Table of Contents

Background

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 two positron sources of unknown strength at unknown locations.

(source: Wikipedia)

References

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

1 Goals

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:

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.

Note on safety: Some students have worried about being around multiple button sources in the lab for hours on end. From what I can tell, the risk is pretty much nonexistent for what we're working with. Occupational safety limits suggest that eating 400$\mu$C activity a year worth of sodium is the acceptable limit for exposure. Considering that students would have to crack open every single source we own and consume it to come close to that, they should be fine. Especially considering that it has a biological half-life of about 10 days.

2 Prelab Research

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

3 Positron emission tomography technique

3.1 Positronium

How are positrons produced and used?

The positron emitter that we will use in this experiment is sodium-22, which decays to an excited state of neon-22 by either electron capture or by positron emission. The neon later decays to its ground state by the emission of a 1.27 MeV gamma. (See Fig. 1.)

Figure 1: Nuclear decay scheme for sodium-22. (Source: C. Michael Lederer, Jack M. Hollander, and Isadore Perlman, Table of Isotopes, 6th Edition, John Wiley & Sons, 1967.)

The emitted positrons are slowed down and are captured by electrons in the source to form an electron-positron bound state called https://en.wikipedia.org/wiki/Positronium, a hydrogen-like “atom.” The ground state of positronium, which has a binding energy of 6.8 eV, has two possible configurations depending on the relative orientation of the electron and positron spins. The state with anti-parallel spins has net spin equal to 0, and is variously referred to as the singlet state, para-positronium, or, in spectroscopic notation, the state $1S^0$. This state decays into an even number of photons, with the most likely result being two back-to-back photons with equal energy and oppositely directed momentum. The state with parallel spins has net spin equal to 1, and is referred to as the triplet state, ortho-positronium, or the state $3S^1$. This state decays into odd numbers of photons, most commonly three.

For reasons having to do with the lifetime of the two states and with the likelihood of triplet states flipping into singlet states, the two photon decay is much more likely. Since the rest masses $m_0$ of the electron and positron are converted to energy in the annihilation process, each of the resulting two photons has energy $E = m_0c^2 = 511 \,\textrm{keV}$ , and are created simultaneously.

3.2 Positron emission tomography technique

How do we use positrons to get a signal?

The photon pairs which PET detects are produced in the following manner. (Letters correspond to features illustrated in Fig. 2.)

  • A positron emitting radionuclide decays. (a)
  • The emitted positron travels only a short distance (typically ~1 mm) through living tissue before it binds with an electron forming positronium. (b) Positronium is a bound state of a positron and an electron, analogous to a hydrogen atom. Positronium is unstable, with a lifetime less than $10^{-9}$s.
  • Although positronium has more than one decay mode, the most likely is a decay to two photons. Assuming that the positronium was at (or nearly at) rest when it decays, conservation of momentum and energy dictate that the two photons produced are emitted 180° apart and that each photon carries half of the total rest mass energy of the positronium (c). Since the rest mass energy of the positronium is just the sum of the rest masses of the electron and positron, each decay photon (gamma) has an energy of 511 keV.
  • At these energies, the gammas are very unlikely to interact before leaving the tissue and can be detected by a pair of detectors (NaI+PMT for our experiment) placed along the line defined by the trajectory of the decay gammas (d). This line is called the line of response, abbreviated LoR.

Figure 2: Schematic of PET technique

Medical imaging PET scanners utilize a large number of gamma-ray detector pairs, typically arrayed in an annulus to produce high resolution scans. (See Fig. 3.) The reconstruction of an image from these scanners is can be somewhat computationally intensive, requiring the use of https://en.wikipedia.org/wiki/Tomographic_reconstruction#Reconstruction_algorithms. Furthermore, processing all of the signals from such a ring of detectors would be daunting to someone just learning how this technique works.

Figure 3: PET scanner detector configuration. (Source: Wikipedia.)

