The giant magnet (MRI) — setting the stage: Equation of Motion

Fig. 1: MR Scanner 1.5T (source: https://www.sqhpl.com/mri-scanners.php )

In our previous blog, we discussed what a spin is and how it contributes to the NMR signal. When a proton is placed in an external magnetic field, it aligns itself either parallel or antiparallel to the external magnetic field. Let’s refer to this external magnetic field as B₀. Additionally, it precesses around the axis of the main field with the Larmor frequency. We then defined the Larmor equation as ω₀ = γB₀.

But where does this equation come from? To answer this, we need to look to classical physics.

The precession of protons creates a tiny current loop around the axis of the B₀ field (Fig. 2).For a current loop of cross-sectional area A and a tiny current 𝒾, a small magnetic dipole moment μ is created. This magnetic moment is perpendicular to the loop, and its direction is determined using the right-hand screw rule. The direction in which the screw advances determines the direction of the moment. Every spin has a magnetic moment associated with it.

Fig. 2: Magnetic Moment created by a current loop (source: https://mriquestions.com/magnetic-dipole-moment.html )

Since the magnetic moments are in an external magnetic field, they will try to align themselves with the B₀ field. Therefore, we can say that a torque is applied to the magnetic moment by the B₀ field. This torque (N) is given by the cross product of the magnetic moment and the external B₀ field, as shown in the equation

The potential energy of the magnetic moment is given by

The negative sign in the equation accounts for the fact that the energy is lower when μ aligns with B₀ compared to the reference energy in the unaligned state.

We need to note that the equation of torque mentioned above is applicable to objects with zero angular momentum. However, this does not hold true for our spins. Since our spins rotate about their axis, they possess angular momentum (J). The torque on a particle is determined by the rate of change of angular momentum over time.

Furthermore, the relation between the magnetic moment and angular momentum is given by μ = γJ. γ is called the gyromagnetic ratio, and it depends on the type of particle.

Differentiating both sides, we get

Substituting (1) in the equation above, we get

This is the fundamental equation of motion for precession and this serves as the basis for other equations that we will come across in subsequent blogs. It is important to note that there will be significant corrections to this equation due to the interaction of spins with their environment.

In a previous discussion, we made a comparison between precession and the behavior of a spinning top. A spinning top rotates counterclockwise under the influence of gravity, as gravity pulls it downward. However, in our case, since the B₀ field is in the opposite direction, both the torque and precession are in the opposite direction. As a result, the complete representation of a single spin is depicted in Fig. 3.

Fig. 3: Precession model for a single spin in external magnetic field (source: Haacke, C. (1999). Magnetic Resonance Imaging: Physical principles and sequence design. Wiley-Liss.)

If dφ is the angle subtended by dμ, and θ is the angle between μ and B₀, then

From equation 4,

Comparing both sides of the equation, we get

dφ/dt is the angular velocity given by ω. This is our Larmor frequency.

Therefore, ω = γB₀ . While deriving this equation, it is important to note that the term B₀ has been consistently used. The strength of the main magnetic field is crucial for various reasons in this field. Depending on the requirements, higher or lower strength magnets can be utilized for the main field. Currently, most clinical scanners operate at 1.5T and 3T. Although higher field strengths are approved for human use, they are not as widely employed. Higher fields, such as 9.4T or 11.2T scanners, are predominantly utilized in a preclinical setting for research purposes.

In our upcoming blogs, we will utilize the equations discussed in this blog to delve into further principles in MRI. These equations will serve as the foundation for exploring various concepts and phenomena related to magnetic resonance imaging.

Thank you for reading. I intend to write a series of blogs explaining different concepts of Magnetic Resonance Imaging, since its easier to read it this way. As we go through the series, you will be able to tie everything together and hopefully get interested in MRI.
I learn new concepts in MRI, as I am writing this blog, so if there is any error, please do let me know.

Thanks to my mum, my dad, all my friends who took time to read this blog and my PI, Dr. Zhongliang Zu, who has something new to teach me everyday.

References:

1. Elster, A. (n.d.). MRI questions & answers; MR Imaging Physics & Technology. Questions and Answers ​in MRI. https://mriquestions.com/index.html
2. Haacke, C. (1999). Magnetic Resonance Imaging: Physical principles and sequence design. Wiley-Liss.
3. Schild, H. H. (1999). Mri made easy: (… well almost). Berlex Laboratories.