This is the last of our four introductory lectures. We still have some loose ends, and in today s lecture, we will try to tie up some of these loose

Transcription

1 This is the last of our four introductory lectures. We still have some loose ends, and in today s lecture, we will try to tie up some of these loose ends. 1

2 We re going to cover a variety of topics today. We ll talk about relating wavelength of radiation to the photon energy, that is, relate the wave nature of electromagnetic radiation to the particulate view of electromagnetic radiation. We ll look at the various components of the electromagnetic spectrum. We ll talk a bit about how we refer to energies within the electromagnetic spectrum. We ll look at a little bit of relativity and be able to relate kinetic energy to relativistic energy and look at rest mass in terms of energy. Finally, we are going to look at exponential behavior, which is a behavior common in radiation medicine, and identify quantities like half-life and average life. So, there are a several loose ends that we will try to tie up in this lecture, and not everything is going to be related. 2

3 The first loose end is characterization of the quantum nature of radiation. We have been talking about electromagnetic radiation as waves. We all know that from the quantum theory that photon radiation has a dual nature; we can look it as either waves or as particles. There are a number of particle properties that a photon radiation beam will have. We can relate the energy of the photons to the frequency and/or wavelength by the relationship E = h or E = hc/λ. E is the photon energy and h is Planck s constant, which is Joule-seconds. The quantity is the frequency of the waves, λ is the wavelength, and c is the speed of light, which equals meters per second; c is really 3.00, or more precisely, meters per second. In some circumstances we are going to look on electromagnetic radiation as a beam of particles. We call these particles photons. Photons have no mass and photons carry energy. The energy is going to be related to the frequency or wavelength of the electromagnetic radiation. 3

4 Let s connect wavelength to energy. We often express the energy in electron volts, which is a very common unit of energy. An electron volt is the energy obtained by an electron accelerated through a potential of one volt. We often express wavelengths in Angstrom units. It is very useful at times to relate the wavelength in Angstrom units to the energy in electron volts. So let s plug in some numbers. The constant h, we said, is Joule-second and c is 3 X 10 8 meters per second. So if we express the wavelength in meters, the conversion to Joules is That s a very hard number to remember, and it is not very useful because we rarely express wavelengths in meters. But if we now convert units, we divide that by to convert Joules to electron volts and multiply by Angstroms per meter, we find the conversion factor is a very easy number to remember, So if you can express the wavelength in Angstrom and divide the wavelength into 12.4 you will have the energy in kiloelectron volts. I m giving you this equation because it is an easy one to remember as there will be times when you want to convert a wavelength to energy or vice versa and just remember that is energy in kiloelectron volts is equal to 12.4 divided by the wavelength in Angstrom. That s a useful relationship. 4

5 Let s look at some conventions for expressing energy. When do we use electron volts and when do we use volts? When do we use kiloelectron volts and when to we use kilovolts? Electron volts, kiloelectron volts, and megaelectron volts are quantities used to express energy. They are typically used to describe an electron beam or a proton beam, which are nearly monoenergetic beams. These units are also used to describe a monoenergetic photon beam. Kilovolts peak and megavolts peak are quantities used to represent the maximum energy of photons in a polyenergetic photon spectrum. Volts, kilovolts, and megavolts are quantities that describe electrical potential. We use these quantities to describe the accelerating potential in an X-ray machine or in a linear accelerator. 5

