In today’s lecture, we will talk some about definitions and measurement of exposure and dose.
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So, the order of business will be to talk a little more about exposure, then about the quantity W/e for air, next the relationship between exposure and air kerma, and finally about dose. When we talk about dose, we will finally introduce the Bragg-
Gray cavity theory.
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I gave you a very quick definition of exposure the last time. But here is the official definition that the ICRU gives. To quote the ICRU, exposure is “the quotient of dQ by dm
, where the value of dQ is the absolute value of the total charge of the ions of one sign produced in (dry) air when all the electrons liberated by photons in air of mass dm are completely stopped in air.”
That’s a lot of words for a relatively simple concept. Exposure is given by dQ /dm
, and if you want to think about it visually, exposure is the ionization equivalent of the collision kerma in air for photons. In exposure we look at ionizations, whereas in collision kerma we look at energy transfer.
So you can go back sometime and look at what we said about collision kerma.
Remember, however, that when we speak of exposure the material we are interested in is always is air. Exposure and kerma both are only defined for photons and no other kinds of ionizing radiation, just X-rays or gamma rays. We will come back and look at the definition more closely in order to understand it better.
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Let’s talk about
W/e in air. As the electrons slow down in gas, they lose energy by ionizing the gas. The electron goes through a material and it ionizes the material.
In this case, because we are interested in the electron going through air, the electron ionizes the air, and creates a positive and negative ion pair. The quantity W is the mean energy expended in the gas per ion pair formed. It’s usually expressed in terms of electron volts per ion pair. For air this quantity is found to be constant; it’s independent of electron energy above a few keV. It’s very difficult to do the measurements at very low energies, and there is some indication that at very low keV the energy required to create ion pair increases. But that issue is of no interest to us, because it is out of the energy range that we are interested in.
So, for dosimetry purposes, for cavity theory, and measurement of exposure,
W is taken as a constant and equal to 33.97 electrons volts per ion pair. This value has been determined by the national calibration laboratories around the world.
Occasionally, not so much now, but in the past, this value would be updated a little bit from time to time. And all your dosimetry would change by a fraction of a percent. Last time they updated the
W value, it was about a 1% change. So one has to be aware that occasionally the national calibration labs might reevaluate
W
.
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For dosimetry purposes we generally look at the quantity (
W/e
), and we express it in
SI units, joules per coulomb. In SI units it has the same value 33.97, but this is now joules per coulomb.
Sometimes we look at it as
W/e
; other times as e/W
. When it’s
W/e
, it is the number of joules of energy deposited in the air per coulomb of charge released. When you create an ion pair you have the charge released, and this just gives the amount of energy in terms of joules of energy deposited in the air per coulomb of charge released. You can look at this inverse as the number of coulombs released per joule of energy deposited in the air.
We make use of both relationships.
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The definition of collision kerma states that it is the energy transferred (less the radiative losses) to charged particles per unit mass. So if we have the collision kerma and we multiply it by e/W
, the number of coulombs of charge created per joule of energy deposited, we will have the charge created per unit mass of air. And that if you went back to the definition, that is exposure. Take the collision kerma and multiply by e/W to get exposure.
We saw last time that collision kerma is given by the particle fluence times the mass energy absorption coefficient.
Finally, when charged particle equilibrium exists, the exposure is related to the dose times the quantity e/W
.
The expressions on this slide, then, give the exposure in terms of the collision kerma, the photon fluence, and the dose. Exposure is related to dose only when we have charged particle equilibrium, but in most cases, we try to arrange for that to be the case.
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We can look at the relationship between collision kerma and exposure just by changing the order around. The collision kerma in air is
W/e multiplied by the exposure. If you want not just collision kerma but total kerma, then you need to include the part due to the Bremsstrahlung losses.
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The amount of energy radiated is not included in exposure because exposure is a measure of the ionization finally produced by the secondary electrons. So if you have energy going off into Bremsstrahlung, that energy is assumed to escape the volume of interest and gets out of the way in the diagrams we had. If we want to include the Bremsstrahlung and use the total kerma, we have to put the g back in there.