For the purposes of our experiment, we will use a single detector pair to create a two-dimensional image of a sample containing two sources. This simplified technique allows us to clearly illustrate the basic principles of PET while making use of a very simple tomographic reconstruction algorithm. The downside is that we are effectively collecting our individual scans in series (having to do one angle at a time) rather than parallel (using dozens of detectors). Nevertheless, a medical PET scanner operates on the same set of principles which apply to our scaled-down setup. A schematic of our PET scanner with the single pair of NaI+PMT detectors, a sample container and a sample guide is shown in Fig. 4.

Figure 4: Left: A schematic of our simplified 2-detector PET scanner. Right: Our physical apparatus setup for the PET experiment

The line connecting the two PMT+NaI detectors is the line of response (LoR). Positronium decays which occur anywhere along the LoR will produce back-to-back 511 keV gamma pairs. Some of these 511 keV gamma pairs will strike the two PMT+NaI detectors. If they are equidistant from the source, the two detectors will be hit by these decay gammas nearly simultaneously. Conducting a scan amounts to recording the rate of simultaneous hits at the two detectors as a function of position as the source is moved across the LoR along one of the source axes. This constitutes one slice of the sample. (See, for example, the x-axis slice corresponding to the blue lines in Fig. 5.) The source can then be rotated, for example by 90º, and the process repeated along the new y-axis (the red lines in Fig. 5) producing a new slice. If the rate of coincident hits is measured at 5 equally spaced locations along each axis, we would create a grid as shown in Fig 4. If, for example, the sample contained two unequal strength sources at locations (X2, Y2) and (X4, Y4.5), then the slices would produce projected intensity profiles along each axis as shown. Figure 5: An example of x- and y-axis intensity profiles for two unequal strength sources. For the case of two or fewer sources, we can use the following algorithm to reconstruct a two-dimensional image from one-dimensional slices. If there are $m$  projections along the x-axis and $n$  projections along the y-axis, there will be a total of $m \times n$  intersection points. We can assign a value to each intersection point $A_{xy}(i,j) = I_x(i) * I_y(j)$ , where $i$  runs from 1 to $m$  and $j$  from 1 to $n$ , and where $I_x(i)$  and $I_y(j)$ are the measured intensities of the ith and jth projections along the x-axis and y-axis, respectively. Figure 6 shows a contour plot of the values of $A_{xy}(i,j)$  that would result from the x and y slices of Fig. 5. Figure 6: The contour plot of the values of $A_{xy}(i,j)$ that would result from the X and Y slices of Fig. 4. Note that the image reconstruction (Fig. 6) shows four sources instead of two. This is an example of an artifact called shadowing which is present to some degree in all tomographically-reconstructed images. More data and additional processing is needed to suppress the shadows and more clearly reveal the locations of the actual sources. For our simple sample containing only two sources, we need to add only one additional slice, taken at an angle different from the previous two. We choose to define a new axis called Z as shown in Fig. 7. Figure 7: A third axis, called Z, is added to the grid at an angle to both X and Y. It is necessary that the LoR's of the Z slice intersect with all of the intersections $A_{xy}(i,j)$  one (and only one) additional time. Accordingly, there are $q=m+n-1$  LoRs along the z-axis with measured intensities $I_Z(k)$. The value at each intersection point is now calculated as $A_{xyz}(i,j,k)$  = $A_{xy}(i,j)*I_z(k)$  where $k$ runs up to $i+j-1$ The resulting contour plot is shown in Fig. 8. Notice how the two shadows have been suppressed and the signals corresponding to the actual source locations have been enhanced. Figure 8: The new contour plot of $A_{xyz}(i,j,k)$. Note that now only two sources appear instead of four. If one were to try using this simple tomographic analysis method for more than two sources, you would need to continue adding more slices along new axes in order to suppress false signals caused by shadows. As a result, the computing requirements would quickly become unimaginable. For this reason, medical PET scans use much more sophisticated algorithms for image reconstruction. Even so, the computing requirements are still enormous, and are the primary cause of the high cost associated with this form of medical imaging.

3.3 Image reconstruction

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.

We're still working on adapting material from an old version of this lab, so more info may be forthcoming.

