Positron Emission Tomography (PET) is a medical imaging technique which is commonly used to map out metabolic activity in the body. Typically, a $\beta^+$-emitting (positron-emitting) radionuclide (such as ${}^{18}\textrm{F}$) attached to a glucose molecule is injected into the body where it is taken up by tissues in proportion to their metabolic activity. Positrons produced by the decay of the radionuclide usually travel less than 1 mm in human tissue before they bind with an electron, and quickly annihilate. Most of the positron-electron annihilations result in the emission of a back-to-back pair of 511 keV photons. These photon pairs leave the body and can be detected. Areas of the body where there is high metabolic activity – such as cancer cells or active regions in the brain – will be more intense emitters of these 511 keV photon pairs. In a PET scanner, arrays of scintillator+PMT detectors are used to measure the intensity of this radiation along well defined planes (referred to as slices) passing through the patient's body. Tomography is the process of reconstructing a three-dimensional image of positron emission intensity from these slices.

In this experiment, you will use PET to create a two-dimensional image of a sample box containing several positron sources of unknown strength at unknown locations.

(source: Wikipedia)

Kerstin Sonnabend, Wolfgang Bayer, Peter Mohr, and Andreas Zilges, "A simple experimental setup to demonstrate the basics of positron emission tomography," American Journal of Physics 70, 929 (2002).

Principles of Tomographic Imaging

In this experiment you will use positron emission tomography to identify the positions and relative intensities of two unknown positron emitters inside a sealed box. Specifically, your goals for this experiment include the following:

• to understand the formation of positronium and the annihilation of electron-positron pairs;
• to interpret the physical meaning of a one-dimensional coincidence scan
• to understand how one-dimensional scans are combined into a two-dimensional image
• to investigate what factors affect scan results, and the limitations of the results.
• to use your knowledge of the method to locate areas of high positron emissions in an unknown sample, and estimate their relative intensity.

The last goal here represents a typical use case for PET scanning: a chemical called Fluorodeoxyglucose that contains Fluorine-18 is administered to a patient, which the body treats as regular glucose. Cells with rapid metabolisms (e.g. cancer) take in glucose much more rapidly than other tissues, and thus will end up with higher concentrations of beta-emitting ${}^{18}$F.

PET scans and other radioimaging techniques are often performed/overseen by medical physicists; for information on this career path look into the American Association of Physicists in Medicine or the University of Chicago Medical Physics Ph.D. Program. On another front, some relatively recent research has begun to look into PET imaging using Na22 laden nanoparticles. These have the upside of combining the (relatively) long 2.6 yr half-life of sodium-22 with the lesser bio-availability of nanoparticles. This both ensures that the radioactive material doesn't accumulate in the body and eliminates the need of on-site generation of radio-isotopes.

To start out, you should do a bit of background research. Some suggested topics are the following:

• What is a positron emission tomography (PET) scan?
  • What are they used for typically?
• What is special about positron decay?
  • How do the gammas produced by a decaying positron differ from those produced by other forms of radioactive decay? 
  • What property do they have that is useful for distinguishing them from other gamma rays?
• What sort of signals might we be looking for with our NaI+PMT detectors?
  • How can multiple PMT detectors be used to detect when a positron has decayed?
• What energy of gamma rays will be be expecting from the decay of sodium-22 (Na-22)?
  • How picky will we be when looking for coincidences?
• What will we do with the data we gather to create an image?

A good introduction to the basic techniques we'll use for processing data is this handout on Tomographic Image Reconstruction, from the 41st American Association of Physicists in Medicine conference. We will use what is essentially a discretized version of an unfiltered backprojection operation, but reading further on what other options exist may be interesting.

Between 5-11 minutes of the following video also do a decent job of introducing our basic tomographic reconstruction technique.

Figure 9: Apparatus for measuring correlated pair-annihilation gammas.

A schematic of the apparatus is shown in Fig. 9. Individual components are described below. 

The general process of doing a PET scan is the following:

• Prepare a sample with positron emitters distributed throughout a space
• Incrementally move the sample between a pair of scintillation detectors
  • Count the rate of coincidences at each increment
• Rotate the sample and repeat

A single scan (moving of the sample between detectors) gives you essentially a 1d plot of coincidence rate versus position. By rotating the sample between scans, you gather information about the distribution of sources that could be obscured at some angles (e.g. if some of the sources are co-linear with the detectors they will show up as a single stronger source). By combining data from multiple scans using an algorithm we can reconstruct the spatial positions of the positron emitters without having to disturb the sample or have physical access to the interior. When the sample is, say, a person, they tend to appreciate not having their interior disturbed.

We have set up a motorized control stage and digital coincidence counting for this experiment using an instrument called a Red Pitaya. The Red Pitaya combines a Linux computer with a Field Programmable Gate Array (FPGA) and high-speed oscilloscope inputs in one package. One is currently monitoring the air quality on the international space station.

