Transcription
Welcome everybody to the start of our MRI physics journey. I'm Dr. TMD here. I'm so excited to bring this all to you. We're going to laugh, we're going to cry. We'll probably cry more than we'll laugh, but at the end, I promise you, you're finally going to understand what all those spinning arrows mean.
But first, a word of warning: MRI physics is hard, like really hard. As a great philosopher, Prometheus Lionheart, once said, "Nobody understands Amari physics." And usually, those pretending to know show something like this: here's a diagram with arrows, some are aligned one way, some are aligned the other way. Here's a diagram with a bunch of squiggly lines. We'll talk about the terms frequency encoding and phase encoding. We'll maybe show a picture like this and we'll say that from all of that, we generate this. Any questions?
But there actually is a logical story behind all of this, and I've spent hundreds of hours searching out every single resource available to bring the full story to you. I wanted to start this lecture series by looking at something that first seems totally unrelated, but it will highlight an important point. This is a picture of an F-22 Raptor, the most advanced airplane ever built. In fact, it's so advanced, it can't even fly. Now, I know what you're thinking, it has to fly, or that would just be ridiculous. And yes, it actually does fly, but if you shut off all the electronics on this plane, it would tumble out of the sky. Every subsystem of this plane is likely a postdoctoral level subject. The person who designs the wings doesn't know what's going on with the engines. The person in charge of the control surfaces doesn't understand the electronic systems. And when you see an image from a modern MRI machine, such as this, it's the same situation. These machines are so advanced that there's an expert designing every single component: the main magnet, the transmit coil, the receiver coil.
So how in the world do you begin to learn something so advanced? When it comes to airplanes, you would start by examining something like this. This is a picture of the Wright Flyer, the very first time humans took to the sky in sustained flight in 1903. Now, this is something you can actually calculate by hand. So if you want to begin learning MRI physics, where do you start? How about here? This is the first clinical MRI of a live human being, taken in 1977. And this is something more manageable to begin learning. And most MRI physics resources out there are teaching you things related to this level of physics, not the fancy images we see today. But remember, this plane has to follow the same physics that this plane follows, and this MRI machine has to follow the same physics that this one does. So if we build a solid foundation, it will go on to serve all of your future educational ventures into MRI physics. And that's exactly what we're going to do with these lectures.
So, warning: you're about to be hit with a ton of bricks. The good news is, you may have heard and know a lot of the terms we're going to talk about. The bad news is, they haven't been taught in an intuitive way. And the ugly is, everything else about MRI physics. So this is the outline we're going to follow. We're first going to talk about nuclear magnetic resonance and this amazing phenomenon we take advantage of in order to build these images. We'll then move on to discuss signal localization, which is essential for us to be able to build the picture. And finally, we'll discuss image contrast, where it comes from and how it makes the pictures look like they do. And the good news is, this is 99% of what you need to know to be able to be a good, God-fearing, and or respectable radiologist scientist.
But wait, there's more. We have this unique concept to MRI physics: the dreaded k-space. It's such a confusing topic because it just comes out of nowhere. People just start talking about k-space and they don't explain why it even exists. Why do we have k-space? The story is actually fascinating, and in order to explain it, we need to understand four things: the MRI equation, and then how this relates to building an image. And these constitute the full, complete story of MRI physics. And if you're interested in this, let's get nerdy together.
So, the goal: we want to take advantage of a complex physical property and build an image. In order to do this, we have this machine that we call an MRI machine. It's going to receive some inputs and it's going to produce an output, and from that, we're going to be able to build this image. So first, let's start by talking about the phenomenon that makes this all possible: nuclear magnetic resonance.
So, the story of nuclear magnetic resonance begins with the proton. For MRI imaging, this is a diagram of a hydrogen proton. Notice that the proton is charged, right? Has a positively charged nucleus, and it also has a spin and has a radius. Now, one of the interesting things is, since it has a charge, a spin, and a radius, that you can model this as a current flowing through a loop at radius R from the axis. Think about it, what's an electromagnet? It's a charge spinning through a loop. And since this behaves like a little magnet, it's going to follow magnetic field lines, just like a compass trying to align itself with the Earth's magnetic field lines, as shown in this picture. Our proton will want to do the same. It behaves like a magnet, so any magnetic field we put it in, it will try to align itself with that field.