4 Experimental procedure

4.1 Apparatus and setup

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

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

[A] PMTs

The NaI(Tl)+PMT detectors are mounted in carriages. One detector is fixed in position (defined a $\theta = 0^\circ$ ) and the other is free to move around a circular track but should be set to 180º, directly opposite the fixed PMT. Anode signals from the PMTs are connected to the Red Pitaya for signal processing.

[B] Red Pitaya

The Red Pitaya is a Field Programmable Gate Array (FPGA) that serves to process signals from the PMTs [A], send signals to control the motors via the Logic Level converter [C], and stop the motor at fixed positions determined by the Feedback Sensors [G]. We can access a Python program to let us interface with the Red Pitaya without needing sophisticated knowledge of its inner workings for now.

[D] Motor Driver

The motor driver acts both as a power supply for the motors as well as an interpreter for the incoming control signals from the Red Pitaya [B]. There are inputs that control the motor direction as well as one that will rotate a motor by a small angle for every individual pulse received from the Red Pitaya.

[E] Sample Platform

The clear acrylic disk will hold the sample that you are scanning for the various parts of this experiment. There is a motor beneath it that can rotate the sample, as well as a small brass flag that is used to ensure that the Feedback Sensors [G] are able to reset the platform to a consistent position.

[F] Linear Stage

This stage consists of another motor connected to a long worm gear, which holds the entire Sample Platform [E]. It translates the sample between the two detectors with better than mm precision thanks to the fine control from the Motor Driver [D].

[G] Feedback Sensors

There are three separate feedback sensors here: one photodetector that senses when the brass flag attached to Sample Platform [E] is aligned, and two switches that keep the linear stage from traveling too far in either direction. While the setup could operate without them, it would reliably resetting the position much more difficult to do.

==== Starting the experiment ==== Make sure that the electronics are wired as shown in Fig. 9, and turn on the high voltage (if not already on).

To turn on the high voltage supplies, complete the following procedure:
  1. Start with both switches on each supply turned off (switched to the left). 
  2. Flip on the left-hand switch. 
  3. After about 30 seconds, you will hear a click and the standby light will come on. Now, flip the standby switch on and look for a negative deflection of the meter.
  4. The PMTs should now be powered.

===== 4.2 Looking at signals on the oscilloscope =====

To start, it is a good idea to get an idea of the characteristics of the pulses from the PMTs. This can help you make better decisions about what voltages and coincidence times are reasonable during the experiment.

To this end, we've set up a Jupyter notebook that will let you pass the input signals from the Red Pitaya straight through to the outputs. There should be a link on your computer's desktop to a notebook named FPGA_Signal_Observation Make sure you use a 50 $\Omega$ terminator in either case to avoid distorting the signal. It may be easiest to just place one of the Na22 sources near the detector rather than trying to work with moving the platform at this point.

Upon running the notebook, you'll be presented with a screen like the following:

There are a lot of moving parts here, so we'll go through them one by one:

  • Counts: Shows a count of how many incoming pulses have met the lower threshold condition.
    • Lower Threshold: Shows the digitized value a pulse must exceed (i.e. be more negative than) to be counted.
    • Upper Threshold: Shows the digitized value a pulse must exceed to be excluded from being counted.
  • Reading: Shows the current digitized value seen at the Red Pitaya input.
    • Pulses happen on too short of a timescale to reliably appear here.
  • Coincidences: Counts how many pulses from the two detectors have arrived within ~8 ns of one another
    • Resets ~once a second.
  • Timeouts: Counts the number of times a pulse registered that did not have a coincident partner in the other detector.
  • Overthreshold Counts: Counts how many times pulses have exceeded the upper threshold.
  • Out N Select: Selects what signal the Red Pitaya will output for external use.
    • Input: Outputs raw input to the channel, i.e. the PMT pulses.
    • Low Threshold: Outputs the lower threshold value as a voltage.
    • High Threshold: Outputs the upper threshold value as a voltage.
    • Both Thresholds: Oscillates between the two thresholds.

===== 4.3 Understanding system resolution =====

With any experiment it is important to understand the operating characteristics of your apparatus. In a PET scan, the goal is to determine – as well as possible – the location of $\beta^+$emitting radioactive sources by detecting the simultaneous emission of back-to-back 511 keV gammas. Do we really need to do this complicated tomography procedure, or do we have good enough timing and timing and spatial resolution to find the sources directly?

