Magnetic Resonance physics question: RF pulse role and 'in-phase'

[Astrocyte]

Hi, after I majored in physics, I moved to the field of medical physics. However, I cannot understand some of their explanations. One of them that is bothering me a lot is RF pulse and in-phase.

What we measure in MRI is the magnetization of water particles. The magnetization comes from the sum of magnetic moments of protons in our body. Due to the Zeeman effect, the energy band of the proton splits into two in a magnetic field. Following statistical dynamics, it shows us a paramagnetic effect. That's how many protons in our body shape a magnetization.

In MRI, we often measure the horizontal component of the magnetization. Since the proton's magnetic moment precesses around the B0 magnetic field (horizontal magnetic field), the horizontal component of the magnetic moment can exist. Before hitting the RF pulse, the phase of the xy plane is different from proton by proton.

That is, protons' horizontal components are so random that they are canceled out in terms of magnetization. We call these various phases of protons' horizontal component of magnetic moment out-of-phase. However, after hitting the RF pulse, they are flipped in the opposite direction, and their vertical components of magnetic moment are canceled out in terms of magnetization.

This is because RF pulse has the same energy as the gap of the two split energy band. At the same time, their phase becomes the same. Therefore, the proton's horizontal components are not canceled out. Finally, we can observe the horizontal component of the magnetization of protons. For me, the change from out-of-phase to in-phase by RF pulse feels like magic because I cannot figure out an exact reason for this. Does Somebody know how can we explain the effect of RF turning out-of-phase to in-phase?

 

[Dale]

I will use the terms “longitudinal” and “transverse” instead of “vertical” and “horizontal”.

You appear to be missing the longitudinal magnetization. If you initially have fully relaxed magnetization, then after the RF excitation it is the longitudinal component (before) that produces the in phase transverse magnetization (after).

 

[marcusl]

Magnetization M in the absence of an RF field is longitudinal (parallel to the B0 field), as Dale mentioned. Application of an RF magnetic field B1 that is circularly polarized in the plane perpendicular to B0 cause the spin system to precess, thus generating transverse magnetization. When the frequency of B1 equals the Larmor resonance frequency, the transverse magnetization M is in phase with it. It is then convenient to view M in the rotating frame, a coordinate frame that rotates with B1. The magnitude of transverse magnetization varies sinusoidally during the time that B1 is applied, with the pulse length needed to generate maximum transverse magnetization called a "pi/2 pulse" because the net magnetization has rotated from longitudinal to transverse in the rotating frame.

 

[Astrocyte]

I see, but I want to understand why applying an RF pulse makes transverse magnetization from out-phase to in-phase.

Below is what I understand so far.

From one nuclear perspective, it would do a precession when exposed to external magnetization. This precession makes the energy level of the nuclear split into two levels. When the axis of precession is the same as the external magnetic field, the energy level is lower than one without the external magnetic field. On the other hand, when the axis is opposed to the magnetic field, the energy level is higher than usual. We call this phenomenon the Zeeman effect. Also, since it does a precession, it has a transverse element of magnetic moment. However, because there are a bunch of nuclear, with the B0 field, they are out-of-phase. Namely, the transverse element of the macroscale magnetization is canceled out due to the random phase of the precession. That’s why we cannot observe the transverse one.

An RF pulse flips the axis of precession from the lower level to the higher one. Due to the certain energy gap between the two levels, only a certain frequency called Larmor frequency does work. That’s what one nuclear experiences in MRI as far as I understand.

However, just flipping the axis of precession cannot make a transverse magnetization. This is because what flip changes is the longitudinal one. Nonetheless, we can observe the transverse element. So, it seems like an RF pulse turns the out-phase of precession into the in-phase. Here is where I am stuck in. I cannot see any possible reason why it could happen.

My PI said that the RF pulse makes a B1 field, so the axis of the precession itself slightly leans. This leaned axis allows us to measure the transverse magnetization even with the out-of-phase of each nuclear. However, I cannot be 100% sure of this reason.

BTW, I can understand the basic MRI concepts based on the classical view of one huge magnetization M. That’s not what I would like to know. Sorry.

 

[Dale]

I want to understand why applying an RF pulse makes transverse magnetization from out-phase to in-phase.

Are you specifically asking about the excitation RF pulse or the refocusing RF pulse? Your description is a little off either way, but I want to make sure that I am explaining the thing you are actually interested in and they are physically different scenarios.

The axis of precession is always the longitudinal axis. The axis of precession does not substantially change.

 

[Astrocyte]

Are you specifically asking about the excitation RF pulse or the refocusing RF pulse? Your description is a little off either way, but I want to make sure that I am explaining the thing you are interested in and they are physically different scenarios.

The axis of precession is always the longitudinal axis. The axis of precession does not substantially change.

Thank you for your reply.
Could you explain both: the excitation RF pulse or the refocusing RF pulse? I cannot tell the difference. If you cannot explain both, could you focus on the excitation RF?
I want to know the proton’s point of view and combine perfectly the micro point of view of one proton with the macro point of view of many protons. That is, how can non-synched protons’ precession get synched to make a transverse magnetization by an RF pulse?
If you find any misconceptions about my explanation, do not hesitate to point out them. I really want to know what’s going on under MRI. So far, for me, it seems a magic.

Again, I really appreciate it!

 

[Dale]

If you cannot explain both, could you focus on the excitation RF?

Ok, I will focus on the excitation.

I can understand the basic MRI concepts based on the classical view of one huge magnetization M.

