Saturday, November 28, 2020

Magnetic Resonance Imaging

[This is a transcript of the video embedded below. Some of the text may not make sense without the animations in the video.]

Magnetic Resonance Imaging is one of the most widely used imaging methods in medicine. A lot of you have probably had one taken. I have had one too. But how does it work? This is what we will talk about today.


Magnetic Resonance Imaging, or MRI for short, used to be called Nuclear Magnetic Resonance, but it was renamed out of fear that people would think the word “nuclear” has something to do with nuclear decay or radioactivity. But the reason it was called “nuclear magnetic resonance” has nothing to do with radioactivity, it is just that the thing which resonates is the atomic nucleus, or more precisely, the spin of the atomic nucleus.

Nuclear magnetic resonance was discovered already in the nineteen-forties by Felix Bloch and Edward Purcell. They received a Nobel Prize for their discovery in nineteen-fifty-two. The first human body scan using this technology was done in New York in nineteen-seventy-seven. Before I tell you how the physics of Magnetic Resonance Imaging works in detail, I first want to give you a simplified summary.

If you put an atomic nucleus into a time-independent magnetic field, it can spin. And if does spin, it spins with a very specific frequency, called the Larmor frequency, named after Joseph Larmor. This frequency depends on the type of nucleus. Usually the nucleus does not spin, it just sits there. But if you, in addition to the time-independent magnetic field, let an electromagnetic wave pass by the nucleus at just exactly the right resonance frequency, then the nucleus will extract energy from the electromagnetic wave and start spinning.

After the electromagnetic wave has travelled through, the nucleus will slowly stop spinning and release the energy it extracted from the wave, which you can measure. How much energy you measure depends on how many nuclei resonated with the electromagnetic wave. So, you can use the strength of the signal to tell how many nuclei of a particular type were in your sample.

For magnetic resonance imaging in the human body one typically targets hydrogen nuclei, of which there are a lot in water and fat. How bright the image is then tells you basically the amount of fat and water. Though one can also target other nuclei and measure other quantities, so some magnetic resonsnce images work differently. Magnetic Resonance Imaging is particularly good for examining soft tissue, whereas for a broken bone you’d normally use an X-ray.

In more detail, the physics works as follows. Atomic nuclei are made of neutrons and protons, and the neutrons and protons are each made of three quarks. Quarks have spin one half each and their spins combine to give the neutrons and protons also spin one half. The neutrons and protons then combine their spins to give a total spin to atomic nuclei, which may or may not be zero, depending on the number of neutrons and protons in the nucleus.

If the spin is nonzero, then the atomic nucleus has a magnetic moment, which means it will spin in a magnetic field at a frequency that depends on the composition of the nucleus and the strength of the magnetic field. This is the Larmor frequency that nuclear spin resonance works with. If you have atomic nuclei with spin in a strong magnetic field, then their spins will align with the magnetic field. Suppose we have a constant and homogeneous magnetic field pointing into direction z, then the nuclear spins will preferably also point in direction z. They will not all do that, because there is always some thermal motion. So, some of them will align in the opposite direction, though this is not energetically the most favorable state. Just how many point in each direction depends on the temperature. The net magnetic moment of all the nuclei is then called the magnetization, and it will point in direction z.

In an MRI machine, the z-direction points into the direction of the tube, so usually that’s from head to toe.

Now, if the magnetization does for whatever reason not point into direction z, then it will circle around the z direction, or precess, as the physicists say, in the transverse directions, which I have called x and y. And it will do that with a very specific frequency, which is the previously mentioned Larmor frequency. The Larmor frequency depends on a constant which itself depends on the type of nucleus, and is proportional to the strength of the magnetic field. Keep this in mind because it will become important later.

The key feature of magnetic resonance imaging is now that if you have a magnetization that points in direction z because of the homogenous magnetic field, and you apply an additional, transverse magnetic field that oscillates at the resonance frequency, then the magnetization will turn away from the z axis. You can calculate this with the Bloch-equation, named after the same Bloch who discovered nuclear magnetic resonance in the first place. For the following I have just integrated this differential equation. For more about differential equations, please check my earlier video.