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This value of g is very small. As you can see for cobalt 60, it is about a 1/3 of percent (0.003); for cesium-137 gamma rays it’s even less; and for orthovoltage it is
0.0. So it may be a small correction factor for cobalt-60, but nevertheless we have to take it into account.
If we plug in the numbers for g in the expression relating air kerma to exposure for cobalt-60, we find that the air kerma in Gy is equal to 8.79 x 10 -3 times the exposure in Roentgens. We don’t actually use Roentgens anymore; we just use joules per coulomb and that changes the equation again.
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For cesium-137, the conversion factor from Roentgens to air kerma changes a little bit, and for orthovoltage, it changes a bit more. In this case, however, we are not going to be worried about cesium-137 or orthovoltage. It’s really just the expression for the cobalt-60 that we are more interested in.
However, we will see air kerma again in Med Phys III when we talk about brachytherapy source strengths.
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Let’s look at the problem of the measurement of exposure.
In an earlier lecture we talked about the free-air ionization chamber, but now we want to talk about it in some more detail.
The figure is a schematic of what a free-air ionization chamber looks like. The chamber is in a leadlined box, because you don’t want any stray radiation coming into it. Inside of the box you have a parallel-plate ionization chamber. The top plate is solid, while the bottom plate is divided up. The center part of the bottom plate is called the collecting electrode, and on either side of the collecting electrode are guard electrodes or guard rings. There is a small gap between the collecting electrode and the guard electrodes.
One applies a high voltage from the top plate to the bottom plate. The reason the guard electrodes are there is that you need to define very precisely the volume of air from which you are going to collect the charge. If you have a guard electrode, the lines are forced to be quite straight. You have to make this gap between the collecting electrode very small, but it has to be large enough so that you have insulation. It doesn’t show here but the guard wires are sort of wrapped around with wires and the potential of those wires drops from the high to the low. All of that is done to ensure that these lines are forced to be as perpendicular to the electrodes as possible. The whole alignment of this is very critical, because a small change in these lines of force (and they can sometimes bow out and sometimes bow in) will change the volume. And that is going to change your measurements and you don’t want that to happen.
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The x-rays come into the ionization chamber through a diaphragm. The diaphragm is extremely carefully machined so that not only is the area of the opening known but also its shape.
X-rays enter the chamber and interact with the air, producing electrons. The electrons go off and produce further ionization.
You take an electrometer and measure the charge that is collected. The definition of exposure says we are interested in the electrons that are produced in the mass of gas of interest. So we are interested in the electrons produced in the shaded volume.
We must collect all of the charge that is released even though the charge may be released outside of the volume of interest.
You might have an electron produced outside the collection volume getting in, but you arrange so that if an electron is produced outside the collection volume and it gets into the collection volume, it is balanced by an electron produced in the collection volume and getting out. That’s the charged particle equilibrium condition.
To ensure charged particle equilibrium, the distance of air surrounding the collection volume must be a little greater than the range of the secondary electrons produced.
If we make this dimension just greater than the range of the secondary electrons produced by the x-rays, it gives you a good approximation to charged particle equilibrium and electrons produced outside the volume will just make it in. But they will be compensated for by electrons produced in the collection volume, which just get out.
In that way you get charged particle equilibrium.
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The other thing that you have to be careful about is you need to collect all of the charge produced by an electron. The electron is producing positive and negative charges as it comes to a stop. So you need to collect all of the charge produced that way.
If the chamber is too small, a secondary electron could go and hit one of the collecting electrodes directly and this is going to give you problems. So the separation of the plates is determined again by the range of the secondary electrons.
And if you think about it, ignoring the fact that maybe these electrons are a little lower in energy than the ones coming in the forward direction, the separation of the plates has to be about twice the range of the secondary electrons. Thus an electron produced in the collection volume can stop before it gets to a plate. So the dimensions of the ion chamber are determined by the energy of the x-rays that you are measuring. Eventually that will create a problem for us.
The exposure that we measure is then equal to the charge that is collected divided by ρ , which is the density of the air. A
D is the cross sectional area of the diaphragm and L is the distance of the length of the collecting electrode.
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Let’s add a slight complication:
The definition of the Roentgen is in terms of the mass of air. The use of the term
“air” implies dry air. If you have moisture then the value of W/e does change a little bit.
W/e is defined for dry air and so you have to make corrections for humidity.
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When we measure ionization, we typically have defined a fixed volume of air, but exposure is defined for a fixed mass of air, so we need to make corrections for temperature and pressure, which change the mass of air in a fixed volume.
The normal temperature pressure correction factor is shown here, but this again is based upon dry air. In terms of the perfect gas law working, really in this day and age we should put in SI units but we still use millimeters of mercury and degrees
Kelvin.
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We typically do not make humidity corrections on a regular basis when we do calibrations, but the national labs certainly do. If there is humid air, it changes the gas law and the temperature-pressure correction value.
We need to put in a term which takes into account if there is moisture in the air to make sure we can get the mass of the gas accurately.
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The collection volume is defined by the entrance aperture and the length of the collecting electrode. If we went back to the diagram, the cross sectional area of the beam is at point P, which is inside the chamber, and A d of this area at the measurement point, rather than A
D would be the cross-sectional the area at the entrance aperture. Note that this is a lower-case d rather than an upper case D. Then the exposure at this point would be given by Q/ ρ A d
L.
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We can’t measure A d accurately; there is no easy way of getting to that point inside the chamber. But if we say the exposure at the entrance of the diaphragm is related to the exposure at the point P inside the chamber, it will be related by the simple inverse square relationship. S
2 chamber. S
1 is the distance from the source to point P inside the is the distance from the source to the entrance of the diaphragm.
So that is just the application of the inverse square law.
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We are still looking at the cross sectional area in the center of the chamber. But that cross sectional area is related to the cross sectional area of the diaphragm again by the inverse square relationship, which is now the other way around.
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It’s just very convenient because when you put those in together those two inverse square relationships cancel out. And you get the exposure at the entrance diaphragm is related to the charge collected inside the chamber. We divide the charge collected by ρ times the cross-sectional area of the diaphragm times the length of the collecting electrode.
When you build a free-air ionization chamber for standards work these things are very carefully machined. The length of the collecting volume and the crosssectional area of the diaphragm are both known very precisely, and you have to determine the density of the air very precisely as well. If there is moisture you must take that into account, Q is just the measurement of charge. You measure the charge very precisely. With that information you can determine X, the exposure, in the middle of the entrance diaphragm. And that is how exposure was measured and still can be measured using free-air ionization chambers.
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This kind of chamber, however, is not really very convenient to use, and you can’t build a free-air ionization chamber that can be easily moved around.
The question then is how do we get around this problem, and build an ionization chamber that is more mobile?
To answer that question, we must keep in mind that the standard chamber has no wall to it. The wall, if you like, is an air wall. The basis of an air wall chamber is that the purpose of the air surrounding the active volume is used solely to stop electrons that are produced in the measurement volume.
Consider what would happen if the air wall were “condensed” into a “solid air” wall material. A
“solid air” wall thick enough to provide electronic equilibrium would permit measurement of exposure with a cavity chamber that would be far more mobile than a free-air ionization chamber.
We now replace the solid air wall with a wall material that interacts with radiation in the same manner as air interacts. That is, a solid material whose radiation parameters, such as the mass energy absorption coefficient, stopping power ratio, stopping power values, are the same as that for air. And you get certain materials that come very close. Carbon by itself acts very much like air when you look at the interaction parameters. This condition means that the mass energy absorption and mass attenuation coefficients for the material should be the same as that of air.
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It’s important to make sure that the walls are equilibrium thickness, but you have to correct for wall attenuation. To correct for wall attenuation, measurements must be made for a number of wall thicknesses. So you have now a chamber which has a wall made of air-equivalent material but you have caps of different thickness. And you keep putting additional thicknesses on and look at the ionization. The ionization will initially start to increase, because of increased production of secondary electrons, but then when you reach charged particle equilibrium, it starts to decrease because of attenuation. The desired wall thickness will occur where you reach the maximum ionization, and you haven’t begun to see the absorption begin to dominate. However, there is absorption in the equilibrium thickness of wall, so you plot ionization vs wall thickness, and extrapolate the curve back to zero wall thickness to obtain the true ionization value.
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Let’s go back to this slide just to show you some problems we have using these chambers. We’re only good for energies up to the orthovoltage range.
