Resolution of a PMMA phantom using. 200 MeV Protons. Maureen Petterson. University of California, Santa Cruz. June 23, 2006

Transcription

1 Resolution of a PMMA phantom using 200 MeV Protons Maureen Petterson University of California, Santa Cruz June 23, 2006 Professor Hartmut Sadrozinski Thesis Advisor Professor David Belanger Department Chair 1

2 Abstract The feasibility of using Proton Computed Tomography in the imaging of tumors is explored by means of an experimental set up that uses 12 blocks of Polymethyl methacrylate (PMMA) to simulate human tissue. One block has holes drilled out, giving a density difference within the material. The exit energies of protons that have traversed the apparatus are measured with a crystal calorimeter time converter. Resultant energies of the particles traveling through the holes are distinctly different than those traveling through solid PMMA, yielding a 2 d image of the simulated tumor. The minimum dose necessary to resolve a volume element of 1 cm and.6 cm in diameter tumor is found by reducing the number of protons included in the analysis until the image can no longer be resolved. 2

3 I Introduction When a person is stricken with a malignant tumor, there are several options for treatment: chemotherapy, surgery, radiation therapy, or a combination of all three. Chemotherapy and radiation in particular completely ravage the body, often causing complications that can be worse than the cancer itself. Radiation treatment typically uses high energy photons in the hundred kev range (corresponding to x-rays) in order to damage the cancerous cells, thus reducing the size of the tumor over successive treatments. This method of therapy can have horrendous side effects due to the massive amount of healthy tissue that gets also gets irradiated. Though the past few decades have seen huge advances in the medical physics field, X- ray radiation therapy continues to have a high rate of damage to healthy tissue. This has necessitated new developments in treatments that are able to damage the cancerous cells while leaving healthy tissue as unaffected as possible. High energy protons have proved to be more accurate at delivering lethal amounts of radiation to the cancer cells while minimizing damage to the surrounding tissue. Proton treatment centers are becoming more popular as the effects of protons on cancer cells are becoming better understood and as the technology to implement this method is improving. In order to find the exact location of the tumor within the patient, an image of the body must be taken, usually with X-rays. Surprisingly, the X-ray image sometimes gives the position of the tumor with a relatively high uncertainty up to 1 cm depending on where in the body the image is taken (Schulte). Considering the massive dose of radiation administered, 1 cm is a quite a large area of healthy tissue that can be damaged. Recently, scientists and researchers have been studying the use of protons for both imaging and treatment. A wide, lower intensity proton beam could find the exact location of the tumor and then a higher precision, high intensity beam could deliver the radiation necessary to damage the cancer cells. Loma Linda University Medical Center (LLUMC) was the first proton treatment center open in the world and is where we conducted our experiment to test the feasibility of using protons to image various sized tumors. Proton Computed Tomography (pct) is the general term for computerized imaging using protons (as opposed to X-Ray CT). Several different hardware and software components were integrated to create an accurate simulation for imaging a tumor. This experiment, conducted by The Santa Cruz Institute for Particle Physics (SCIPP) and done with collaboration with Loma Linda University Medical Center, looks at the resultant energies of the protons after they have traversed through the body to find out where the tumor is located. This involves analyzing the energy, entrance point, and exit position of every proton that leaves to body to get a composite image. The focus of this paper is analysis of the energy measurement obtained by the calorimeter. II Background Information Proton Computed Tomography The use of high-energy protons in cancer therapy is not new Loma Linda started utilizing the unique properties of protons in order to treat cancer well over 20 years ago. The idea to actually image with protons was first introduced in 1963 by Cormack. The basic technology behind imaging, no matter what the particle, is basically the same. The first apparatus designed to image with protons was quite rudimentary a beam in front of the object being imaged and a photographic plate behind it to capture the particles that passed through the object; much akin to an X-ray apparatus. This yields a very two dimensional image that gives little information on the composition of the tissue since there is no way to distinguish between particles that 3

4 exited with a higher energy from those with a lower energy. A trivial method to distinguish between materials of varying density is to measure the energy of the exiting particle. A higher energy corresponds to a lower density. Using protons for imaging seems like it would be accurate for determining the density differences within the body, but it is not without its setbacks. The biggest problem with proton imaging is multiple coulomb scattering, which will greatly displace a protons exit position from its entrance point. This can yield an incorrect image, as protons will appear to be losing energy in the wrong place (the same way stellar parallax will give the impression that the star is located in a different position in the sky). In the past ten years the errors associated with this anomaly have been reduced due to improved technology, making pct a viable alternative to x-ray CT. pct as compared to X-Ray CT Imaging with protons is very similar to imaging with x-rays; both essentially look at the particles after they traverse through material. Any density difference will show up as a relative change in energy from the surrounding particles that went through a different material. Looking at the energy change of protons gives a much more detailed density map of the body than a typical x-ray image. X-ray machines are fairly simple a photon beam is projected toward the body and the number of exit photons is measured with a detector. The softer tissue in the body is composed mostly of atoms with a smaller Z (where Z is the number of protons in the nucleus), making them less likely to absorb the high-energy photons. Denser materials, such as bone, are composed of high Z atoms that readily absorb the x-rays. The resulting image is very black and white, for the most part showing only areas where X-rays are absorbed and not absorbed. While X-ray imaging is quite good for most medical purposes, it s more difficult to see very minor density fluctuations within the body. The high energy of the x-ray beam also creates more problems due to the possible adverse effects of ionizing atoms within the body. Figure 2-1: a) Proton energy loss in different materials found in the body and b) the X-Ray attenuation in those same materials. Proton energy loss is much more uniform through all materials, while the plot for X-rays differs slightly for bone. (Sadrozinski, et al, 2004) For most medical purposes x-rays are just as effective as protons, but there are unique cases in which protons would be advantageous because of their unique interactions with matter. Unlike photons protons have mass, and therefore interact with cells in a different way than x-rays. The x-rays used in medical imaging interact via Compton scattering, where the photons loses a fraction of its energy in a collision with an electron, and the photoelectric effect, where the photon transfers enough energy to the atom to eject an electron. A high-energy incoming photon is likely to deposit all it s energy within a couple collisions, leading to a very high radiation dose within the first few centimeters of penetration. By contrast, a proton will exhibit multiple coulomb scattering and collide off many particles, depositing a little of it s energy each time. This allows the protons to travel further in the tissue than a typical X-ray. 4