What you see here is the magnetization that points in the z-direction, so that’s the direction of the time-independent magnetic field. And now a pulse of an electromagnetic wave come through. This pulse is not at the resonance frequency. As you can see, it doesn’t do much. And here is a pulse that is at the resonance frequency. As you see, the magnetization spirals down. How far it spirals down depends on how long you apply the transverse magnetic field. Now watch what happens after this. The magnetization slowly returns to its original direction.

Why does this happen? There are two things going on. One is that the nuclear spins interact with their environment, this is called spin-lattice relaxation and brings the z-direction of the magnetization back up. The other thing that happens is that the spins interact with each other, which is called spin-spin relaxation and it brings the transverse magnetization, the one in x and y direction, back to zero.

Each of these processes has a characteristic decay time, usually called T_1 and T_2. For soft tissue, these decay times are typically in the range of ten milliseconds to one second. What you measure in an MRI scan is then roughly speaking the energy that is released in the return of the nuclear spins to the z-direction and the time that takes. Somewhat less roughly speaking, you measure what’s called the free induction decay.

Another way to look at this process of resonance and decay is to look at the curve which the tip of the magnetization vector traces out in three dimensions. I have plotted this here for the resonant case. Again you see it spirals down during the pulse, and then relaxes back into the z-direction.

So, to summarize, for magnetic resonance imaging you have a constant magnetic field in one direction, and then you have a transverse electromagnetic wave, which oscillates at the resonance frequency. For this transverse field, you only use a short pulse which makes the nuclear spins point in the transverse direction. Then they turn back to the z-direction, and you can measure this.

I have left out one important thing, which is how do you manage to get a spatially resolved image and not just a count of all the nuclei. You do this by using a magnetic field with a strength that slightly changes from one place to another. Remember that I pointed out the resonance frequency is proportional to the magnetic field. Because of this, if you use a magnetic field that changes from one place to another, you can selectively target certain nuclei at a particular position. Usually one does that by using a gradient for the magnetic field, so then the images you get are slices through the body.

The magnetic fields used in MRI scanners for medical purposes are incredibly strong, typically a few Tesla. For comparison, that’s about a hundred thousand times stronger than the magnetic field of planet earth, and only a factor two or three below the strength of the magnets used at the Large Hadron Collider.

These strong magnetic fields do not harm the body, you just have to make sure to not take magnetic materials with you in the scanner. The resonance frequencies that fit to these strong magnetic fields are in the range of fifty to three-hundred Megahertz. These energies are far too small to break chemical bonds, which is why the electromagnetic waves used in Magnetic Resonance Imaging do not damage cells. There is however a small amount of energy deposited into the tissue by thermal motion, which can warm the tissue, especially at the higher frequency end. So one has to take care to not do these scans for a too long time.

So if you have an MRI taken, remember that it literally makes your atomic nuclei spin.

31 comments:

  1. The cost of MRI may come down substantially if the machines can use a higher temperature superconductor. Most still require < 10Kelvin temps.

    ReplyDelete
  2. About 6 years ago I was getting a persistent pain in one knee while walking. It put me in a bit of a panic being someone who enjoys hiking and cycling. It got to the point that I couldn't walk even one hundred yards without having pain. So I elected to have an MRI done on that knee. It seemed to take forever, as I lay prone with my lower body deep in the machine that was humming away steadily. In a follow up trip, accompanied by my two brothers, the nurse explained the situation as being nothing serious. That was a relief. In the following six years I've pedaled almost 12,000 miles (19,312 km.) on a bicycle with an occasional bit of soreness that goes away after rest. I'm planning to get out today and tomorrow with mild temps to add to this year's accumulation of 2522 miles so far, hoping to hit 2650 before snow shuts the season down.

    ReplyDelete
  3. Interesting. In what form is relaxation energy detected?

    ReplyDelete
  4. Two other scientists deserve mention: Raymond Damadian, who developed the first medical MRI device, and Bill Oldendorf, who came up with the method for reconstructing planar cross-sectional images from the mass of otherwise useless data these machines produce (applicable to both CT and MR imaging).

    ReplyDelete
  5. Given that this is from you, and your usual harping on quantum mechanics, its interesting that you use pure classical mechanics (and electromagnetism) to describe this. Lets hear you describe it with quantum mechanics! (I'm not serious, since I know the answer.) Lots of NMR specialists actually truly can't!