To control the device and gather data, we've developed a way for the Red Pitaya to take in a list of commands and run them in order. The commands are:

• File
  • BaseFileName defines the save file prefix, a time/date will be appended to the end of the name.
• ResetLateral
  • Moves the platform to the leftmost edge of the track
• ResetRotation
  • Rotates the platform to a known $0^\circ$ position.
  • Only works after a lateral reset as the required sensor is on the far edge of the track
• Move
  • Moves the platform a # of motor steps left or right
    • Dir is 0 for left and 1 (or anything else) for right
    • Dist is the # of motor steps
• Rotate
  • Rotates the platform a # of motor steps clockwise or counter-clockwise
    • Dir is 0 for clockwise and 1 for counter-clockwise
    • Dist is the # of motor steps
      • 25600 motor steps correspond to one full rotation
• Scan
  • Counts coincidences between detectors at the current position and saves to a file
    • Time is the time in seconds
• LOOP
  • Performs a list of commands repeatedly
    • Times is the number of times to repeat
    • List is a list of commands to loop through
• Threshold
  • Channel defines which input is being set
  • Kind may be either:
    • Lower for the voltage that needs to be exceeded to begin detection or
    • Upper for the voltage that needs to be exceeded for an input to be rejected
  • The values are from 0 to $2^{14}-1 (16383)$, which map to inputs of 1 to -1V

Don't worry, we don't expect you to be writing command lists from scratch. We've provided several to perform basic tasks, but it will be up to you to tweak the settings to do what you want.

To use these commands, you'll open up the FPGA PET Command Interpreter notebook linked on your computer's desktop.

In deciding how far apart to space one's measurements, several several factors need to be taken into consideration. Spreading out the scans too much will result in poor resolution of the location of the sources. More closely spaced scans will improve resolution, but at the cost of requiring more time to acquire the data as well as increased computing requirements. However an upper limit on how closely to space the scans can be set by measuring the spatial resolution of the detector system perpendicular to the LoR.

Your goal for this section is to use a single Na-22 button source translated orthogonally across the LoR as shown in Fig. 12, to determine the spatial resolution of the system.

Figure 12: Different source positions perpendicular to the LoR.

We'll briefly go through the behavior of the command interpreter notebook here:

• The top of the cell shows your current thresholds and offset settings, just to keep you aware of what you're using.
• The 'Current' and 'Previous' command areas show what just finished running and what is happening at the moment.
• Counts shows the number of coincidences that have been recorded in the active scan.
• Scan Time shows a progress bar for the current scan (not the entire run).
• Below all of these is a textbox that will show:
  • The estimated run time.
  • The recorded data after each individual collection run.
  • A link to the data file upon completion of the script.

After you've established how wide the signal distribution is from a single source, you should also empirically determine how close a pair of identical Na22 sources can be before the signal becomes indistinct from a single source.

- How close two sources can be before their signals are no longer distinct?

Now that you've got a good idea of the limitations of the system, it's time to scan a known sample from multiple angles and use the data to try out our reconstruction techniques. Set up a configuration like the one shown below:

A system with 3 sources of intensity I1,I2,and I3. I1 is centered, whereas I2 and I3 are separated from it by some know distances d1 and d2.

The specifics aren't critical, but you should measure out the positions of the sources beforehand so that you can calibrate your data reconstruction. The resulting plot of intensities versus position and angle is known as a sinogram because of the tendency of point sources to form sine curves in this format

An example sinogram. There are three sources here, two with low activity and one with high activity.

You will want to run at least two scans for the known configuration:

1. One quick scan to test how the setup works.
2. One overnight scan (between days 1 and 2) to get some high-resolution data.

You may use either one for the exercise, but be aware that you may want to work on the plotting between lab sessions. The notebook for reconstructing configurations can be found in the next section

After you have a good idea of what parameters are needed to resolve our Na22 sources you'll move onto the core piece of the lab: scanning a sample with $\beta^+$ emitters in unknown locations. One such sample is shown below:

The unknown samples with the tops sealed. Note the mark that can be used to align the sample on the platter so you can reliably put it back in the same position from day to day.

It is suggested that you take a quick, low-resolution scan first to have some idea of what you're dealing with, and then to set up a longer scan to run overnight to capture fine details in the setup.

To create a 2-dimensional image from your data, we've created a Python script that will do the basic reconstruction, which can be accessed here:

pad_amt = 7
new_size = len(shifted_data[0])+pad_amt
padded_data = np.zeros((len(shifted_data),new_size,new_size))
for index,item in enumerate(shifted_data):
    padded_data[index] = np.pad(item, (pad_amt,0))
    
    
fig,ax = plt.subplots()
ax.imshow(padded_data[0])

There are instructions on how the code works and what it is doing at each step, and reading through it will be much more helpful than trying to run all of it at once.

You will be able to produce a few different sorts of plots with this code, and are free to modify it as needed.

For the last day of the lab, we have an optimization task for you. We have another unknown sample prepared for you to scan today. This time, you have the following constraints:

1. You may only run one script to scan the sample.
2. You may only use a total of 30 minutes (estimated) time for the scan
3. You should locate the position of the strongest source to within 5mm
   1. If you can't, justify the precision you do have

You should write a short summary of the capabilities of the system used for this experiment. It should include:

You should include plots relevant to how you decided any of the above quantities.

Present your reconstruction & analysis of the unknown configuration. This should include:

Describe your performance in the challenge task. This should entail: