Table of Contents
| — | — | Positron Emission Tomography (PET) is a medical imaging technique which is commonly used to map out metabolic activity in the body. Typically, a β+-emitting (positron-emitting) radionuclide (such as 18F) 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. | [[https://en.wikipedia.org/wiki/Positron_emission_tomography#/media/File:PET-image.jpg|{FIXME ${/download/attachments/195462212/400px-PET-image.jpg?version=1&modificationDate=1542125163000&api=v2}$]]
//(source: [[https://en.wikipedia.org/wiki/Positron_emission_tomography#/media/File:PET-image.jpg|WikiPedia]])// |
| * 1References | ||||
| * 21 Goal | |||||||
| * 32 Positron emission tomography technique | |||||||
| * 3.12.1 Positronium | |||||||
| * 3.22.2 Positron emission tomography technique | |||||||
| * 43 Experimental procedure | |||||||
| * 4.13.1 Apparatus and setup | |||||||
| * 4.23.2 Looking at signals on the oscilloscope | |||||||
| * 4.33.3 Time calibration | |||||||
| * 4.43.4 Understanding system resolution | |||||||
| * 4.4.13.4.1 Measurement of spatial resolution along the LoR | |||||||
| * 4.4.23.4.2 Measurement of spatial resolution perpendicular to the LoR | |||||||
| * 4.4.33.4.3 Detector coincidence efficiency | |||||||
| * 4.53.5 Scan X, Y and Z slices | |||||||
| * 54 Analysis | |||||||
| * 6Rubric | |||||||
| * 6.1Autumn quarter | |||||||
| * 6.2Winter quarter |
References
1 Goal
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 understand how to use timing electronics and NaI(Tl)+PMT detectors to make coincident gamma detections;
- to investigate the spatial and temporal resolution of you PET apparatus, and to make time-of-flight measurements; and
- to complete a full 3-axis scan of your sealed box and use simple tomography techniques to reconstruct a 2-dimensional source intensity map.
2 Positron emission tomography technique
2.1 Positronium
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.)
{
${/download/attachments/195462212/Na-22_toi%20copy.png?version=1&modificationDate=1542125160000&api=v2}$\\
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 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 1S0. 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 3S1. 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_0_c_2 = 511 keV, and are created simultaneously.
2.2 Positron emission tomography technique
| 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 ©. 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 l ine of response, abreviated LoR. | { ${/download/attachments/195462212/PET_principle.png?version=1&modificationDate=1542125163000&api=v2}$ Figure 2: Schematic of PET technique |
| — | — |
Figure 4: Schematic of our simplified 2-detector PET scanner.
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.
{ ${/download/attachments/195462212/PET_Slice.png?version=1&modificationDate=1542125163000&api=v2}$
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 x 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 _i_th and _j_th 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.
{ ${/download/attachments/195462212/PET_xy_example.png?version=1&modificationDate=1542125163000&api=v2}$
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.
{ ${/download/attachments/195462212/PET_Slice_z.png?version=1&modificationDate=1542125163000&api=v2}$
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.
{ ${/download/attachments/195462212/PET_xyz_example.png?version=1&modificationDate=1542125163000&api=v2}$
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 Experimental procedure
3.1 Apparatus and setup
{ ${/download/attachments/195462212/PET_equipment.png?version=1&modificationDate=1542125163000&api=v2}$
Figure 9: Apparatus for measuring the angular correlation of pair-annihilation gammas.
A schematic of the apparatus is shown in Fig. 9. Individual components are described below.
[A] – Sample Box and Guide: A sample box is provided containing two sources of unknown strength at unknown locations. The sample box sits in a wooden guide allowing it to be translated across the LoR defined by the two PMTs.
[B] – NaI(Tl) and photomultiplier tubes (PMTs) : The NaI(Tl)+PMT detectors are mounted in carriages. One detector is fixed in position (defined as _θ _= 0°) 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 constant fraction discriminators, [C].
[C] – Constant fraction (CF) discriminators: Discriminators are electronic modules commonly used in particle detection experiments for the purpose of rejecting small amplitude noise pulses from detectors. Constant fraction discriminators are designed specficially for applications where high resolution timing of detector pulses is required. Our CF discriminators have the ability to set both an upper and a lower bound on the detector pulse sizes for which an output is generated, though we will use only the lower-level discriminator in this experiment.
[D] – _Analog time delay: _ This box uses coaxial cable of different lengths to delay pulses by different times. For reference, a cable of length 0.6 foot delays pulses by about 1 ns. Delay times of up to 60 ns are set by toggle switches.
[E] – _Time-to-amplitude converter (TAC): The TAC has s_tart and stop inputs. A start pulse initiates a linearly increasing voltage ramp in the TAC. The ramp stops increasing when either a pulse arrives at the stop input, or a preset time limit (range) is exceeded. If a stop pulse arrives before the range is reached, the TAC generates an output pulse whose voltage is proportional to the time between the start and stop pulses. If the range is exceeded the TAC resets itself and produces no output pulse. |
[F] – Pulse height analyzer (PHA): The PHA integrates the charge in each pulse (proportional to energy of the gamma) and displays a histogram of pulse charges on a computer. For more detailed information, see NaI Detector Physics and Pulse Height Spectra. |
[G] – Scalers: The scaler module counts input pulses for a time set by the accompanying timer. Note that the timer sets times in the format N x 10_M_-1 seconds, where N is the number on the first dial, and M is the number on the second. (Note the “-1” in the exponent).