So how does this come into play when we stick a human inside a magnetic field? In this diagram, you can see that we have our stick figure here, laying inside these magnets, and the magnetic field they're generating is going in this direction. If we could take a cube of matter and zoom in on it and look at what the protons are doing, we might see something like this. Notice that within this cube, we have two options. We have protons aligned with the magnetic field, and we have protons aligned against the magnetic field. And notice that there's slightly more protons aligning with the magnetic field than against. Now, why does this happen? It turns out the direction aligned with the magnetic field is slightly more energetically favorable than that aligned against the magnetic field. If you were to add all of these up, you would have a sum total going in this direction with the magnetic field, and that actually creates a magnetic vector in that direction. We are literally creating a magnet inside the body by having these aligned slightly more with the field than against. Now, this effect is small. We're talking a few protons out of a million. But if you think of how many protons there are in the body, billions, maybe trillions, then the effect magnifies, pun intended. And the stronger the magnetic field, the greater this effect.
So throughout this lecture, we're going to ponder some conceptual questions. A lot of these are questions I had when I was trying to figure this out. So, does the net magnetization immediately develop as soon as the object is put into the magnetic field? Or maybe a better question, does anything just instantly happen in life? So the answer is no. It takes time for it to reach its maximum value, growing as a simple exponential with a time constant called T1. More on this to come. You may have even been shown a graph that looks something like this. Here we have net magnetization along the y-axis, time along the t-axis, and we can tell that something is growing with respect to time, reaching a plateau at some maximum value.
So let's illustrate this a little bit better. Here's our stock model. I think his name is Noah. If not, we'll still call him Noah. The guy needs a name. So Noah here is waving to us, and what we're going to do is again look at a little cube of matter inside of Noah. In its natural state, we're going to have protons arranged randomly within this cube of matter. Now, we're going to take some electromagnets, we're going to run some current through these coils, and we're going to produce this magnetic field that we're just going to call B0. We move Noah into the magnetic field, and let's see what happens to the protons inside his body. They're going to align, as we discussed before, right? Some protons are going to align with the magnetic field, some against. There's going to be more aligning with than against, and this is going to produce a net magnetization along the same direction of our applied magnetic field. We call this magnetization M_net. But on its own, nothing happens. We haven't imparted any extra energy into the system to make this move in any particular direction.
So let's see what happens if we are able to somehow push this magnet into a different orientation. Notice how all the protons inside of this initially keep their respective alignment, all turning together as the net magnetic vector is reoriented. So what happens when we let go? The net magnetic vector we created in Noah's body begins to spin, or precess, as we call it, with respect to the applied external field B0. Now, the amazing thing is, this precession is predictable and occurs at what we call the Larmor frequency, which is a very simple formula that's going to govern everything that we do in MRI physics. The formula consists of a gyromagnetic ratio, which for hydrogen protons is constant at 42 megahertz per Tesla, and then the strength of the applied external magnetic field. For a quick example, if we have a 1.5 T magnet, the Larmor frequency will be calculated as 1.5 Tesla times 42 megahertz per Tesla equals 63 megahertz. For comparison, a local AM news radio station may broadcast at somewhere around 68 megahertz. This has important implications, which we'll talk about shortly.
So here's an important question: How do we push the net magnetization away from our main magnetic field B0? Do we simply stick our hand in and start poking the body? I wouldn't recommend that on chair practice. So let's think about this a little bit further. We kind of gave the answer away in the slide before. We introduce a new oscillating or rotating magnetic field at the Larmor frequency through the use of an electromagnetic coil. So think of an electromagnet, as we discussed before. We have this loop of wire, sometimes it's around something that magnifies the effects, but we flow some current through this loop of wire, and it generates a magnetic field. If we apply a constant current through it, we'll get a constant magnetic field. If we apply alternating current through it, we'll get an alternating magnetic field. And if you think about it, what is an electromagnetic wave? It's an alternating magnetic field. For a 1.5 T magnet, the Larmor frequency will be 63 megahertz, which is, as we said, within the radio frequency range. So we will need a radio frequency pulse.
Another conceptual question: Why does the radio frequency pulse have to be at the Larmor frequency? So actually, all electromagnetic waves will impart a degree of energy to the system. However, things applied at the natural frequency of the system, in our case the Larmor frequency, will most efficiently transfer the energy of the wave to the system. I want you to think about this example. I think this is a little bit more tangible. You've likely heard the concept of an opera singer who can break a wine glass with his or her voice. So this is a picture where we have a wine glass and we have a sound producing machine, and notice that it is perpendicular to the wine glass. We can play sounds in front of this wine glass all day without it breaking, just like you talking in front of a wine glass, right? These don't spontaneously break. But if you can produce a pitch that's at the same frequency that this wine glass naturally wants to resonate at, something interesting happens. The walls of the wine glass start flexing, and they flex more and more until finally the wine glass shatters. This is the same concept that we're using in MRI physics. You can send different frequencies of electromagnetic waves or alternating magnetic fields all day long, and we won't do much to that magnetic field within the body. However, if we apply just set the frequency that this magnet inside the body wants to rotate at, the Larmor frequency, then we'll be able to impart that energy and get it to start precessing.