Let’s make sure of that before we go to the quantum picture. Some of your comments make me suspect that you do not fully understand the classical picture.

applying an RF pulse makes transverse magnetization from out-phase to in-phase

Classically, it does not do this. The phase in the fully relaxed state is undefined. The longitudinal magnetization is not in phase (because the phase is undefined) but there is a net magnetization which points in a single direction, the longitudinal direction. When that single direction is tipped down it is necessarily in phase.

So it is not taking out-of-phase magnetization and making it in-phase. It is taking longitudinal magnetization and making it transverse. That transverse magnetization is inherently in phase

Does that make sense classically?

I will describe the quantum picture next, but it is important to understand the classical picture first because it shows you where to look in the quantum explanation. You must understand where the classical longitudinal magnetization arises.

 

[Astrocyte]

Classically, it does not do this. The phase in the fully relaxed state is undefined. The longitudinal magnetization is not in phase (because the phase is undefined) but there is a net magnetization which points in a single direction, the longitudinal direction. When that single direction is tipped down it is necessarily in phase.

Oh, I see. Classically, the net magnetization is parallel to the external magnetization when it is fully relaxed.
Then, I carefully guess that when the net magnetization is initially exposed to the external magnetic field, its longitudinal net magnetization gets bigger, and it would rotate following Bloch equations. Also, an RF pulse tips down the net magnetization because an RF pulse provides B1 field.

I thought that a particle with magnetic moment keeps precessing in the magnetic field, but it sounds like it does not. But, that's make no sense considering Zeeman effect. Maybe, here is where quantum mechanics have to come. Thanks.

 

[Dale]

Classically, the net magnetization is parallel to the external magnetization when it is fully relaxed.

Yes.

I carefully guess that when the net magnetization is initially exposed to the external magnetic field, its longitudinal net magnetization gets bigger

Yes. This happens with a rate constant T1. So, it is fully relaxed within a few seconds for most biological tissues.

Also, an RF pulse is tipped down because an RF pulse provides B1 field.

Yes.

I thought that a particle with magnetic moment keeps precessing in the magnetic field, but it sounds like it does not. But, that's make no sense considering Zeeman effect. Maybe, here is where quantum mechanics have to come. Thanks.

So all of the above was classical. Meaning not "a particle" but "an ensemble of particles". To go to "a particle" you must use quantum mechanics.

An ensemble of particles does not keep precessing in the magnetic field. Instead, it relaxes. This does not contradict the Zeeman effect in any way. The Zeeman effect places no restrictions on an ensemble of particles. All the Zeeman effect does classically is to determine the magnitude of the longitudinal magentization in the fully-relaxed ensemble.

A quantum treatment will follow in my next post.

 

[Dale]

The usual quantum treatment expresses the state of an individual spin in a "spin up" "spin down" basis. $$\Psi=a \uparrow + b \downarrow =\frac{\hbar}{2} \left( \begin{array}{c} a \\ b \\ \end{array}\right)$$ Where $a$ and $b$ are complex coefficients that are normalized such that $|a|^2+|b|^2=1$ .

Now, a single spin does precess about the longitudinal direction in the following very narrow sense. The Hamiltonian is $$H=-\frac{\hbar}{2}\left( \begin{array}{cc} \omega_0 & \omega_1 e^{i\omega t} \\ \omega_1 e^{-i\omega t} & -\omega_0 \\ \end{array}\right)$$So for $\omega_1=0$ (no RF pulse) we can solve the Schrodinger equation to get $$\Psi = \frac{\hbar}{2} \left( \begin{array}{c} a_0 e^{i \omega_0 t/2}\\ b_0 e^{i \omega_0 t/2} \\ \end{array}\right)$$ Note that the magnitude of $a$ and $b$ is constant, but there is a phase that accumulates over time. So the state, the wavefunction, of the spin has a phase that changes over time.

However, I do not particularly like to call this precession as it is not a classical precession, but an evolution of the wavefunction. Importantly, if you start with a relaxed spin in the state $a_0=1$, $b_0=0$, then although the phase of $a$ evolves, the probability of measuring the spin on either the $x$ or $y$ axes is zero. So this is not a precession in the sense of a little magnet with a definite north and south pole whose magnetic moment points in a definite direction at any point in time. When we speak of spins being out of phase, it is not this phase that we are describing.

Now, if we have an on-resonance RF pulse then we can solve the Schrodinger equation to get $$\Psi= \frac{\hbar}{2} \left( \begin{array}{c}
\left[ a_0 \cos(\omega_1 t/2) + b_0 i \sin(\omega_1 t/2) \right] e^{i \omega t/2} \\
\left[ b_0 \cos(\omega_1 t/2) + a_0 i \sin(\omega_1 t/2)\right] e^{-i \omega t/2}\\ \end{array}\right)$$ If the amplitude and duration of the RF pulse is set for a 90 degree RF pulse then at the end of the RF pulse the relaxed state has evolved from $a=1$, $b=0$ to $a=b=\sqrt{1/2}$.

Now, this is the state of a single spin, not an ensemble. In this state, the probability of measuring the spin on $y$ and $z$ is zero, and now the probability of measuring the spin on $x$ is 1. Under the free-precession Hamiltonian, this state will now evolve to gain some phase in the wavefunction as before. However, now this is legitimately precession in the sense that measurements of the spin on some axis will fluctuate sinusoidally. This is what we mean when we talk about in phase and out of phase.