[I] – High voltage (HV) supplies: Each PMT is powered by its own external high voltage supply. The PMTs in this experiment can be safely run up to -2500 V.
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:
- Start with both switches on each supply turned off (switched to the left). - Flip on the left-hand switch. - 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. - The PMTs should now be powered.
Verify that the HV supplies for each PMT are set as follows:
- Fixed PMT V = -2300 V
- Movable PMT V = -2100 V
3.2 Looking at signals on the oscilloscope
In order to understand what each element of the electronic circuit does, it helps to look at input and output signals on the oscilloscope, both individually and simultaneously with other, related signals.
With a Na-22 source (not the sample box) at the center of the table, start by looking at the anode pulses from one of the PMTs on the scope. Use a 50 Ω terminator at the scope input.
NOTE: When using the bare Na-22 source, place it on two layers of foam so that it is raised up to a greater height. This will place the source more in line with the center of the detector and therefore raise your count rate. The mystery sources in the smaple box are likewise located at an appropriate height inline with the detector center.
NOTEBOOK: Sketch the pulses in your lab notebook (to scale) and record typical amplitudes and pulse widths.
Continue through the electronics chain, sketching and commenting on what you see. When possible, compare the timing and voltage of input signals to the time and voltage of output signals. Notice which circuit components produce uniform output pulses and which produce outputs that vary in voltage or time.
NOTEBOOK: At each output through the chain of electronics, look at the pulses on the scope and sketch them (to scale) in your lab notebook. Comment on what you observe.
3.3 Time calibration
The TAC measures the time between detections by the fixed and movable detectors. The output of the TAC is recorded in the form of a pulse height spectrum. In order to calibrate the pulse height spectrum of the TAC we will use the output of a single CF discriminator to generate both a start pulse, and a delayed stop pulse. Configure the electronics as shown in Figure 10. Place a BNC “tee” on the delay box input. Connect the output of the discriminator to one side of the tee and the TAC Start input to the other side of the tee. Now the same pulse, from the CF discriminator, both starts and stops the TAC. The only difference between the start and stop pulses is the amount of delay added to the stop pulse. Connect the output of the TAC to the direct input of the PHA and make sure that “Direct Mode” is selected in the software.
{ ${/download/attachments/195462212/ang_corr_schem3.png?version=1&modificationDate=1542125164000&api=v2}$
Figure 10: Block diagram for calibrating the output pulses from the TAC.
Start the PHA and observe the effect of the delay box toggle switches. Disconnect the inputs to the start and stop inputs on the TAC and look at them simultaneously on the scope. Use the cursors on the scope to directly measure the time delay between the start and stop pulses for a range of delay settings on the delay box. Now reconnect the start and stop pulses to the TAC and use these settings to calibrate the x-axis of the PHA in units of time.
NOTEBOOK: Sketch a typical start/stop pulse pair, noting carefully the amplitude and width of each pulse and the time delay between them.
NOTEBOOK: Create a table with columns for delay box setting, measured delay time, and PHA channel. This data will be used to calibrate the TAC output spectra.
3.4 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 β+-emitting radioactive sources by detecting the simultaneous emission of back-to-back 511 keV gammas. Thus, it is appropriate to first determine the timing and spatial resolution of the setup.
3.4.1 Measurement of spatial resolution along the LoR
Our technique relies on being able to count simultaneous detections (hereafter referred to as coincidences) on two NaI(Tl)+PMT detectors. Since the decay gammas from positronium annihilation are emitted back-to-back, the source of any such coincidences must have been somewhere along the LoR connecting the two detectors. Given that it takes time for each of the photons to travel the distance between the source and the detector, it should in principle be possible to determine where along the LoR the source lies through a_ time-of-flight measurement_: the detector which is closer to the source will be struck first and the time difference between hits at the two detectors can be used to calculate the location of the source along the LoR. This time-of-flight measurement is limited by the ability of the experiment to resolve time differences.
Rewire the apparatus according to Fig. 9 and connect the TAC output to the PHA input, making sure again that the software is in “Direct Mode”. Place a single Na-22 button source at different positions along the detector LoR as shown in Fig. 11. At each position measure the distance of the source from the midpoint of the LoR and record a pulse height spectrum from the TAC. Select a wide region of interest (two or three times the width of the peak) in software.
NOTEBOOK: For each position, keep the region of interest fixed and record in your notebook the gross counts, live time, full-width half-maximum (FWHM), and peak centroid. Estimate the uncertainty in peak centroid by eye. Save the pulse height spectrum in both *.tsv and *.spu formats.
NOTEBOOK and REPORT: How does the peak change as you move the source?
