Transcription
In the previous talk, I spent quite some time examining the linear attenuation coefficient, the proportion of x-rays that are removed from an incident x-ray beam as it travels through a set distance in tissue. And we saw that negative exponential relationship between our transmitted x-rays and the distance that that incident x-ray has traveled through the tissue.
Now, if you haven't watched that talk, I'd highly recommend going to watch that first, prior to watching today's talk, because the concepts that we cover in that talk segue nicely into the concepts that we're going to cover today.
Now, today we're going to be looking at the half-value layer. And in its most simplest terms, the half-value layer is the thickness of tissue required to reduce the x-ray beam intensity by half. It's the thickness of tissue that will reduce the photon number of the x-ray beam by half, will remove half of those photons over that distance.
Now, when we looked at the linear attenuation coefficient, we saw that it was inversely proportional to our half-value layer. As our linear attenuation coefficient increases, our half-value layer decreases. And that makes sense. In this tissue, if our linear attenuation coefficient was high, if it took a large proportion of x-rays out of the beam quickly, then the distance required to remove half of those x-rays would be smaller. Our half-value layer could be smaller. You can see our half-value layer is measured in units of distance, the distance required to remove half of those x-ray photons.
Now, you'll be familiar with this from the previous talk, that our number of transmitted photons is determined by our incident photons and the negative exponents of our linear attenuation coefficient and the distance that is traveled through tissue. And I mentioned specifically in that talk that the distance required to attenuate our beam by half is the same as we go into that tissue. This distance here will remove half of our x-ray photons. That same distance will again remove another half of those photons. And that's how we get this negative exponent here. We asymptote with zero photons. We never actually truly get to zero photons because we are halving the number of photons each time.
The relationship between our half-value layer and our linear attenuation coefficient is inversely proportional. We've seen that as our linear attenuation coefficient increases, our half-value layer decreases. And you might be wondering how we get to this strange 0.693 divided by our linear attenuation coefficient. And actually, we can go about calculating using this formula, this equation that we've created here.
Now, don't run away when you see all of this math. You don't need to be able to reproduce this, but I want you to actually understand what is going on, how our linear attenuation coefficient is related to our half-value layer.
Now, the distance X here in this equation, we can substitute for our distance being our half-value layer. A half-value layer is an actual distance, a distance in millimeters or centimeters. So, our half-value layer distance will result in one x-ray incident photon being reduced to half. So, if we had a hundred incident photons, we would have 50 transmitted photons. We've reduced that proportion by half. So, we can substitute our N naught and our N here for one and a half.
Now, if we want to isolate this negative exponent, we can take the natural log of both sides of this equation. And the natural log of this equation here, e to the minus linear attenuation coefficient times our half-value layer, which is an actual distance. So, the natural log of a half equals minus 0.693. And the natural log of this equation will equal minus our linear attenuation coefficient times by our half-value layer. And if we isolate our half-value layer here, we get this equation. We can see that our half-value layer is inversely proportional to our linear attenuation coefficient.
So, what have we actually calculated here? How can we use our linear attenuation coefficient and our half-value layer to get an idea of how x-rays are interacting as they pass through matter and how they are being attenuated as they pass through the tissues of our patients? Well, we can use the linear attenuation coefficient as a good proxy for how specific tissues remove x-rays from their being. We've seen that bone, because it is more dense than water, removes proportionally more x-rays. So, the linear attenuation coefficient gives us an idea of how a specific material removes x-rays from a beam.
Now, the half-value layer is a distance. The distance a set of x-rays must travel in order for the intensity of those x-rays to be halved. Now, we can use half-value layer as a good proxy for our x-ray beam quality, the average energy of our x-ray beam. The higher the average energy of our x-ray beam, the further they will travel through tissue before the intensity is halved. So, as our half-value layer increases, so does quality, or the average energy of our x-ray beam.
When we're calculating linear attenuation coefficient and our half-value layer, we're using three separate parameters: our x-ray beam energy, our material density, and the atomic number of our material. Now, when we're dealing with one patient as a whole, the entire thickness of the patient, the density of that entire thickness, and the atomic number of the thickness of that patient hasn't changed. So, as we change our photon energy, our x-ray energy, our half-value layer will change in proportion to that x-ray beam quality. So, we can use half-value layer as a proxy for our x-ray beam quality, and we can use our linear attenuation coefficient to get an idea of how well specific tissues attenuate an x-ray beam.
Now, I want to test your understanding a little bit here, because in exams, this is where I see people getting confused the most. And this is a common question that comes up in exams. And if you are prepping for a radiology physics exam, I've linked a question bank that I've curated in the top line in the description. Highly recommend going to check that out.
Now, in this example, we have eight x-ray photons interacting with a thickness of tissue that is equal to two half-value layers. Now, I want you to calculate how many photons will be transmitted through this tissue. I've got eight on this side. It goes through one half-value layer, another half-value layer. How many photons will be transmitted?
Now, the common mistake that people make is they say that no photons will be transmitted. They say that a half of the beam will be removed here, and the other half will be removed here, and no photons will be transmitted. And this is where our understanding of that negative exponent of x-ray removal, that half-value layer halving the number of x-rays per half-value layer, comes into effect here. So, in fact, two x-ray photons will make it through. Four photons will be removed in this first half-value layer. Half of these x-ray photons, those four photons, will then interact with our next half-value layer, in which half will be removed, and two photons will make it to the other side.
If we want to calculate the factor by which these photons will be reduced as it goes through a set number of half-value layers, we can use this equation here. Two to the power of the number of half-value layers will equal the factor of reduction. Here, we've gone through two half-value layers. Two to the power of two is four. We've reduced eight to two. We've reduced it by a factor of four.
Now, this is the concept that most people get confused. They look at the linear attenuation coefficient, and it's called linear. You think of it as a linear attenuation, that we're getting the same number of x-rays removed per set distance. And we know that's not the case. It's a negative exponent. We proportionally take out the same proportion of x-rays. We are taking out 50% of the x-rays per half-value layer. We're not taking out the same number. We're not taking out four photons and four photons. We're taking out 50% of photons and 50% of photons.
So, I hope that helps you to understand how x-rays are attenuated as they travel through tissue.
Now, our x-rays have made it out of our patient. They are finally ready to go towards our x-ray detector. And that's what we're going to be looking at next. We're going to be looking at screen-film radiography, computed radiography, as well as direct and indirect digital radiography systems, how we go about capturing the information that we've created from our x-rays that are released from the x-ray tube, interacting with matter, transmitted and scattered through that patient, finally creating our radiograph that we can actually use to make a diagnosis within our patient. So, I'll see you all in that next talk, where we're going to go over an overview of the x-ray detection systems. Until then, goodbye everybody.