==== 4.3.1 Selecting the right signal ====

We could design our system to register a coincidence any time the signals from both detectors simultaneously differ from 0V, but in practice there are quite a few flaws with that plan.

  1. There's a certain amount of general noise on the PMT output even without a source present, so we don't want to be using that to gather data.
  2. There are lots of low-energy x-rays emitted by the source that don't correspond to positron annihilations. If we include these in our comparisons, then just by random chance we'll have some signals arriving coincidentally even though they don't represent the phenomena we're looking for.
  3. The same applies to the high end of the energy spectrum as well, since Na22 emits 1.27 MeV gammas as part of its normal decay.

To address this problem, we've coded the software on the Red Pitaya to limit its search for coincidences only to signals that exceed one voltage (the low threshold) without going past another value (the high threshold). One of your goals is to find a set of threshold values that filter out most signals that aren't due to the 511 keV gammas.

You'll be using the FPGA_Signal_Observation notebook for this again.

We'll use a Python script on the Red Pitaya that will let you see the pulses from the PMTs alongside the voltage thresholds you've selected for filtering. This, alongside your spectrometer, will allow you to determine the settings needed to filter out most spurious coincidences. You'll need to calibrate both channels, as the PMT responses might not necessarily be identical.

While the default settings will work, they aren't particularly optimized. You should adjust values to get a sense of how they affect the resulting coincidence counts, and record what you'll want to use in the future.

4.2 Controlling the apparatus

To make it more manageable to change settings, move the sample, and collect data using the Red Pitaya, we've developed a way for it to take in a list of commands and run them in order. They are:

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.

Don't just run this notebook all at once, it needs you to input the file you'll use partway through.

4.2.1 Measurement of spatial resolution perpendicular to the LoR

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.

Using a single Na-22 button source translated orthogonally across the LoR as shown in Fig. 12, determine the spatial resolution of the system perpendicular to the LoR.

Figure 12: Different source positions perpendicular to the LoR.

More specifically, you should try and determine:

  1. How the rate of coincidences from a single source behaves as a function of distance from the LoR
  2. What position in the software corresponds to the platform being centered

Use the linearscan.json file as a template for this part of the lab

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

4.2.2 Two Source Spatial Resolution

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?

Again the linearscan.json file will serve as a basis for this part of the lab

4.3 Scanning a known configuration

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.

Now you can use the Three.json file as a template for this part of the experiment

4.4 Working with Unknown Samples

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:

An example setup, containing five Na22 sources. The sample with the top closed. 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.

There should be four sample containers in the lab. By the end of the first day, you should set up a configuration for use by another lab group.

Samples will be rotated as follows:

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.

3.4.2 Thresholding

One other parameter we can control is the threshold voltage that the code will accept as a pulse. This code uses both an upper and lower limit. The lower limit helps us screen out spurious x-rays that all samples will emit, and the upper limit will help screen out 1.27 MeV gammas that are also emitted by the Na22 sources. This is similar to setting a region of interest in our spectrometer software, except it just ignores all pulses that don't meet the criteria.

You should experiment some with the threshold settings to make sure that the count rate is zero (or close to it) when no source is present, and that some coincidences are detected when a source is present

Currently, there is an issue where one of the PMTs has a bit of an offset voltage on its signal. As a result, setting the lower threshold too low can result in the software ignoring all pulses. If you see this occur, make a note of when it happens and make sure to keep the lower threshold above that value.

Access your data remotely from this link

5 Analysis

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:

pet_plotting_2022.ipynb

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.

Ultimately, you should identify the position of all sources within a sample as well as the relative intensities between them.

Rubric

Analysis Rubric

Each item below is graded on a 0-4 point scale. Full, partial and no credit correspond roughly to the following:

All rubric items carry the same weight. The final grade for the analysis will be assigned based on the average (on a 4.0 scale) over all rubric items.