5 Figure 2-2: X-ray interaction with matter. Sufficiently high energy x-rays also exhibit pair production, but those energies are not used in imaging. The protons used in our experiment were on the order of 10^8 ev, while typical x-rays are only around 10^5 ev! It s the unique property of proton interactions that allow such high energies to be used without significantly damaging the tissue. The Bragg peak of protons also makes it possible to control where the majority of the energy is deposited; the higher the energy of the beam, the larger the depth of maximal energy deposit. This is particularly advantageous in cancer treatment since it is very easy to change the energy of the beam so that the maximal dose is administered exactly where the tumor is located. Knowing the exact position of the tumor is of the utmost importance when treating a cancer patient with such high doses of radiation. Initial studies have shown that pct more accurate than X-ray CT and can cut the positional uncertainty up to 60% (Schulte). Figure 2-3: (a) Dose per depth for x-rays and protons (Kraft, 2000), (b) placement of tumor relative to depth of dose (Schulte, 2004) III System Unfortunately cancerous tumors do not reside right underneath the skin, nor are they easily distinguishable from healthy tissue. There were several constraints that the experimental set up needed to reflect in order to accurately determine whether pct is a viable alternative to X-ray CT. 5

6 Silicon Detectors and Data Acquisition Our detectors are 4.5 cm x 4.5 cm x.05 cm silicon strips detectors. Each detector has 192 strips that are connected to 6 chips on the electronics readout board. A module consists of two silicon strip detectors placed back to back and mounted onto the readout board. These modules were placed inside a 30 cm x 30 cm box along with PMMA, a human tissue substitute. Placed directly behind the box was the calorimeter used to measure the energy of particles traversing through it. The calorimeter is type MQT300A, measures 6.35 x 6.35 x 18.5 cm, and was built by Vladimir Bashkirov, a researcher at Loma Linda University Medical Center. Figure 3-1: Schematic of the experimental set up for a run with 12 PMMA. The beam is incident from the left. It travels through the first telescope, then though half the PMMA, through the roving detector, the rest of the PMMA, the last telescope, and then finally through the calorimeter. There are several components to our electronics readout system: the modules, fanout board, FPGA board, NI DAQ card, and the translator board. When a particle with energy above a predetermined value enters the calorimeter, a signal is sent to the FPGA board to tell it to start acquiring data from the detectors and calorimeter. The data from the detectors is readout through the modules into the fanout board as a analog signal. Here the data signal is transformed into low voltage differential signal (LVDS) before being read out to the FPGA board. The FPGA board processes this data and attaches a time stamp to each data set before sending the signal to the translator board. The signal is changed once again into CMOS before finally reaching the NI DAQ card inside the computer. An executable program compiles all the data into a.root file, which will be manipulated and analyzed after the experiment is complete. With a proton spill of about 400 events per second, the data acquisition must occur fairly quickly so that the FPGA can process all the data and send it to the translator board before it s buffer fills up. When the buffer does fill up the overflow data is lost. A more comprehensive description of our data acquisition hardware can be found in a paper by Heimann, J. Calorimeter and Charge to Time Converter 6

7 While the data from the silicon detectors is being read out to the FPGA, the calorimeter is sending the information of the particle energy to a charge to time converter (CTC). The basic function of a CTC is to measure the time it takes for a certain amount of charge to build up on a capacitor s plates and then relay that information to the FPGA. The charge to time converter actually has 4 time stamps that can be analyzed to get the necessary energy measurement. CTC12 is the time between the first and second pulse, CTC13 is the time between the first and third pulse, and CTC14 is the time between the first and fourth pulse. We were looking at the CTC13 measurement to extrapolate an energy because the third pulse is the one that is the most sensitive in the MeV range. This means that it will give a more accurate reading of the energy since it s value respond more to a change in energy than the second or fourth pulses. The CTC transmits this data to the FPGA to be processed with the positional information from the detectors. Calorimeter Figure 3-2: Electronics Readout Schematic Figure 3-3:The four charge to time converter pulses. The pulses are counted from the left. Phantom and Experimental Apparatus In order to simulate human tissue we used Polymethyl methacrylate (PMMA), a clear plastic that responds similarly to human tissue in terms of particle scattering and energy loss. We had a total of 12 blocks of PMMA, each measuring approximately 10cm x 5.5cm x 1.25cm. One of these PMMA blocks had holes of varying depths and diameters drilled out to give a difference in density within the "tissue". To find out exactly where the voxels resided relative to our detectors, precise measurements were made so that a strip on the x or y detector plan could be correlated to the beginning and end of the voxels. A measurement in millimeters is also given: the x distance was measured relative to the left side of the phantom while the y distance was measured relative to the top of the phantom. The results of the measurements are shown in Table