Two things happened as the energies went higher. First, the plate separations got much greater to ensure charged particle equilibrium, and if you go above a plate separation of about a meter, the chambers became inordinately large. And if the chambers were too large you were not able to get good straight lines of electrical force to define the collection volume. So you cannot define the collection volume very well. In addition the distance between the diaphragm and the collection volume gets very large, and you start to get significant attenuation of the x-rays in the air in front of the collection volume. And so for energies above about 300 or 400 keV, the size of chambers just got so large it was impossible to measure exposure with any kind of accuracy, and so another approach had to be found. And that’s when we turned to the cavity ionization chambers.
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Small spherical chambers were constructed of high-purity graphite that attenuates radiation in a manner very similar to air except graphite is more dense than air.
Moreover, graphite conducts electricity, so you can use it for one of the electrodes.
Closely fitting shells could be used for wall absorption measurements. National labs made these small spherical ionization chambers with series of shells to look at the wall absorption and to determine the correct wall thickness for charged particle equilibrium. Several chambers were made with their shells and they could get several exposure rate determinations and then take the average.
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To go from a measurement with a cavity chamber to measure exposure, you are required to use the Bragg-Gray relation. In the next lecture, we are going to look at the Bragg-Gray relationship in much more detail. But basically the Bragg-Gray theory relates the energy absorbed in a medium to energy absorbed in a cavity in that medium.
In the equation that we’ve shown here, the medium to be is taken to be w, the chamber wall, and it’s also the medium surrounding the cavity. So we are not going to distinguish between the two. The symbol g refers to the gas.
The energy absorbed in the gas is equal to the number of ion pairs that are produced in the gas,
J g
, times the average energy to create the ion pair, which is
W/e
.
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In order to obtain
D w
, the absorbed dose in the material w, that is, the wall of the chamber, we need to multiply the dose in gas by the ratio of the mass stopping power of the electrons in the material w to that in the gas g.
The ratio of the mass stopping powers allows us to relate the energy transferred from the charged particles to the material to the energy transferred from the charged particles to the gas.
Thus we take the measurement of ionization charge, divide it by the mass of gas in the chamber, multiply it by
W/e
, a known quantity, and multiply it by the stopping power ratio, also a known quantity. This gives us the dose to the medium.
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Let us now relate the dose in medium to exposure in air.
In an exposure measurement, the medium w is the wall of the chamber when the wall is thick enough to provide electronic equilibrium. So if we were to replace the chamber with air, we can obtain the dose to air in the chamber cavity by multiplying the dose in the wall of the chamber by the ratio of the mass energy absorption coefficients, air to wall.
Recall that the dose equals the fluence times the mass energy absorption coefficient under charged particle equilibrium. We just made use of that relationship to get this ratio. If the fluence is the same then the ratio of dose in one material to the dose in the other material is given by the ratio of the mass energy absorption coefficients.
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So combining the two equations gives us this equation for dose to air. It’s equal to the charge per unit mass of gas that you collect, multiplied by
W/e
, which is a constant, multiplied by these stopping power ratios and mass energy absorption coefficient ratios, which are all calculated values.
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We still have to relate that to the dose in the air.
We can measure the ionization in the chamber with a cavity ionization chamber but we want to determine what the exposure is. The air is the gas in the cavity, so the g may be replaced by air. So we are now just going to talk about air rather than just gas in general. For almost all measurements we do in dosimetry the gas is air.
Except if you ever get involved in neutron dosimetry then you might put another material or a different gas in there. There are times in medical and radiological physics where you are not necessarily going to have air in your ion chamber. In that case,
W/e changes and other things may change as well. You probably won’t have to worry about that but you can go through and see what values you need.
So here is the current definition of the Roentgen, which is 2.58 x 10 -4 coulomb per kg of air. That’s the definition of the Roentgen in SI units. We don’t talk about the
Roentgens anymore; we talk about coulombs per kilogram. So we can now rewrite the previous equation giving us the exposure in terms of the measurement that you make with your cavity ionization chambers. That measurement is the quantity
J g
The 2.58 × 10 -4 is a conversion factor. The next two factors, the ratios of mass
. stopping powers and mass energy absorption coefficients, are calculated quantities.
Finally, we have to include a series of correction factors.
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Just a word about these correction factors, because these cavity ion chambers are not suspended in air. They generally have a stem, which is in the beam, and you can get scatter from the stem of the ionization chamber. You have to correct for that, but it’s a small correction. You do have to take care of that and do measurements to determine what the stem scatter correction is.