Item Description
Calibration and Setup Pulses due to $\beta^+$ annihilation are identified and characterized.

Parameter choices for thresholds are recorded and the methodology for finding them is described.
Structured Investigations Spatial resolution of point sources perpendicular to Line of Response (LoR) is investigated, with appropriate plots and/or fits.

Spatial resolution between pairs of sources is investigated, with appropriate plots and/or fits.
PET Scan Data Sinogram plots for single source, paired sources, and an unknown configuration.

Relevant parameter values for all scans are documented.

Parameter choices are (briefly) explained/justified.
Tomographic Reconstruction Plots showing tomographic reconstructions for important configurations are created.

Sources are identified in the reconstructions and locating methods are described.

(relative) intensities are estimated and compared. Note that you are not expected to calculate uncertainties rigorously, as doing so for two dimensional functions is beyond the scope of the lab.

Report rubric

Each item below is graded on a 0-4 point scale.

All rubric items carry the same weight. The final grade for the report will be assigned based on the average (on a 4.0 scale) over all rubric items.

Item Description
Presentation of Data Presents plots of data as needed and uses them to support the narrative of the report.

Properly labels plots, and makes presentation clean and clear.

Uses error bars where appropriate.

Includes captions that provide appropriate context.

Presents all numerical values with appropriate units and significant figures.

Clearly formats numbers, equations, tables, etc.
Data Handling Describes how the raw data was processed including with uncertainties.

Details fit functions and provides sample fits (if appropriate).

Details other calculations/considerations and provides sample calculations or reasoning (if appropriate).
Discussion of Uncertainties Identifies relevant sources of uncertainty in measured quantities, and quantifies values when possible.

Describes how uncertainties were assessed and incorporated into the analysis.

Identifies potential sources of systematic bias and describes how they are accounted for in the analysis or eliminated.
Presentation of Results Final results are presented clearly.

Data tables and plots are used where appropriate and are properly labeled and annotated.

Measured and calculated quantities include units and uncertainties where appropriate.
Conclusions Makes clear final conclusions that are fully supported by the experimental results and discusses the overall take-aways of the experiment appropriately.

Properly accounts for or contextualizes measurement uncertainties and potential sources of systematic bias.

Archive - Students can't see ===== 3.3 Collecting and processing data =====

First of all, Don't Panic. You should not have to write any Python code from scratch to complete this project. You may have to tweak values in existing scripts, but you don't need to know a ton of programming to be able to do this project.

The Red Pitaya we're using here has most of the default software installed, but we'll be working primarily with Python via Jupyter.

When you've opened Jupyter, click on the notebook link PET User Interface at the top to get to the code we'll use for this lab.

From that page, you can run the linked notebook to set up the Red Pitaya and create a user interface. It will take a little while for the code to finish running, but the end result will be something akin to the interface shown below:

PET user interface. Note that you need to press ENTER to update values entered manually.

Your data will be a two-dimensional array of intensity values as a function of position and angle. For instance, in the plot below showing a scan of three sources, we can see that one is much stronger than the others, and that the stronger source is aligned with each of the others somewhere between 70-100 degrees. The weak source at 15 cm that isn't moving is at the center of the setup, as the coincidence count rate doesn't depend on angle at all.

A typical sinogram produced by the scanning system.

Your data will be saved on the Red Pitaya as it is gathered. If you have a scan that ends after your time in lab, you may also download it here: Box PET Data

Note that this updates only once an hour, so files won't be instantly available.

Notes on motor control

The motors used to move the sample are not continuous, but is instead driven by electrical pulses. Our current setup will move the sample ~0.5 mm laterally for every 1600 pulses the motor receives. (This distance is determined by the dimensions of the worm screw and the pulses set by the motor controller.) We have not tested the behavior for sequences of very few pulses, so beware that real world factors such as friction and backlash may become causes of systematic uncertainties if you drive the sample over small distances.

The rotational controls likewise allow the sample to be rotated one complete revolution per 25600 pulses. Again, some assumptions may break down when attempting fine angular control.

This isn't to say you shouldn't try to move the sample in small increments, but rather that you should not trust that it will behave properly without first testing.