8 ifigure 3-4: The Phantom PMMA. The entire block is 10 x 5.5 x 1.25 cm w with holes of varying depth and diameter drilled out. (Lucia, 2006) Table 3-1: dimension of holes in phantom PMMA Voxel Depth X begin mm X end mm Y begin mm Y end mm first x strip first y strip First X Strip Last X Strip First Y Strip Last Y Strip A Empty B Half C Empty D Empty E Empty F Half G Empty Taking the data In order to get a good idea of how well the density fluctuations could be measured, several different experimental set ups were used. There were 5 distinct runs: 6 modules and no PMMA, 5 modules and 12 PMMA (roving module), 4 modules and 12 PMMA, 4 modules with 11 PMMA, and 4 modules with no PMMA. The first four set ups had a beam energy of MeV while the last set up had an incident beam energy of 100 MeV. The 11 PMMA runs consist of 11 uniform blocks of PMMA, while the 12 PMMA runs are the 11 solid blocks plus the one phantom PMMA. The distance between the notches on figure 3-5 below is 3 cm. A table showing all the relative distances of the detectors and PMMA is given in the appendix. 8

9 Figure 3-5: schematic of different set up. From the left: 6 modules and no PMMA, 5 modules with 12 PMMA, 4 modules with 12 PMMA, and 4 modules with 11 PMMA. The distance between notches is approximately 3 cm. The run with all six modules and no PMMA was done to get a good benchmark of how our hardware and software responded to the beam, as well as calibrate the calorimeter so a relation between proton energy and calorimeter counts (CTC values) could be established. There was also a run with 4 modules and no PMMA that is not show above, but the experimental set up is similar to the run with 6 modules. Immediately after taking a set of data a profile of the beam and the calorimeter data was checked. This verifies that our experimental set up is correct and that all electronics equipment is working correctly. Figure 3-6: Beam profiles from raw data. The profile on the left is the y plane and on the right is the x plane. The position of the beam peak is confirmed by Figure 4-1 in the Analysis section. The first image is the y plane (strips oriented horizontally to give an effective y coordinate) and the second is the x plane. Note that the peak of the beam is not centered this is consistent with the color 2d histogram of the beam shown in figure 4-1. It also implies that many protons could have flown out of the range of our detectors as they were traversing the PMMA. The channels with an extremely high number of hits correspond to hot channels. These anomalies are due to strips that are constantly firing even though they were not hit by a particle, leading to an absurdly high number of events for one channel. The events from these channels will need to be completely cut out before any real data analysis can be done; the method for this occurs after we have finished taking data at LLUMC and will be explained in section 4-2. What we are interested in is the data from the charge to time converter, which is given in a 1d plot. The expected CTC graph is a Gaussian distribution of the different proton energies calculated from the calorimeter; the x coordinate corresponds to the energy and the y coordinate is the 9

10 number of events with that entry. Upon first viewing the CTC13 plot, the long tail of the Gaussian was a bit perplexing how could some of the protons have an energy half of what was expected? These events correlate with protons that for some reason or another (mostly due to scattering out) did not deposit all of their energy into the calorimeter. Figure 3-7: charge to time conversion values before being converted to energy. From left to right: CTC12, CTC13, and on the second row CTC14. The image on the right is a close up of CTC13, the value we used in our analysis. The lines correspond to cuts made for a peak to tail graph, explained in a later section. IV Analysis Alignment The amount of raw data that we were able to get is not enough to do a precise alignment of the beam, which involves analysis of the beam angle and spreading using the positions of the protons at various distances in the apparatus. A detailed description of this is outlined in a thesis by N. Blumenkrantz. In this experiment we had no more than a few hundred events per small subsection of the detector, so we can only look at the position of the beam relative to the other equipment. There are three main components to having our set up aligned relative to the detectors: the beam placement, calorimeter placement, and phantom placement. The majority is the beam resides within the detector area, so we did not lose too many events. However, the displacement of the beam from the center of the detectors does become problematic with the absorber. The scattering of the protons will inevitably lead to some events leaving the detector area. The beam was not perfectly centered, as shown on the plot below. The color scale gives number of events, with red pixels having the most events and blue pixels having fewer events. The first two graphs are before the PMMA; the last two are after the PMMA. The beam spreading is approximately the same even when there is no absorber. 10

11 Figure 4-1: Beam passing through 11 PMMA. This plot was made after the data had been cleaned up using a process detailed below. It corresponds exactly to the one-dimensional beam plots shown in figure 3-6. Each of these plots has the same number of events, so even though it is apparent that some protons did hit some detectors and not others, they are not included in any element of the data analysis. The method for excluding these is detailed in the next section. The calorimeter was centered fairly well with the detectors. We were able to find the effective center of the calorimeter by looking at the ratio of peak events to tail events. The latter are all events with a CTC value between 380 and 840, while the peak events are those that fall between In the one-dimensional histograms that showed the energy of the beam, there was often a long tail. These unexpected energies correspond to particles that for some reason or another did not deposit all their energy into the calorimeter. So the areas of the detector that record more events in the tail than the peak are towards the edge of the calorimeter. Figures 4-2: The peak to tail graphs. From right to left: the 2d plot of peak to tail ratios for the entire detector area, a CTC13 plot looking at an area outside the calorimeter center, a CTC13 plot looking at events inside the calorimeter center. Note the difference in the size of the tail end. The gradual change of peak to tail ratios on the left side is exactly what is expected. The slight non-uniformity of the right side shows that there might be some anomalies with the calorimeter itself density differences, impurities, etc. From this graph, we can deduce that the 11

12 calorimeter is a bit right of center relative to the detectors. The odd response of the right side of the calorimeter is something to keep track of when analyzing the energy and uncertainties in our measurements. The PMMA was aligned using a custom made cradle that was tightly secured at the same level of the detector. Below is a picture of the 12PMMA run with the roving module. The absorber was placed very close to the roving module to allow for maximum precision in calculating the position of a particle. Figure 4-3: 12 PMMA with roving module. The phantom can be easily identified by the holes drilled in it. Cuts on hot channels and clustering Not all events made it through the entire experimental set up the first detectors register a noticeably higher number of events than the last detector. No important data can be gained by analyzing these events- in fact inclusion of these can actually alter the data, so they must be cut out. The raw data files were run though two separate programs: clusterandreorder4.exe and nate4.exe, respectively. The first program was used to cut out the hot channels mentioned above, create new variables that could be manipulated later for more analysis, and organize the data by selecting clusters. The process of creating and selecting clusters involves looking at the areas of the detector where multiple neighboring strips had been hit and then forming a cluster of these hits. The cluster gets bigger when more neighboring strips are hit and stops increasing in size when there are no hits on the adjoining strip. (Feldt, J.). If a hot channel that has been cut out is inside the cluster, it takes on the value of the neighboring strips, so there are no empty channels in the beam profile. On the two dimensional beam plot shown in the Alignment section, the blue horizontal and vertical lines that cross the image are actually due to channels being cut out of the data. The second program cuts out the events that did not hit all the detectors and creates more variables to analyze the entry and exit angle of the particle (analysis that can be done independently of the energy). A more detailed description of the clusterandreorder and nate4 program can be found in papers by J. Feldt and N. Blumenkrantz. 12

13 Figure 4-4: beam profile after going through cluster and reorder. It is still consistent with the 2 d beam plot in figure 4-1. There are still a few hot channels that did not get masked out, but Analysis of the data from the Calorimeter It s quite obvious from looking at the CTC graph that the calorimeter does not give a direct measurement of the energy. Rather, it gives a CTC count a number that corresponds to the energy. The exact CTC value used for a given run is determined by fitting a Gaussian to the front end of the CTC plot. The events with the higher energy (the ones displayed at the front end) are the events that went all the way through the calorimeter and most accurately represent the exit energy of protons. The back end hits correspond to protons that left the calorimeter before all their energy was deposited. The mean of this Gaussian, or the x value of the peak, is our energy it just needs to be converted from a CTC value into MeV. This energy scale was determined by calculating two values: the CTC value for a run with only 6 modules and a beam energy of MeV and the CTC value when there is no incident beam. The actual exit energy for the beam was found by looking at the energy loss of protons in silicon in the NIST database. The stopping power for protons of a specific energy is given the actual energy loss is calculated by multiplying that stopping power (de/dx) by both the density of the material and the distance the protons have traversed. The energy loss is recalculated every.1 cm within the PMMA to obtain a more accurate result. Since de/dx is proportional to the incoming energy, the energy loss through the first cm of the PMMA is different than the energy loss in the last cm of PMMA, and hence needs to be recalculated often. de de = ρ Δx Equation 1 dx de is energy loss, de/dx is stopping power (obtained from NIST), ρ is the density of PMMA ( g/ cm ), and Δx is the distance that the proton has traveled. The MeV incident beam ends up with an energy of after it has gone through 6 modules. In the case where there was no beam, the energy is considered 0 MeV. These two CTCenergy values are plotted with a fitted trendline to determine the exact relation between the two. 13

14 Figure 4-5: The charge to time conversion plots for the runs with a MeV beam, no beam, and 100 MeV beam. Also note that the Gaussian is fitted only to the front end of the distribution. The 100 MeV run had fewer events so the peak of the Gaussian is not well determined, and thus was not used in the CTC-energy calibration. Table 4-1: The CTC value and energy for the three cases without PMMA which are being fit to acquire a ctc-energy conversion. run ctc expected energy error pedestal y = x x Quad CTC calibration 5Si12PMMA 4Si12PMMA 4Si11PMMA 5Si11PMMA y = x x Quad CTC calibration 5Si12PMMA 4Si12PMMA 4Si11PMMA Figure 4-6: The graph of the CTC values for the absorber runs plotted with the fit line for the no absorber CTC values. Left, a magnified view of the CTC values in relation to the line. From the data we get that, 2 CTC value =.0002x x Equation 2 Once we have this relation we can easily convert all the CTC counts into energy values, and compare these to the theoretical results of energy loss in PMMA and silicon. 14

15 Table 4-2: theoretical final exit energy and measured final exit energy for the different runs Type of Run Initial Energy (MeV) Final Energy (MeV) Measured Energy 5 modules, 12 PMMA modules, 12PMMA modules, 11 PMMA Initial estimates actually show the theoretical energies disagree with the experimental values found using the CTC-energy conversion described above. The CTC values for all the runs are very well determined, with an uncertainty of at most.6. The theoretical energies are also well determined, so the discrepancy in values is unexpected. A plot of the actual CTC values of each run is plotted with respect to the line fitted in figure 4-6. All the points fall the same distance from the line, implying that there is a systematic behavior not being accounted for. 11 PMMA Graphs Once the data has been cleaned out, in depth analysis of the energy can be done. The first step is to create a two dimensional image of the energy. We first looked at the 11 PMMA spectrum to make sure it was uniform. All the blocks were identical, so the energy reading over different areas of the detectors should be the same. All of the one and two-dimensional plots are created with the same algorithm. The detector is subdivided into several equal parts and the energy for each section is calculated and plotted. For the graphs shown here, the detector was divided into sections that were either 10 strips x 10 strips wide or 5 strips x 5 strips wide, resulting in a plot with either 100 or 400 pixels. Each pixel is representative of the CTC13 value calculated from the Gaussian fit over the events in that section. As shown in figure 4-7 below, most of the detector area is uniform, but there are some energy values that vary significantly. Figure 4-7: energy and sigma from the 11 PMMA run (each pixel is 10x10 strips) 15

16 Notice that the mean is fairly uniform in the center and varies greatly at the edges. This is due in part to the method of Gaussian fits. The value for the energy (CTC value) is determined by automatically fitting a Gaussian in ROOT, but not all subdivisions have enough events to fit a well-defined Gaussian. The result is a graph with energy values that are several sigmas away from where they should be. The increase in efficiency by doing the automatic fit causes a decrease in resolution and accuracy. In the plot above, the image on the left is the energy for the entire detector area while the image on the right is the sigma (width of the Gaussian) for the same run. The areas of extremely high sigmas correspond very well to the areas that have energy values far from what is expected. The general area of uniformity also corresponds to the beam peak. Figure 4-8: Incorrect Energy calculated due to a bad gaussian fit. The mean is displayed at 34 even though it is closer to 650. Figure 4-9: mean energy for 11 PMMA run. The pixels are 2.36mm x 2.36mm wide. After making sure that there are no anomalies that are not understood, we can cut down the area of the detector that we are analyzing so that inaccurate data does not affect our final results. The first and last 5 strips from the x plane detector were cut out, and the first 5 and last 15 strips of the y plane detector were also cut. The final image of the energy for the 11 PMMA run is shown above with pixels that are 10x10 strips wide. It is almost perfectly uniform, exactly as expected. To obtain better resolution (which is the ultimate goal of imaging), the detector is subdivided into 5 strip by 5 strip wide sections and then replotted. With such small pixels, often there were not enough events for the automated Gaussian to fitted correctly. This resulted in a far worse image, with almost 15% of the data fitted incorrectly. One could go in and hand fit the mean for each pixel, but with over 400 pixels, that would be quite inefficient. To remedy this, an equation for a Gaussian was constructed with parameters that constrained the mean and width of the Gaussian to within certain values (these were slightly different for each run). The equation used is 2 x β σ 2 = α e f ( x) Equation 3 The parameter α is the normalization constant and β is the mean. Here the sigma is fixed to be 12, the average sigma found on the well fitted gaussian graphs. The mean is constrained between and the normalization constant is constrained between These are small enough to give an accuate CTC value, but large enough to keep the results unbiased. 16

17 Figure 4-10: (a) 11 PMMA energy with the general Gaussian, (b) and the custom gaussian Now that the accuracy of our set up and analysis has been validated, we can look at the runs with the phantom PMMA. 12 PMMA Graphs The expected result of the 12 PMMA runs is a fairly uniform background, similar to what the 11 PMMA run looked like, but with some raised areas that correspond to the protons that went through the voxels and have a higher energy as a result. Figure 4-11: The 12 PMMA energy looking at the events on the roving detector plane with each pixel 10x10 strips wide. There are two choices when analyzing the 12 PMMA runs analyzing the events that hit the roving module, which was placed directly behind the phantom PMMA, or looking on the last module. The advantage of the former method gives a much sharper picture of the voxels due to less multiple scattering, making the analysis of energy and resolution more accurate. However, a tumor is 17

18 not often at the surface of the skin so looking at the events hitting the last module would give a more realistic image. This paper uses the first method for energy calibration and dose. Subsequent analyses will use theoretical modeling of the most likely path of a proton through tissue in order to reconstruct the trajectory of the proton (see Bluenkrantz, 2006). This will allow a sharp image to be created from the data on the backplane, further illustrating the practicality of proton computer tomography. Figure 4-12: (a) The image of the phantom using events on the last plane (b) and the roving plane (right)! On the first image, it is easy to see where the voxel is, but the image is not sharp enough to detailed analysis. With the image on the right, both the 1 cm diameter and.6 cm diameter voxels can be easily seen. The pixels here are 5x5 strips wide. Some of the pixels in the plot still register a slightly higher energy than expected. This is due in part to the gaussian fit still not being perfect, especially when there is a lack of events in a particular pixel. Using 5 strip by 5 strip pixels reduces the number of events significantly in certain areas (especially near the detector edge). The voxel is easily seen with the current method of analysis, so now it is desired to look at just the voxel versus the background to see the energy difference between the two. Three areas were selected for comparison a 35 x 35 strip area inside the voxel, and two 35x35 strip areas directly to the right and left of the voxel. The last two control areas give the energy of the background, the energy the protons would have if they had traversed through uniform density tissue. We are most interested in the difference between these two measurements and the relation to the uncertainty on the mean of the energy of the protons that went through the voxel. The difference between the two measurements needs to be at least a factor of two greater than the uncertainty on the mean in order for an accurate distinction to be made between the two energies; this is known as the 2σ significance. Mathematically, 18

19 M C M V σ M 2 Equation 4 Mc is the mean in the control area (background), Mv is the mean in the voxel, and uncertainty in the mean in the voxel. σ M is the Figure 4-13: The areas selected for in depth energy analysis. The value of the small voxel in the upper region of the detector area was observed as well. The values for the mean of all three sections were calculated by obtaining the CTC values for the events in the selected areas and plotted with a hand fitted Gaussian. The peak of the gaussian is the average mean energy of the values, and each of these has it s own parameters associated with the fit, including the error on the mean, sigma, and error on the sigma. The results for the three Gaussian fits are listed below in table

20 20

21 Figure 4-14: The Gaussian fit results for the area inside the voxel and the area directly to the left and right, as shown on figure The energy difference between the voxel and background areas is readily apparent. Table 4-3: the parameters from the Gaussian fits for the 35 strip x 35 strip voxel and two control areas. Area # entries mean error on mean sigma error on sigma voxel control L control R It is also interesting to see how much each individual pixel in one of the areas deviates from the mean energy of the entire area. This is useful in order to see how uniform the energy in each pixel is in the particular area. Looking at the deviation of the pixels from the total average was also done on the same areas on the 11 PMMA runs (figure 4-16) to check for any inconsistent behavior. Distribusiton of mean energy in voxel pixels More Figure 4-15: The deviation of the mean of individual 5 strip x 5 strip pixels from the average mean over the entire voxel. A histogram also shows the same deviation. Frequency 21

22 Histogram - 11 PMMA voxel More Figure 4-16: The same figure but for 11 PMMA this shows that the result from the 12 PMMA is consistent with the 11PMMA runs. Frequency Bin V Medical Interpretation Dose Reduction by Limiting the Number of Events Ultimately, we are concerned with the dose of radiation necessary in order to resolve the image of a tumor with some finite precision. The energy fluctuations of the voxel versus the background are quite noticeable in figure 4-12b, so why not reduce the number of protons until the image can no longer be resolved? This would be equivalent to finding the dose at which the tumor cannot be distinguished from the surrounding tissue. The following figures are the same plot as in figure 4-12, but the number of events has been reduced. The reduction of events was done by counting every second event, every fourth event, every eighth event, etc in order to keep any bias out of the cutting. N = 2 corresponds to cutting the events in half, n=4 is selection the total number of events by a factor of four, all the way up to n=128. Since each pixel is quite small, only 5 x 5 strips, cutting events increases the odds that there will be no or very few events in the pixel. The result is a very blurred image, as in the figures where the events were cut by a factor of 32 and 64. With so few protons, the resolution of the voxel is marginal at best. 22

23 Figure 5-1: The mean of the detector area with reduced events. The upper left corner has half of the original events, while the lower right corner has the events cut by a factor of 64. The white pixels in some of the plots are due to a lack of events in the area. With no events, no Gaussian can be fit, and hence there is no energy value. As the events per pixel are reduced, the Gaussian fit become less accurate, often giving a CTC value that deviates noticeably from the previous CTC value in the plot with all events. The result is a seemingly random pattern of differing energies, as shown above. The average number of protons per pixels is less than one on the data set with entries reduced by a factor of 128. The high incidence of zero events in the pixels is described by the Poisson distribution if the mean number of events in a pixel is close to 1 or 2, then the probability of getting 0 in a pixel is quite high. The mean can actually be estimated from the number of zero event pixels out of the 1080 pixels in the entire figure. Table 5-1: Number of entries for each n, where n is the factor by which the number of events were cut. When n = 128, there is only about 1 proton per pixel inside the voxel. n # entries voxel #entries per 5sx5s pixel # entries control L #entries per 5sx5s pixel # entries control R #entries per 5sx5s pixel

24 Analysis of the 1 cm voxel By looking solely at the images above, it appears that the image cannot be definitively resolved if n is greater than eight or sixteen, but other methods of analysis can improve this. One possible solution to a badly resolved image is to increase the size of the pixels so there are more events to fit a Gaussian to. Below are energy plots with each pixel 35 x 35 strips in size, starting with the full number of events in the upper left corner and a reduction by a factor of 200 in the lower right corner. With this method of analysis, the energy of the exiting protons is easily calculated and has a smaller error (since the error scales with the square root of number of entries). Evidence of the smaller voxel is also apparent in the center pixel on the top row. 24