    ReplyDelete
    Replies
    1. Quantum mechanics is kind of unnecessary here because you only need the equation for the collection of *all* spins. It kind of enters if you want to derive the Bloch equation, which is essentially the expectation value for all the spins. It also becomes relevant if you want to understand where the relaxation times come from and why nuclei have this or that magnetic moment. But the Bloch equation in and by itself is fapp a classical equation and can be dealt with as such.

      Delete
  6. I know someone who did their dissertation on the relaxation of NMR, nuclear magnetic resonance. MRI is as you say a new moniker for marketing purposes.

    ReplyDelete
    Replies
    1. The two types of machines are completely different from a mythological point of view.

      Delete
    2. As an undergraduate I measured the Lamor frequency. It was as Sabine says a case of kicking a top so that it precesses. This happens when the EM field you apply is resonant with the Lamor frequency.

      The magnetic field is from a Helmholz coil, where near a center the field is nearly constant. However, it has deviations, which means the Lamor frequency differs from point to point. Then with a spectrum of EM waves you get a differential resonance response.

      The Lamor frequency in MRI is as I recall for carbon nucleus. So this images in effect the distribution of organic compounds.

      Delete
    3. It would have to be C13 then, because the C12 nucleus has spin zero. (The normal abundance of C13 is about one percent.) As Sabine mentioned, hydrogen nuclei are far more numerous.

      Delete
    4. That would image a lot of water. Later today I will look this up.

      Delete
  7. Sabine said: "If you put an atomic nucleus into a time-independent magnetic field, it can spin."

    By conflating the terms "spin" and "precession" you are creating unnecessary confusion. A nucleus with spin will never stop spinning; only the rate of precession will vary with the applied magnetic field. These processes have vastly different time scales. In the case of the Earth the precession period is 26000 years. If you were to take away the Moon, this time would become longer, but day and night would still repeat every 24 hours.

    It's nice to present the classical picture of NMR, but as you know, it's just an approximation. And to my mind, the quantum explanation is even easier to understand. The magnetic field causes a kind of Zeeman splitting, and the transverse electromagnetic wave causes transitions between the different angular momentum states. The energy differences are minute, and the populations of spin up and down states disagree at very much less than the percent level. It's just a statistical effect, but one that is reliably detectable through the induced polarization (diamagnetic currents).

    Aside from the magnetic field, the Larmor frequency depends only on the mass of the electron. It is way too large (Bohr magneton vs. nuclear magneton), and Larmor's theorem would suggest that every substance should precess at the same frequency. But as you said, the differing magnetic moments of atomic nuclei permit a glimpse at the chemical composition.

    ReplyDelete
    Replies
    1. Yes, sorry, I misspoke this. You seem to be conflating EPR with MRI.

      Delete
    2. Dear Sabine,
      I know you are doing a great service by explaining such technologically important ideas to non scientists. I also understand that to make the ideas understandable to general readership, you have to take some liberty with rigorous physics. But it should not be totally wrong. So I agree with Werner that conflation of spin with precession is totally wrong. It may not be possible to fix video, but at least please fix the following paragraph
      "If you put an atomic nucleus into a time-independent magnetic field, it can spin. And if does spin, it spins with a very specific frequency, called the Larmor frequency, named after Joseph Larmor. This frequency depends on the type of nucleus. Usually the nucleus does not spin, it just sits there. But if you, in addition to the time-independent magnetic field, let an electromagnetic wave pass by the nucleus at just exactly the right resonance frequency, then the nucleus will extract energy from the electromagnetic wave and start spinning."
      As you know Larmor frequency is frequency of precession. Also no one has power to stop a particle from spinning! A small point, although semiclassical Bloch's equation is fine, detailed calculations need QM and density matrix. I have done these calculations for NMR biophysics. Also you might mention that Lauterbur and Mansfield got Nobel Prize in physiology and medicine (2003) for invention of MRI method. Thanks.

      Delete
    3. I have put a note in the info below the video. It is not possible to edit or replace videos on YouTube.

      Delete
  8. Speaking of nucleons, it is odd that the half-life of a proton is at least 1.67 x 10^34 years while that of a free neutron is only about 10.2 minutes. It would be interesting to learn more about the quantum dynamics of that process in a future post.

