θ = 0◦.

(b) Now assume the radionuclide in part (a) is replaced by a positron emitting radionuclide with the same concentration. Assume the linear attenuation coefficients in the two regions are the same as in….

Give the mean and the variance of the reconstructed image, mean[ˆ f(x, y)] and var[ˆ f(x, y)].

Ignoring the inverse square law and attenuation, an approximate reconstruction for SPECT imaging is given by

where c˜() =  {||W()} and W() is a rectangular windowing filter that cuts off at = 0. Suppose we use M projections (θ1, θ2, …, θM) uniformly spaced over the range [0, π), and N + 1 (odd) ray paths per projection. Assume the spacing between detectors is T, and gij = gθj (iT). A discrete approximation to the reconstruction of ˆ f(x, y) can be written as

(a) The observation gij is proportional to the number of photons hitting the detector i at angle θj, Nij, that is, gij = kNij. Assume Nij is a Poisson random variable with mean Nij and is independent for different i and j. Give the mean and the variance of the reconstructed….

Find the numerical responses in each to an event in crystal C(4, 6).

Suppose a PET detector comprises four square PMTs (arranged as a 2 by 2 matrix) and a single BGO crystal with slits made in such a way that it is divided into an 8 by 8 matrix of individual detectors. Assume that the PMTs and the detectors cover the exact same square area and that each PMT is 2 in by 2 in in size. This geometry is shown in Figure P9.5. The response of a PMT to an event occurring in a particular subcrystal depends on the distance from the center of the PMT to the center of the subcrystal, r, as follows

where τ = 1 inch.

(a) Find a general expression for the response in PMT(i, j) to an event in crystal C(k, l).

….

Find the detector thicknesses for NaI(Tl) and BGO.

Both NaI(Tl) and BGO can be used as PET detectors. The linear attenuation coefficient of BGO at 511 keV is 0.964 cm−1 as opposed to 0.343 cm−1for NaI(Tl). BGO does not convert gamma ray photons to light photons as efficiently as NaI(Tl), and is in fact roughly 13% as efficient. In this problem, suppose two detectors are designed, one from NaI(Tl) and one from BGO. Assume both are designed to stop 75% of the 511 keV photons that strike the crystal.

(a) Find the detector thicknesses for NaI(Tl) and BGO.

(b) A light burst from NaI(Tl) has a higher intrinsic SNR than that from BGO. Find the ratio of intrinsic SNRs, NaI(Tl) to BGO.

Explain how you would use the X and Y signals coming out of the opposing Anger cameras.

You decide to build a PET scanner from a dual-head SPECT camera you own. It has two 30-cm square Anger cameras mounted facing each other 1.5 m apart.

(a) Would you use the low-energy collimators, the high-energy collimators, or no collimators? Explain.

(b) What significant piece of electronics would you have to add to the SPECT circuitry to make this work as a PET scanner?

(c) Explain how you would use the X and Y signals coming out of the opposing Anger cameras. Would you use the Z-pulses? Why or why not

(d) You will have to rotate the Anger cameras at some point during the scan. What is the minimum number of angular positions of the two cameras that would be required in order to get full….

Explain how to compensate for attenuation in a PET scanner in order to reconstruct an accurate image of the radionuclide concentration.

Consider the two-dimensional cross section shown in Figure P9.6 consisting of three separate compartments R1, R2, and R3.

Figure P9.6 Object geometry for Problem 9.9.

(a) Suppose a solution containing a 511 keV gamma ray emitting radionuclide with concentration 0.3 mCi/cm3 fills only R2; R1 and R3 contain nonradioactive solutions. Let the linear attenuation coefficients (at 511 keV) in the three regions be μ1 = 0.2 cm−1, μ2 = 0.3 cm−1, and μ3 = 0.1 cm−1, respectively. Suppose we image the radioactivity usinga (2-D) SPECT scanner outside the object. Compute the projected radioactivities gSPECT(, 90◦) and gSPECT(, 270◦).

(b) Now assume the radionuclide in (a) is replaced by a positron emitting radionuclide with the same concentration. Assume the linear attenuation coefficients in the three regions are the same. This time the body is….

Explain why resolution of the PET scanner is typically worse away from the center of the scanner

Suppose a PET scanner has 1,000 detectors packed tightly around a circle of diameter of 1.5 m.

(a) What is the approximate size (width) of each detector? Explain the tradeoff between using deep (long) and shallow (short) detectors.

(b) What is the purpose of coincidence detection in PET? What is a nominal time interval defining a coincidence ‘‘event window’’ for PET? Explain why is it undesirable to make the event window (1) smaller or (2) longer.

(c) Assume there is no ‘‘wobbling’’ or ‘‘dichotomic’’ motion of the PET gantry. What is the line integral sampling interval (which has the symbol T in CT) in the center of the scanner? Assuming the usual sampling ‘‘rule of thumb’’ from CT, how many pixels would be in a typical PET image…..

Write an equation for the number of coincidence events Nc arising from positron annihilations at the center of the circle that will be detected as a function of N0, N+, and N−.

A PET-CT system is shown in Figure P9.7. There are N0 positron annihilations at the origin of the circle whose gamma rays travel on the line L(0, 0◦). N+ of these positron annihilations reach detector A and N− reach detector B

Figure P9.7 See Problem 9.11.

(a) Write an equation for the number of coincidence events Nc arising from positron annihilations at the center of the circle that will be detected as a function of N0, N+, and N−. For parts (b)– (d): Suppose that the radioactivity is the same everywhere and the value of g(1, 0◦) is 2/3 of the value of g(0, 0◦).

(b) Write an equation representing the relationship between μsquare and μcircle.

(c) What is the ratio g(2, 0◦)/g(0, 0◦)?

(d) What is the local….

Show that the measured pressure in the far field is approximately p(r, t) = zf(t − c−1r)/r2.

An acoustic dipole can be modeled by two point sources (monopoles) residing close together but vibrating exactly out of phase. Suppose point source 1 is located at (0, 0, −d) and point source 2 is located at (0, 0, d), where d is very small. Further suppose that source 1 is generating the narrow-band signal f(t), leading to a spherical pressure wave given by p(r1,t) = f(t − c−1r1)/r1 and source 2 is generating the narrow-band signal −f(t), leading to a spherical pressure wave given by p(r2,t) = −f(t − c−1r2)/r2, where the two radii r1 and r1 are measured from their respective sources.

(a) Show that the measured pressure in the far field is approximately p(r, t) = zf(t − c−1r)/r2.

(b) Sketch isopressure lines in the x-z plane….

Show that the time delay for a point (x0, y0, 0) on the flat face is approximately as below

A transducer can be focused by curving the crystal to conform to the shape of a sphere, as shown in Figure P10.1. To analyze the field pattern, however, it is convenient to assume that the transducer is flat but that we can independently control the time delay of each element on its face. Assume that the point (xm, ym, 0) is the point farthest out on the flat face.

(a) Show that the time delay for a point (x0, y0, 0) on the flat face is approximately

where  m and d is the desired focal depth.

(b) Use the steady-state approximation and show that at z = d the Fresnel field pattern is

where S(u, v) is the 2-D Fourier transform of s(x, y)

(c) Discuss the….