6 Let me give you an example of this. We will talk of an 18 MV photon beam exiting a linear accelerator. That s actually a nominal energy, a name given to describe the energy of the beam. We could just as well have called them blue photons as well as 18 MV photons. It is just a name to differentiate the photons from a 6 MV photon beam from a linac. But the key point in labeling an 18 MV linac beam is that the electrons in the linac are accelerated to approximately 18 million electron volts of energy. I said approximately because the actual acceleration depends on the design of the linac. These electrons can be removed from the accelerator as an 18 million electron volt electron beam. So the electrons having been accelerated to 18 million electron volts can be removed as an electron beam or we can take these electrons, have them impinge on a target and generate an 18 MV X-ray beam. The beam really consists of a whole spectrum of X-rays that comprise the polyenergetic beam emanating from the linac. In fact, the average energy is typically about 1/3 of that. So roughly, the average photon energy is 6 MeV, but we say we have an 18 MV X-ray beam. Again it s a name maximum energy will be around 18 million electron volts, mean energy will be roughly 6 million electron volts, but calling it 18 MV describes the energy spectrum. If we have a cobalt machine, the energy of the photons are emitted from a cobalt source are 1.25 MeV. So the 1.25 MeV designates a mono-energetic photon beam. The phrase 18 MV, using the units MV rather than MeV, designates a poly-energetic photon beam. We could have used MVp, but as a matter of convention, we don t. 6

7 Let s look at other radiations in the electromagnetic spectrum. When we talk about radio waves we usually characterize the radio waves in terms of frequency. The AM frequency band goes from 550 khz all the way up to 1500 khz, whereas the FM band ranges from 87.9 MHz to MHz. Microwaves are characterized by either frequency or wavelength. 450 MHz microwaves are the frequency that has sometimes been used in hyperthermia; that is, the use of heat for treatment. 3,000 MHz microwaves are produced by the power supply in the linear accelerator. Now, if we look at the wavelength that corresponds to that frequency we find that wavelength is roughly about a centimeter. In fact a cross sectional dimension of the wave guide in the linear accelerator is about 1 centimeter. So that gives us some hint as to the design of the linac. If we want something to have waves of roughly 1 centimeter traveling down this wave guide we need a cross section of about a centimeter. 7

8 Now we will look at infrared, visible and ultraviolet light. Here we characterize the radiation in terms of its wavelengths, which are typically expressed in Angstrom or nanometers. We really should be using nanometers, but I think that out of force of habit, physicists tend to use Angstrom. And finally, when we talk about X-rays, we have the following energy ranges: For diagnostic imaging, energies are in the kilovolt (kvp) range, mammography is 30 to 40 kvp; x-rays used in conventional imaging are about 70 to 140 kvp, and x- rays used in CT scanners are about 120 kvp. For radiation therapy, we actually have some low energy therapy called Grenz rays, an old-fashioned therapy device, less that 20 kvp. Going up in energy, we have contact X-rays. You may have seen some of these units but those devices are fairly old. They operate in the energy range 20 to 50 kvp. Superficial X-rays go from 50 to 150 kvp, orthovoltage are in the range 150 to 300 kvp. Finally megavoltage x-rays, which is the most common radiation used in radiation therapy these days, are in the range 1 MV to around 50 MV, with the most common energies in radiation therapy in the range of 6 MV to about MV. So this gives you kind of a feel for the energies that we use. Here s a question. What are the implications of selection of energy in diagnostic x-ray imaging? It turns out, and we ll see this in more detail later in the course, that lower energy X-rays provide a lot more contrast. In mammography, for example, you are trying to look at very small calcifications in essentially a fatty matrix. So it turns out that the low energy X-rays are needed for breast imaging. As you increase the energy of the x-rays, you increase the penetrating ability. You wind up with less contrast with higher energies but you might want to penetrate thicker body parts, delivering lower dose to the patient at the higher energies, so we go up in the range of 70 to 140 kvp. With the higher energies, there is less attenuation of the x-ray beam, so for the same amount of radiation reaching the image receptor, the higher energies deliver less radiation dose. The specific energies we use in imaging are selected based on the nature of the examinations. 8

9 Now we ll talk a bit about relativity. We know that in classical physics, kinetic energy is one-half mass times velocity squared. You all got that in your kindergarten physics class. What happens when we go to high kinetic energy? In radiation oncology we use electrons with energies of 1 MV, or 10 MV, or something like that. If we try to use the classical equation for kinetic energy, we find that we will have electron velocities that exceed the speed of light. That s clearly a No No. We are not allowed to do that. As the electrons increase in energy the speed of travel of the electrons starts reaching the speed of light so we need to start worrying about relativity. The increase in energy of an electron when we are at these higher energies really comes from an increase in the mass of the electron. Mass increases so we never really get to the speed of light. The energy of the electron now is given by mc 2 where m is now the relativistic mass, not the rest mass. The rest energy is m 0 c 2, in which m 0 is the rest mass of the electron, but now the mass increases with the velocity. So one of the problems with electrons is that if we re dealing with relativistic energy, we have an increase in mass and the energies we deal with are relativistic. This is one of the reasons we don t use a cyclotron to accelerate electrons. A cyclotron is a device that accelerates charged particles in a magnetic field. However, in order for a cyclotron to work, the mass of the accelerated particle has to be constant. For protons used in radiation therapy, the mass is constant, but because of relativistic effects, for electrons in radiation therapy, the mass is not constant. 9

10 So the mass increases with the velocity. Beta is the velocity divided by the speed of light. The mass is equal to the rest mass divided by the square root of 1 β 2. This falls out of relativity. So what is the kinetic energy? The kinetic energy is the total energy, mc 2, minus the rest energy m 0 c 2. So it is m 0 c 2 times 1 over the square root of (1 β 2 ) - 1. That s the kinetic energy. 10

11 Now what we are going to find is that in the limit of small velocity, the nonrelativistic limit, small values of beta, we can actually show that the kinetic energy is one-half times the rest mass times v 2. Note that 1 over the square root of (1 β 2 ) - 1 can be approximated as 1 + ½β 2 1. Remember how to do a Taylor series expansion for small β from first grade calculus. So this quantity in the parenthesis is ½β 2. So the kinetic energy is m 0 c 2 x ½ β 2. The c s cancel out and the kinetic energy then is ½ the rest mass times v 2. Kinetic energy is ½ times the rest mass times v 2 and total energy is relativistic mass times c 2. 11

12 This is a very important table for you to look at. It has beta and total mass for various energies. Notice that for a 100 kev electron, beta is equal to 0.5 and the mass is 1.2 times the rest mass of the electron. So even with a 100 kev electron, an energy we obtain in an X-ray tube, we are starting to worry a little bit about relativistic effects because the electron mass is somewhat more than the rest mass. When we get to a 1 MeV electron, not even the energy we have in a linear accelerator, we find beta is 0.94 and the mass of the electron is 3 times the rest mass. For a 10 MeV electron, beta is and the electron mass is over 20 times the rest mass. So electrons, especially those we use in radiation therapy, are relativistic. For 100 MeV protons, beta is 0.43, and the mass is 1.11 times the rest mass. Typical proton energies in radiation therapy are in the range MeV, so we start looking at relativistic issues for protons, but we don t really have to worry too much about relativistic effects there. For electrons, however, we have to worry about relativity. 12

13 Now, note two important mass-energy relationships. These are numbers you will have to probably memorize. There will not be a lot of memorization in this class. You don t have to memorize a lot of numbers, but just because you use certain things a lot and you don t want to have to stumble through your notes to look up some numbers, here are a few you need to memorize. You need to remember that the rest mass of an electron is MeV. You are going to use this number so often that it will be useful to remember it. Also, remember that one atomic mass unit is MeV. Again, that is a number you will use a lot, so it wouldn t hurt to remember that one as well. 13

14 The last thing I want to look at today, and it is actually going to take a little while, is exponential behavior. We are going to see exponential behavior in a large number of applications in this course. Basically the idea is if there is a quantity that changes by a certain factor in an interval, whether that interval is time or whether that interval is distance, the behavior is exponential. Here are some examples of exponential behavior: Cell growth is exponential. Cell kill is exponential. Radioactive decay, the build-up of radioactive material and attenuation of photons are all events that exhibit exponential behavior. 14

15 What is common to all of these is that they are stochastic events. As we pointed out in the first lecture, stochastic events are controlled by laws of probability they are not deterministic. An isolated event is probabilistic, if it is a stochastic event. And we are going to show now that if you assume that an event is stochastic, when you have a large number of these events, the probability of that event occurring is going to be governed by exponential behavior. Let s use an example. The example is we have n nuclei of radioactive material. Will a given nucleus decay if we watch it? We don t know. It is stochastic. We cannot predict that a particular nucleus decays, but we can determine the probability that say x of these nuclei will decay assuming the probability that any one nucleus will decay is p. So what is the probability that x of these nuclei will decay assuming that the probability of any one nucleus decaying is p? 15

16 Each decay event is an independent event. The fact that one nucleus decays has nothing to do with the fact that another nucleus might decay. The probability of any one event occurring is governed by the binomial distribution. So the probability is given by this quantity: n factorial divided by the quantity x factorial times n - x factorial, all of this times p to the x power times 1 p to the n-x power. That s the binomial distribution. And in your statistics class you are going to see this binomial distribution occurring. We are going to start with this binominal distribution. And, now we ll take the limit of the binomial distribution for a large number of events. 16

17 We want to look at the deterministic quantities that result from sampling a large number of stochastic quantities. All we are able measure are the deterministic quantities. Let s take a very large number of radioactive nuclei. n is very, very much greater than 1. The first thing we are going to do is take this 1 p to the (n x) power and separate it into factors: (1 p) to the minus x power and (1 p) to the n power. Everything else is going to be the same. 17

18 Now let s look at this second factor, n factorial over (n x) factorial. This quantity can be expanded to be n times (n 1) times (n 2) times a lot of other factors times (n x + 2) times (n x + 1). Now we are going to make the assumption that n is very, very large, very much larger than x. If that were true, each of these factors is approximately equal to n, so that this quantity n factorial over (n x) factorial is roughly equal to n raised to the x power. 18

19 If we let μ be the expectation number of decays, that s n times p, the probability of one decay times the number of decays, we have terms in p -x, p x, and n x, so that the probability can be written as 1 over x factorial multiplied by μ x times (1 p) -x times (1 p) n. Now remember that p and x are very small and n is large. So this quantity, (1 p) -x is roughly equal to 1 + px, which is approximately equal to 1, so let s get rid of it. 19

20 Let s look at the limit of 1 p for very small p. We write n as μ divided by p and take the limit of (1 p) n as p approaches 0. The limit as p approaches 0 of 1- p to the nth power, remembering that p is small, is the limit 1 - p to the 1 over p power and raised to the μ power. This quantity is simply 1 over e to the μ power. This is a definition of e, that is, the limit as p goes to 0 of 1 p raised to the 1 over p power. Consequently the limit as p approaches zero of 1 p to the nth power is e -μ. 20

21 So in the limit of small p the probability is e -μ divided by x factorial times μ x, and μ x is equal to e xlog μ. 21

22 Let s look at μ. The quantity μ is a deterministic quantity. It is the mean value of the stochastic variable x, the number of decay events. But the decay events occur over a time interval. So the number of decay events is going to be proportional to the time, μ is equal to λt. The quantity λ is the mean number of decay events per unit of time. 22

23 We want to look at how these decay events are distributed over time. We want to know how many nuclei are present at any specific time, that is, how many have not decayed. So what is the probability that an event has not happened. What is the probability that x is equal to 0. 23

24 We look at the limit of the probability as both p approaches 0 and x approaches 0. This limit is equal to e -λt. That is the probability that decay has not occurred, that is, that we still have a nucleus. 24

25 The expected number of radioactive nuclei present is equal to the original number present multiplied by the probability that no interaction has occurred. So the number present is equal to the original number N 0 times the quantity e -λt, which tells us that no interaction has occurred. This is an example of exponential decay. What we have determined, then, is that if the probability of an individual nuclear decay event is stochastic, then the expected number of nuclei that have not undergone a decay is given by exponential behavior. To generalize we use this same derivation for any stochastic process. The expectation number of something happening is stochastic, independent of whatever else goes on. We are saying that the probability that something has not occurred is governed by exponential behavior. So that for radioactive decay we make the assumption that a decay process is independent of anything. Radioactive decay is independent of pressure, temperature, solubility; it is a strictly stochastic event. Consequently we have exponential behavior for radioactive decay. The probability that a photon will interact with a target atom is a stochastic event. Consequently from this observation, we derive the fact that as a photon beam penetrates through absorbing material, the intensity of the beam or the number of photons that come through is going to be governed by exponential behavior. 25

26 We began with a stochastic approach and demonstrated that exponential behavior is the non-stochastic, or deterministic, average of a stochastic event. So the change in the number of a quantity, whether it is number of radioactive nuclei, number of photons, number of cells, is going to be equal to some proportionality constant times the number present times the interval, whether it s a time interval or a spatial interval. This is the equation; ΔN is equal to ±λ N Δt. If we have a plus sign, then we have growth; if we have a minus sign, then we have a reduction. This is the mathematical description of exponential behavior. 26

27 Let us now look at the differential equation, dn is equal to ± λ N dt, or dn over N is equal to ± λ dt. If we now integrate this expression, we get that N is equal to N 0 e to the power ± λt, plus λt if we have exponential growth, minus λt if we have exponential decay. There is a very important way to look upon this equation, for example, the equation for exponential decay, and interpret the proportionality constant. In the case of exponential decay, the transformation constant λ can be looked upon as the fraction of nuclei decaying, that is, dn over N, in a time interval dt. If we are looking at exponential attenuation, where the equation is now dn over N equals μdx, the linear attenuation coefficient μ is the fraction of photons interacting in a given thickness of absorbing material. This is a very important way to look at exponential behavior. We will come back to this later in this course when we talk about radioactive decay and when we talk about interaction of photons with matter. 27

28 When we have exponential behavior such as exponential decay occurring, we can identify a time when the number is equal to half the initial number. That time interval is called a half-life. For exponential attenuation, we have a thickness required to reduce the beam intensity to half its initial value, and we call this thickness a half-value layer, or a half-value thickness. When t is equal to t ½, N will be equal to ½N 0. Let s figure out how to determine that time interval. Taking logarithms of both sides of the exponential equation gives us that the logarithm of N over N 0 is equal to minus λt. So when N is equal to ½N 0, the log of ½ is equal to minus λ times the half-life. So the half-life is the logarithm of 2 divided by λ. Just in passing, I need you to know that I use the abbreviation log to represent natural logarithms. Some of you may have seen ln for that abbreviation, reserving the symbol log for base ten logarithms. But we never use base ten logarithms in this course. So log means natural logarithms. Since the natural logarithm of 2 is 0.693, the half-life is always equal to divided by the transformation constant λ. Very often we will talk about half-lives rather than transformation constants. I think a half-life gives us a more intuitive feel for what s going on with radioactive material. The half-life of a cobalt 60 radioactive source is about 5.25 years. So we know the activity of a cobalt source, five years from now the activity is going to be half; about 10½ years from now the activity will be one-quarter of what we have now. 28

29 Another quantity we use occurs when for N to be equal to N 0 over e. That time is the average life. So the logarithm of N 0 over e divided by N 0 is the logarithm of 1 over e, and is equal to minus the average life. Because the logarithm of e equals 1, the average life will be equal to 1 over λ, or the mean path is equal to 1 over μ. So those are two things that we are really going to see a lot of when we are dealing with exponential behavior. We are going to be revisiting this several times in this course. In particular when we talk about radioactive decay and in particular when we talk about photon interactions. 29

30 The final item is the plot of exponential behavior. If we plot exponential behavior on linear graph paper, we are going to find that it is represented by a curved line starting at 1.0 (the decimal point does not show up very well here) and never reaching zero. 30

31 Very often we will plot exponential behavior on semi-log paper. The horizontal axis is linear, and the vertical axis is logarithmic. Here we see one decade and we can plot multiple decades which you should be fairly used to doing that on semi-log paper and Cartesian paper. 31