    ReplyDelete
    Replies
    1. And, reading further in Wiki’s general article on the neutron we come across news-to-me that: “The finite size of the neutron and its magnetic moment both indicate that the neutron is a composite, rather than elementary, particle.”
      So, even at this fundamental level, there is a distinction of properties between that which is elemental and that which is composite, with the latter having that of constraint to a particular location in space. I suppose that ‘position’ is still ultimately governed by the uncertainty principle, but must grow more certain upon further aggregation into, say, the tissue being imaged in an MRI.

      Delete
    2. Protons and other hadrons such as the neutron are a composite of quarks and gluons. QCD is a bit odd in that as interaction energy becomes zero the strength of this gauge field increases. This is different from QED where the field strength increase with energy. As a result the low energy state of a proton or neutron is very hard to calculate. It is done approximately with numerical codes on computers.

      Delete
    3. Thanks, I am not sure how this affects the constraint to finite size. It did, however, spark a more philosophical realization that the MRI process is closely analogous to the whole of experimental physics wherein there is a painstaking effort to create some physical device, be it making truly round balls and smooth inclined planes, a computer program or the LHC, all in an effort to illuminate and bring into focus some particular facet of the physical world. Then further, being able to discriminate between whether the result is revelatory of the natural world or simply something conjured by your device.

      Delete
  9. Nice deep and visual presentation! well researched Sabine!!

    ReplyDelete
    Replies
    1. Thanks, this makes me very happy to hear! I spent way too much time on the visualizations. It would have been much easier to just kind of draw something approximately correct rather than insisting on actually integrating the equations.

      Delete
  10. I’m not a Physicist but rather a Geologist. In petroleum geology we use what we refer to as NMR logs, Nuclear Magnetic Resonance to measure the amount of water versus hydrocarbon in the pore space of subsurface rocks. These measurements are calibrated by running rock core samples through old medical MRI which have outlived there lifespan an been retired as the aren’t considered safe for human patients but as we are running rocks through them we don’t have that concern.

    ReplyDelete
  11. Typo in the last sentence. "you" should be "your"

    ReplyDelete
  12. I guess I should have put this in my first message.
    Somebody mentioned Damadian. Damidian never had anything really to do with MRI. What he proposed was the "MR" part, not the very neat idea of gradients to get the "I" part.
    That is, his idea was to use the spins and relaxations to tell tissue types apart. This was indeed a big deal. But his idea was something like little coils that measure volumes the size of the coils. He didn't think of the gradients.

    That was invented by my late colleague and friend Paul Lauerbur, who won a Nobel Prize for it. Damadien was pissed.

    ReplyDelete
  13. A few years ago I came across an article that mentioned in passing that one of the first uses of spinors was to describe spinnong tops. This came as a major suprise to me since spunirs are now so closely associated with wuantum spin that its strange to think that they can be useful in a classical context. Unfortunately there seems to be few references to this use, however, I did find a reference to this early in Kliens and Sommerfelds Theory of the Top.

    Given how mysterious quantum spin is, with even Feynman saying he didn't understand it (actually he said 'we'), I think its important to see how spinirs are involved, in all things, the spinning top, and whuch probanly means any spunning rigid body with an axis of symmetry.

    ReplyDelete
    Replies
    1. Intrinsic spin is a unit of angular momentum, a whole or half of Planck's constant, with no classical sense of something rotating.

      Delete
  14. The interesting thing about this remark is that Feynman knew that he did not understand it. All the others don't even realize this...

    ReplyDelete
  15. And PET - Positron Emission Tomography - scanners produce an image using the gamma rays given off by matter-antimatter annihilation when radioisotopes of elements that occur naturally in the human body - carbon, nitrogen, etc. - are injected. Reading about both MRI and PET makes me feel like we're living in an episode of Star Trek.

    ReplyDelete

PLEASE READ THE COMMENT RULES BEFORE COMMENTING.

Comment moderation on this blog is turned on.
Submitted comments will only appear after manual approval, which can take up to 24 hours.
Comments posted as "Unknown" go straight to junk. You may have to click on the orange-white blogger icon next to your name to change to a different account.