3.1 Fundamental Physics and Nuclear Magnetism

Nuclear Magnetism and the Hydrogen Proton

Magnetic Resonance Imaging (MRI) is based on the interaction of atomic nuclei with an external magnetic field and radiofrequency (RF) pulses. To be active in MRI, an atomic nucleus must possess an odd mass number (the sum of protons and neutrons). Nuclei with an even number of protons and neutrons have spins that cancel each other out, resulting in no net angular momentum or magnetic moment. However, nuclei with an odd mass number possess a net nuclear spin, which gives them a microscopic magnetic field known as a magnetic dipole or magnetic moment.

While several nuclei are magnetically active—including Carbon-13 (13C), Fluorine-19 (19F), Sodium-23 (23Na), and Phosphorus-31 (31P)—clinical MRI almost exclusively targets Hydrogen-1 (1H). Hydrogen is selected because of its extreme abundance in human water (H2O) and fat molecules, and its high gyromagnetic ratio, which maximizes the strength of the induced magnetic signal.

Alignment in a Static Magnetic Field (B0)

In the absence of an external magnetic field, the magnetic moments of hydrogen protons in the body are oriented randomly. In this state, their individual magnetic vectors cancel each other out, resulting in zero net magnetization.

When the patient is placed inside the bore of the MRI scanner, they are exposed to a strong, static external magnetic field, designated as B0. Under the influence of B0, the hydrogen protons align in one of two orientations relative to the direction of B0:

• Parallel (Spin-Up): Protons align in the same direction as B0. This is the low-energy state.
• Anti-Parallel (Spin-Down): Protons align in the opposite direction of B0. This is the high-energy state.

This splitting of spins into discrete energy levels is known as Zeeman splitting. The ratio of spins in the low-energy state to the high-energy state is governed by the Boltzmann distribution, which depends on tissue temperature and B0 strength. The energy difference between the two states is extremely small. Consequently, for every million hydrogen protons, there is only a tiny excess of spins (approximately 3 per million per Tesla) in the parallel low-energy state.

Despite this small ratio, the vast quantity of hydrogen protons in human tissue (roughly 6.7 x 10^22 protons per gram of water) results in a significant macroscopic Net Magnetization Vector (NMV), denoted as M0 or Mz. The NMV represents the net balance of all individual magnetic moments. Because the parallel spins outnumber the anti-parallel spins, the NMV points in the direction of B0 along the longitudinal axis (z-axis). At equilibrium, there is no magnetization in the transverse plane (xy-plane) because the individual spins are precessing out of phase.

Precession and the Larmor Equation

A proton possesses both mass and spin, which gives it angular momentum. Because it is also a charged particle, its spin creates a magnetic moment. When a spinning proton is placed in the external B0 field, the magnetic force exerts a torque on the proton, causing it to wobble around the axis of B0. This secondary, wobbling motion is called precession, and its path describes a cone around B0.

The rate at which the magnetic moments precess is called the precessional frequency (or Larmor frequency), measured in megahertz (MHz). The relationship between the precessional frequency and the strength of B0 is defined by the Larmor Equation:

ω0 = γ × B0

Where:

• ω0 (omega): The precessional frequency of the nucleus (in MHz).
• γ (gamma): The gyromagnetic ratio, which is a constant unique to each magnetically active nucleus (expressed in MHz/Tesla). For Hydrogen-1, the gyromagnetic ratio is exactly 42.58 MHz/T.
• B0: The strength of the external magnetic field (in Tesla).

Technologists must memorize the Larmor frequencies for hydrogen at standard clinical field strengths:

• At 0.5 T, the precessional frequency is 21.29 MHz (42.58 MHz/T × 0.5 T).
• At 1.0 T, the precessional frequency is 42.58 MHz (42.58 MHz/T × 1.0 T).
• At 1.5 T, the precessional frequency is 63.87 MHz (42.58 MHz/T × 1.5 T).
• At 3.0 T, the precessional frequency is 127.74 MHz (42.58 MHz/T × 3.0 T).

This linear relationship means that doubling the magnetic field strength from 1.5 T to 3.0 T exactly doubles the precessional frequency of the hydrogen protons.

Resonance and Energy Transfer

Resonance is a physical phenomenon where a system absorbs energy most efficiently when exposed to an oscillating force that matches its natural frequency. In MRI, resonance is achieved by transmitting a radiofrequency (RF) electromagnetic pulse into the patient. This RF pulse is referred to as the excitation pulse or B1 field.

For resonance to occur, the frequency of the RF excitation pulse must exactly match the Larmor precessional frequency of the hydrogen protons. When an RF excitation pulse is applied at the Larmor frequency, two primary events occur:

1. Energy Absorption: Low-energy protons absorb energy and jump to the high-energy state. This causes the NMV to tilt away from the longitudinal axis (z-axis) toward the transverse plane (xy-plane). The angle of tilt is the flip angle, which is determined by the amplitude and duration of the RF pulse. A 90-degree RF pulse tips the NMV entirely into the transverse plane, resulting in zero longitudinal magnetization (Mz = 0) and maximum transverse magnetization (Mxy).
2. Phase Coherence (In-Phase): The RF pulse forces the magnetic moments to precess in unison, establishing phase coherence.

Signal Induction and Faraday's Law

Once the NMV is tipped into the transverse plane, it continues to precess at the Larmor frequency. Because the NMV is a moving magnetic field, its precession across a nearby conductor (such as an RF receiver coil) induces an electrical current (voltage) within the coil. This process is governed by Faraday's Law of Electromagnetic Induction. The induced voltage in the receiver coil is the raw MRI signal, known as the Free Induction Decay (FID) signal when it occurs immediately after the excitation pulse is turned off.