NOTEBOOK and REPORT: Could you determine the source position only from the peak centroid? How closely spaced can two sources be along the LoR for them to be distinguishable on the PHA? This is the spatial resolution along the LoR. It may be helpful to review the definition of resolution discussed in the Gamma Cross Sections experiment. (HINT: Your PHA measures peak position in units of time, but from the geometry of the problem, you should be able to convert to distance.)
NOTEBOOK and REPORT: Your spatial resolution along the LoR is related to the speed of light, c. Are your results consistent with the known value? There is a special “time-of-flight” tape measure in the lab with distance measured in units of light-picoseconds. Take a look!
{ ${/download/attachments/195462212/Time_res.png?version=1&modificationDate=1542125163000&api=v2}$
Figure 11: Different source positions along the LoR.
Using two identical Na-22 button sources, experimentally determine the spatial resolution both along the LoR by moving the sources until the two distinct peaks appear in the TAC spectrum.
NOTEBOOK and REPORT: Is your experimentally-determined resolution consistent with the resolution determined above? Is it therefore practical to use time-of-flight only to determine peak position in this experiment, or is a full PET scan required?
3.4.2 Measurement of spatial resolution perpendicular to the LoR
In deciding how far apart to space the LoR measurements along each axis, several several factors need to be taken into consideration. Spreading out the LoR's too much will result in poor resolution of the location of the sources. More closely spaced LoR's 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 LoR's 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.
NOTEBOOK: For each position, keep the region of interest fixed and record in your notebook the gross counts, live time, full-width half-maximum (FWHM), and peak centroid (with uncertainty). Save the pulse height spectrum in both *.tsv and *.spu formats.
NOTEBOOK and REPORT: How does the peak change as you move the source?
NOTEBOOK and REPORT: How closely spaced can two sources be perpendicular to the LoR for them to be distinguishable on the PHA? This is the spatial resolution. (HINT: You may need to do a fit of your count rates as a function of position.) Look at the sample box and the slices marked on it. Are the slice separations appropriate given this resolution perpendicular to the LoR?
{ ${/download/attachments/195462212/Spatial_res.png?version=1&modificationDate=1542125163000&api=v2}$
Figure 12: Different source positions perpendicular to the LoR.
3.4.3 Detector coincidence efficiency
The single button sources provided to you for this part of the experiment have a nominal stated activity, but are not calibrated; they come from the manufacturer with activity within ± 20% of the advertised value.
NOTEBOOK: Given that the half-life of Na-22 is 2.603 years, calculate the current activity of the single button sources provided based on the manufacture date and activity stated on the source label. Estimate the uncertainty in this activity.
The efficiency of a detector is typically calculated as the ratio of the number of counts found in the full energy peak of a detector spectrum to the number of photons of that energy which struck the face of the detector. For this experiment, we will define the coincidence efficiency as the ratio of the number of coincidences detected to the number of electron-positron pairs created. This efficiency is highly sensitive to source position and height, as well PMT high voltages; it may vary from group to group.
NOTEBOOK: Using the count rate for a single button source located as close to the center of the two detectors as possible from your experiments above, calculate the coincidence efficiency for your apparatus. In order to estimate the number of electron-positron pairs created, use the activity estimated above and the fact that 10% of Na-22 decays are due to electron capture and 90% of Na-22 decays are due to β+ emission (as shown in Fig. 1.) Estimate uncertainty.
3.5 Scan X, Y and Z slices
As described earlier we will reconstruct a 2-D image of the sample from three slices, each scanned along a different axis (X, Y and Z). Figure 13 shows our sample container with the with markings indicating each of the scan axes and their corresponding LoR's. To create a slice along the x-axis, place the sample container in the sample guide so that the X arrow points in the direction orthogonal to the LoR defined by the two detectors. Position the sample container so that its Ix (_i _= 1) LoR lines up with the LoR defined by the detectors and measure the coincidence rate along this LoR. Repeat for the i = 2 through i = 9 LoRs along the x-axis. To create a slice along the other two axes, simply rotate the sample container and repeat the sequence of measurements.
{ ${/download/attachments/195462212/PET_box.png?version=1&modificationDate=1542125163000&api=v2}$
Figure 13: The 35 LoRs for our 16 cm x 16 cm sample box.
Note that as laid out, a full scan of all three slices will require 35 separate coincidence measurements. Think carefully about how long you plan to collect data along each LoR so that you have time to collect all of the data.
4 Analysis
The best way to visualize your PET scan data is through a contour plot. In order to help you prepare such a plot in python, we provide a contour plot tutorial.
<blockquote>
| * Contour_Example.ipynb</HTML> |
</blockquote></HTML> Adapt the tutorial to plot the functions (Axy)1/2 and (Axyz)1/3. (When plotted this way, the _z-_axis has units of intensity.) Identify the peak positions and intensities, along with uncertainties. State the relative activities of the two sources. Based on count rates for the single gamma button source, estimate for the absolute activity (in Curies) of the two sources inside the sample. (This will be a very rough estimate. What factors contribute to the large uncertainty in the absolute activity?)
Rubric
When writing your report, consult the rubric and notes below for the appropriate quarter.