25 Figure 5-2: the energy for pixels sized 35 x 35 strips. The voxel is easily seen in the first few plots. After reducing events by a factor of 32 the energy of the neighboring pixels fluctuates, making distinction between the background and voxel more difficult. Note the center pixel on the top row has a higher energy this corresponds to the small voxel. From the plot above, it is fairly evident that cutting the events by more than a factor of 128 will severely affect the resolution of the image. Again we look for the reduction at which the 2 sigma significance is reached. Reducing events until the lower limit is reached yields the minimum dose needed. The control mean used for comparison is the left section, as it has a similar number of events per area as the voxel. Table 5-2: finding the minimum dose via calculating the difference on the mean of the voxels divided by the error on the mean. The Gaussian fits are shown in the appendix. n # entries voxel mean error on mean # entries control mean control Mv- Mc/(error on Mv) The means and errors in the table above were found by plotting the CTC values for the 35 x 35 strips sections shown in figure 5-2 in a 1d histogram and extracting the data from a Gaussian fit. When looking at a 1 cm diameter object with pixel size.682 cm^2, the minimum dose necessary corresponds to 1/128 of the incident dose, or 6.29E-05 mgry as shown on the table below. If looking for a higher resolution with a smaller pixel size (such as the images shown in figure 5-1), then the minimum dose would correspond to 1/8 of our beginning dose, or 8.12E-04 mgry. However, the dose is the number of events after our data has been cleaned up using the clustering program described in the Analysis section. It also includes all the events in the pixel (low energy and high energy), not just the events needed to fit a Gaussian. As such, there is actually a lower dose than what is shown in the tables below. 25

26 Table 5-3: Dose per reduction in events. This is for the total number of events, not just the ones used to fit the gaussian Area Detector Area (cm^2) n # of entries Fluence (p/cm^2) de/dx (MeV cm^2/g) Dose (MeV/kg) Dose (mgry) Large voxel 7x E E E E E E E E E E E E E E E E E E-0.5 Table 5-4: The dose in the selected areas of the detectors. The variation is due to the concentration of the beam in the upper left quadrant of the detectors. Area Detector Area (cm^2) # of entries Fluence (p/cm^2) de/dx (MeV cm^2/g) Dose (MeV/kg) Dose (mgry) Large Voxel E-03 Control L 7x E-03 Control R 7x E-03 Each of the pixels can be represented as a 1d plot similar to figure The parameters associated with each Gaussian can be manipulated to check the statistics of our fit. The error on the mean σ M, should be roughly equal to the sigma, σ, of the Gaussian divided by the square root of the number of entries. It is not the total number of entries on the plot, but rather the number of entries that are needed to fit the Gaussian the protons that contribute only to the front end of the plot. Thus, the actual dose needed to determine the energy of the protons is less than the dose given by the total number of entries in each CTC graph. The necessary number of events were calculated by integrating over the Gaussian. This yielded the amount of entries, which is shown to be roughly a factor of 3 less than the total number of entries in the entire pixel. Table 5-5: The dose needed just to fit a Gaussian. This is the true number of event needed to determine the energy of the protons after they have traversed through the body. Area Detector Area (cm^2) n # of entries in peak Fluence (p/cm^2) de/dx (MeV cm^2/g) Dose (MeV/kg) Dose (mgry) Large voxel 7x E E E E E E-04 26

27 E E E E E E E E E E E E-06 5 Now rechecking the dose for n = 128 give a value of mgry for 35 x 35 strip wide pixels. The number of events needed to determine the energy is actually 23, as opposed to the 52 events in Table 5-3. error on mean error on mean vs sigma/sqrtn sigma/sqrtn Figure 5-3: A graph of the error on the mean versus sigma/ N. N is the number of events under the Gaussian. The slope is roughly one, verifying that the statistical data agrees with previously known relation between errors and number of entries. The final value for the number of protons occurs after the raw data has been cleaned up through the clusterandreorder and nate4 program. The number of protons used in the analysis is slightly different than the incident protons. Refinement of both the analysis and experimental setup will increase the percentage of particles used in the data analysis. Information on the original number of events is important, but the most pertinent value is dose calculated solely from the number of protons in the image, as this value is the minimum number of events needed to resolve an image. Different experimental set ups will have varying ratios of incident protons to protons used in the image, but the net number of protons necessary to resolve the tumor will stay constant. Analysis of the 6 mm voxel 27

28 The voxel that measured 6 mm in diameter and 6.2 mm in depth is also of considerable importance. The dimensions alone make it an accurate representation of a tumor, and the imaging done from just one dimension is similar to the image that would be obtained by doing a 3 dimensional analysis. Due to a lack of events, this voxel is a bit more difficult to discern from the background. Analysis must be done with the same method above: by dividing the detector area into 56 pixels (each pixel measuring 20 x 20 strips) and looking at the resultant energy of each pixel. The voxel itself measures approximately 20 x 20 pixels, so the energy of the protons that went through the voxel versus those that didn t is quite noticeable. Figure 5-4: The energy or pixels sized 20 x 20 strips. The upper left corner is for n=1 and the lower right corner is n=32. The.6 cm voxel is easily seen in the n=1 plot as the center pixel in the top row. The pixel directly below it contains a couple strips that were also located inside the voxel, which is why it also has a higher energy from the background. Table 5-6: The mean in the small voxel and in the control area. The maximum factor the events can be cut down by is 8 that is the point at which the energy of the voxel gets noticeably close to the energy of the background. (Mvn mean control mean voxel Mc)/error on Mv