This is a kind of fun one. The Earth has a magnetic field. Can we use it to perform MR imaging? Why not, right? We're in a magnetic field right now, we're standing on the Earth. So the answer to this is actually yes, but you have to remember that the Earth's magnetic field is very weak. And there's a reason. The Earth is 0.5 Gauss. That's the measurement of strength of a magnetic field. And a clinical MRI machine generates a magnetic field in the 1.5 to 7 Tesla range, where one Tesla equals 10,000 Gauss. So our MRI machines are about four orders of magnitude greater than the Earth's magnetic field.
So to summarize, we've placed a body in a very strong magnetic field, and if we could look at the protons inside the body, we've noticed that more of the protons have aligned with the field than against. And if we were to add all of these up, they would generate a net magnetic vector in the direction of the applied external field. We then apply an alternating magnetic field in the form of a radio frequency pulse at the Larmor frequency. When we do this, this net magnetic vector inside the body starts precessing at that same Larmor frequency. So we have figured out our input. Our input to this machine is an RF pulse. So then the next critical question is, what is the output? What does the machine see?
So just to orient us to the axes we'll be using for the rest of the lectures, we'll denote the z-axis as the long axis of the body, and the x and y axises as the cross-section of the body. So here's what we're going to do. We're going to place another coil, or a loop of wire, parallel to the z-axis, and that's what this animation shows. Let's just say this is all of those protons added up, and here's our x and y axes, and then the head and feet of the person will be along the z-axis. And this animation was adapted from the PIRL YouTube channel, so all credit goes to them for the original animation.
So what happens when we apply our alternating magnetic field to this net magnetic vector at the Larmor frequency? You can see that we generate a sinusoidal curve if we measure the current over time. And in particular, as this comes closer to the magnet, we get the maximum current. As it goes away, it goes in the opposite direction, and so on. So now we're going to take a look at what happens when we change the angle of the spinning net magnetic vector with respect to the z-axis. Notice as we change this angle, when it gets closer and closer to being completely in the x-y plane, we get our maximum deflection, we measure our maximum current. And then when it gets as close as possible to the z-axis, we measure the smallest amount of current, almost none. And we call this angle the flip angle.
Finally, in this video, we're going to see what happens to our measured current as we keep the angle the same, but we increase or decrease the magnetic field. Notice when we increase a magnetic field, we get a higher signal, and we also get a higher frequency. And when we decrease the magnetic field, we get the opposite: a lower signal and a lower frequency. Remember, the Larmor frequency directly depends on the strength of the magnetic field. If we increase our magnetic field, we'll get a higher Larmor frequency, as seen on our current graph. And if we decrease it, we'll get a lower Larmor frequency. So as noted, amplitude and frequency change with increasing and decreasing B0.
So this is kind of the "aha" moment. Like the wireless charger in your phone, MRI machines take advantage of Faraday's Law of Magnetic Induction, which states an alternating magnetic field, in this case our precessing net magnetic vector, induces a voltage and thereby a current in a nearby coil. So this is a very expensive wireless phone charger. Let's think about it. What's going on inside your wireless phone charger? Current's going into the charging pad, and then flowing in a circular coil that creates a magnetic field perpendicular to the coil that increases and decreases as the current increases and decreases. And then that induces a current inside a coil inside your phone that charges your phone.
So we figured out what our output is: current. But we have to make some modifications to this. Let's go back to this animation. This is our net magnetic vector casually rotating at the Larmor frequency and generating a sinusoidal current with respect to time. Now, do you think this is really how this works? Or let me ask you this: Let's say your car runs out of gas, and you're on flat ground, and you give it a push. Does that car just keep rolling forever? So the answer is no, right? It'll travel a little distance and then come to a stop. And that's because we have friction between the wheels and the ground. We have these losses that we're never going to get back. We'll lose that energy, and the car will stop moving. So likewise, in MRI physics, once we impart this energy to get this net magnetic vector spinning, it doesn't spin forever. In fact, the MRI machine records a decaying sinusoidal current coming solely from the x-y plane. And that makes sense, right? We impart this energy, and then those protons are going to interact with their environment, and we're going to lose that energy. We lose this current that we capture in our coil, and this current directly represents something that we call T2 decay of our image body part. And that should sound very familiar.
So let's just take another look at this again. This is an animation adapted by the PRL YouTube channel. All credit goes to them. And in this graphic, we're looking down the barrel of an MRI machine. And let's just look at what happens to these individual protons after we excite them. Notice that initially, all the protons are rotating at the same rate. And then they start interacting with each other. Some speed up, some slow down relative to each other. And with that, the signal decays and is lost. Let's just take a look one more time. They're spinning together, they start spinning at different rates, and we lose the signal. And we're not making this up. This is an actual Free Induction Decay curve from a 1.5 T machine by Dr. Elser over at mriquestions.com. But more specifically, it represents the T2 decay curves of each individual voxel summed together. And this is very important. We have a slice through the body, in this case, the face and brain. And this current that we're measuring comes from every single voxel. For example, this voxel containing mostly fat contributes this decay curve. This voxel that is mostly soft tissue may contribute a decay curve that looks like this. And finally, we go back here to the posterior scalp, and this voxel, again containing mostly fat, will give us another T2 decay curve. But we can't see these individual curves. We can only see all of these added up together. And so that's our goal. Our goal is to deconstruct this raw MRI signal back into their individual decay curves coming from each voxel.
Once we find a way to break down this complex signal that our machine records into the signal coming from each individual voxel, we can then choose a time to compare all these individual curves. We'll call the time TE, and we'll use that to calculate the current coming from each voxel at that time point. An example like this: we want to know what these curves are doing at a time of echo of 200 milliseconds. That corresponds at this time point on the graph. So let's say we can measure the amplitude of each individual T2 decay curve at this time point, 200 milliseconds TE. We take a measurement. Let's say the top one is 9 amps, the middle one's 4 amps, and then the bottom one is 9 amps. We can then use these numbers to build a contrasted image, because this is ultimately what we're doing, right? We're plugging numbers into a matrix, and the different values of those numbers give us our contrasted image.
A few more conceptual questions before we end this first lecture. I've been shown charts like this and keep hearing the word "echo." But I'm showing you this, so let's take a closer look. If you look at this chart, which we call a pulse diagram, many of you have probably seen something like this before. You notice that we have two areas of signal here. We have this initial area right after our first RF pulse, and this represents a decaying signal, our Free Induction Decay, which we're showing down here in this picture. The second part of this is something called the echo. We see that we're able to make this signal re-emerge, reach a maximum point, and then decay again. The reality is, this initial Free Induction Decay happens so quick, we're not able to do all the things we need to do in order to be able to break down that complex signal back into its individual T2 decay curves. So we have techniques in order to make this signal re-emerge. We can either hit it with another pulse, we can do fancy things with magnets, but just know that the concepts we're going to talk about will use the Free Induction Decay curve for simplicity, but they can also be applied to this echo. For instance, if you cover up the first part of this echo, guess what? We get a decaying signal curve, just like our Free Induction Decay.
You may have also noted that I said the signal detected by the MRI machine comes solely from the x-y plane. So what about the signal from the z or B0 axis? So through Faraday's Law, the current generated will only come from the component perpendicular to the receiver coil, and thus all detected MR signal solely comes from the component of the magnetic vector in the x-y plane. This is maximum when it lies completely within the x-y plane.
So that was a ton of words. So let's show this more easily with pictures. So looking at the coils of an MRI machine, notice that there's multiple. We have coils within coils, and there's reasons for that that we'll discuss later. But if we were to look at this machine in a cross-section, notice that all the magnets or coils, whatever you want to call them, are kind of layered parallel on top of each other. So the machine will never see our net magnetic vector when it's laying purely along the z-axis. There has to be some component going perpendicular into these coils for us to be able to get that current per Faraday's Law. So in this picture, all of our net magnetic vector is going directly into the coils, and that's when we get maximum current. We also get current if it's oblique to the coils, because remember, if it's oblique, part of it will be in the z component, part of it will be in the y component. So that's why we can only see the signal coming from the x-y plane.
And at this point, I had a big question in my mind. I think you should too. If I can only see the signal on the x-y plane, and this signal comes from T2 decay, then what is this T1 that we always talk about? This actually was an incredibly tough question to find an answer to, but we'll talk about it in the future lecture on contrast. And then finally, is there a limit to the angle we can knock the net magnetic vector into in the x-y plane? And the answer is no. We cannot get 60 degrees, 180 degrees, whichever we choose. We can even knock it back to its original starting point at 360 degrees, or even in circles at 720 degrees. But note that it takes more energy to create a larger flip angle, and that will be important when talking about the heating inside the body that this can generate.
So that's all. Congratulations on completing the first lecture. We talked in depth about nuclear magnetic resonance and what the machine sees. Next lecture, we'll talk about signal localization, all these crazy things we do in order to locate that signal coming from each individual voxel. If you like this lecture, subscribe, like. If you want to support me in my quest to bring truth to MRI physics, consider donating using the links below. Otherwise, here are the images we used for the lecture, a small disclaimer, and that's it. We'll see you next time on Lights On Radiology.