Also, when you have ionization chambers with a voltage across it, not all of the ion pairs may be collected because some of them can recombine. You have to have enough voltage on it to stop recombination and that’s very hard, and so there is a correction factor to account for recombination.
Generally, it’s determined experimentally, and under some circumstances it’s quite difficult to get complete collection of the charge. So the recombination correction factors have to go in there.
Wall thickness has to be just right to give you charged particle equilibrium, but you get attenuation through the wall and you have to take that into account.
Finally, there is something else called the mean center for electron production in the chamber wall. I am not going to talk about that but that is another correction. When you are dealing with the cavity ionization chambers, it’s a small correction, but the center for electron production accounts for the observation that the electrons are being produced over different distances. And you have to take that into account and get the average point from which the electrons are produced and make a small correction for that.
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We can consider the relation between absorbed dose and exposure, 1 Roentgen as I said deposits that, the value of W is 33.97 joules per coulomb. Then one can derive expressions between the Roentgen and absorbed dose or collision kerma, and those I think I have given to you.
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Generally, in SI units one coulomb per kilogram can be rewritten that way and then air exposed to one Roentgen receives 0.876 cGy and air exposed to one coulomb per kilogram receives 33.97 Gy. This is the one we should be really be using today, so if you expose air to one coulomb per kilogram it will receive 33.97 Gy.
Now that is all for air, but we generally are not particularly interested in air, so if some other material is exposed to one Roentgen or one joule per kilogram, the absorbed dose in the medium is going to be the dose in air times the ratio of the mass energy absorption coefficients.
So this equation you are measuring the dose in air and the exposure in air and getting the dose from it. You would have to use this to get back to the medium of interest for general calibration purposes, which is water.
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For an exposure of X in Roentgens the expression for dose to medium in cGy will be 0.876 times the ratio of mass energy absorption coefficients times the exposure.
For an exposure of one coulomb per kilogram, the dose in the medium is going to be
33.97 times the ratio of mass energy absorption coefficients, and will be expressed in Gy.
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This conversion factor, the ratio of the mass energy absorption coefficients multiplied by either 0.876 or 33.97, is assigned a special name and it’s called the ffactor. So if you read the textbooks or the papers on dosimetry you see the f-factor and it refers to the factor that converts from exposure to absorbed dose. And the ffactor obviously is a function of the materials of interest, that is air and the medium, and also this varies with energy so it’s a function of both the medium and the energy of interest.
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One other thing that shows up in the papers and the textbooks is that for higher energy photon beams, an additional factor
A eq is required. When you have higher energy, you’ve designed your chamber with the right wall thickness to give you charged particle equilibrium. But that thickness is thick enough to attenuate the xrays going through it and you have to take that into account. You really would like a wall-less ionization chamber, but in a wall-less ionization chamber you really don’t get charged particle equilibrium. But as you put a wall on your ion chamber, you get attenuation, which you have to take into account. So the quantity A eq accounts for the attenuation of the primary photon beam in reaching d max in the medium. If you are going to relate exposure to dose, you have the f-factor, which takes into account that you are going from air to the medium of interest, and
A eq
, which takes into account the fact that you’ve got attenuation. For cobalt 60 it’s about 3%, but it must be taken into account.
When the national labs found that they couldn’t build free air ionization chambers for cobalt-60, they built these cavity ion chambers. They went through all of these equations and the way they determined exposure for cobalt-
60 radiation was with the cavity chambers and doing the calculation to get back to exposure. And now when they went down to the lower energies they still keep the free-air ionization chambers.
Next time we are going to look in some detail at charged particle equilibrium and go over again a little bit more about charged particle equilibrium, because it is not as simple as it looks in the slides. And I will talk a little bit more about particle fluence and then spend most of next time talking about the Bragg-Gray cavity theory. Then we will talk about Spencer-Attix cavity theory, a theory that came out of the Bragg-Gray cavity theory to take care of the some of the finer details that were in there. Next time, we are really going to look at the Bragg-Gray cavity theory which leads to the Spencer-Attix cavity theory. It’s basically the Spencer-Attix cavity theory that is the basis of modern day calibration protocols.
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