29 For n>8, the 2 sigma significance is still seen, but that is only for the specific control areas selected. Other parts of the detector have changed markedly so the distinction between the voxel and the background is not well defined. 4 Table 5-7: The dose per n for the.6 cm voxel. The dose that corresponds to n=8 is mgry. Area Detector Area (cm^2) n # of entries Fluence (p/cm^2) de/dx (MeV cm^2/g) Dose (MeV/kg) Dose (mgry) Small voxel E E E E E E E E E E E E-04 From the data above, it is easy to see that the events can be cut down by a factor of 8 without losing a significant amount of resolution. Analyzing this data with the same method used for the 1 cm voxel, the true minimum dose can be calculated by looking at the events only as the front end of the gaussian. Table 5-8: The dose measured from including events only at the front edge of the Gaussian. The number of entries is about a factor of 2.5 smaller than in Table 9. Area Detector Area (cm^2) n # of entries Fluence (p/cm^2) de/dx (MeV cm^2/g) Dose (MeV/kg) Dose (mgry) Small voxel E E E E E E E E E E E E-05 29

30 4 The minimum dose needed to resolve the.6 cm voxel is mgry when using pixels that are 20x20 strips. Referring back to figure 5-4, notice that the small voxel cannot be easily distinguished for n greater than 8 when using pixels that are 5 x 5 strips. Implications Putting all the analysis together shows that the minimum number of events needed to image the 1 cm voxel is 10 per square centimeter, amounting to a total of 52 particles in a pixel sized 35 x 35 strips. The.6 cm voxel requires 300 events per square centimeter, amounting to a total of 718 particles in a pixel sized 20 x 20 strips. The dose that corresponds to those values for the 1 cm 5 4 and.6 cm voxels are mgry and mgry, respectively. The values found by Reinhard Schulte, a medical physicist at Loma Linda for resolving smaller voxels with less contrast are between 5.48 mgry and 1.37 mgry. Dose 3.00E E E E E E E+00 Dose per depth Density Difference 1 cm voxel.6 cm voxel Object contrast (%) mgy 1.37 mgy Object diameter (mm) Figure 5-5: (a) Dose per depth. (b)dose per depth plot from LLUMC. The dose is much higher than what we found because the object contrast is much smaller. The graph on the left shows the two minimum doses needed to resolve the 1 cm and.6 cm voxels. More data with difference density differences would yield a fitted line similar to the one in the graph on the left. As the density difference between the tumor and surrounding tissue is decreased, a larger dose is needed to resolve the area. Improvements on future experiments There are several areas that can be improved upon for future experiments that test the viability of proton computed tomography. The necessity for an appreciable amount of data to analysis is apparent since such a high percentage of the data is cut out. Increasing the data we have by a factor of two or greater would allow a more precise alignment of the beam, further improving the 30

31 resolution of our image and allowing greater accuracy when using theoretical modeling of a proton trajectory to image from the last detector. VI Conclusion Proton Computed Tomography is indeed an efficient and accurate method of imaging, particularly in the treatment of cancer. The purpose of this experiment was to resolve the image of a tumor with the minimum number of proton (and thus smaller dose). From the plots and 5 tables above, it is shown that the minimum dose needed is on the order of mgry if 4 looking at large pixels, and as high as mgry if looking at smaller pixels for the 1 cm 3 voxel. The.6 cm voxel needs a higher dose to get the same resolution; from mgry to mgry. The larger voxel requires a smaller dose to resolve the image, which is exactly as expected. An increase in dose would correlate to a higher resolution, allowing the imaging of areas 1 mm and smaller. Higher precision implementation of the beam would also result in less cleaning up of the data. This would increase the number of protons used in the imaging process relative to the number of incident protons, thus reducing the overall dose. More in depth analysis, such as tracking the proton trajectories with the theoretical most likely path would further sharpen the image and allow the image to be resolved on the last detectors. The typical dose for x-ray imaging coupled with the immense damage that x-rays inflict upon tissue make pct a much safer and effective alternative. Appendix A1: Table showing the distances of the different components within the box. All measurements are in mm. 31

32 Run beginning of first module beginning of second module end of second module beginning of PMMA end of first chunk of PMMA 12 PMMA roving z = 0 z = 30.0 z = z = 50 z = 125 beginnin g of roving module z = end of roving module beginnin g of second chunk of PMMA end of PMMA beginning of third module beginning of fourth module z = z = 145 z = 220 z = 240 z = PMMA 4 mod z = 0 z = 30.0 z = z = z = z = 240 z = PMMA 4 mod z = 0 z = 30.0 z = z = z = z = 240 z = 270 A2: Table showing all the parameters for the 35x35 strip sections with reduced entries Area n # entries # entries in gaussian peak sigma error on sigma mean error on mean Sigma /sq.rt n Mv- Half of Mc/Mv entries sigma/sq.rt n Ratio of sigmam/( sigma/sq rtn) large voxel Control L Control R A3: Table showing the parameters for the 20 x 20 strip sections with reduced entries. entries error on error on sigma/sq.rt error on Area n # entries in peak sigma sigma mean mean entries/2 n mean/(sigma/sqrtn) voxel

33 A4:Histograms showing the frequency of means within the 35 strip by 35 strip sections, with pixel size of 5 strips x 5 strips. Frequency Histogram - 11 PMMA voxel Bin More Frequency Histogram - 11 PMMA controlr Bin More Frequency Histogram - 11 PMMA controll More Bin A5: 1d plots for the.6 cm voxel 33

34 A6: 1d plots for all the events from each PMMA run 34