Chapter 15: Magnetization & Relaxation

Welcome to the MRI part. The X-ray CT of the previous chapters counts photons transmitted through the body and measures line integrals of the attenuation coefficient $\mu$. MRI (magnetic resonance imaging) measures something entirely different: the faint signal emitted by the magnetization of hydrogen nuclei (protons) sitting in a strong static magnetic field after being shaken by radio-frequency (RF) pulses. There is no ionizing radiation, and tissue contrast comes not from $\mu$ but from the time constants T1 and T2 introduced in this chapter, together with proton density.

Nuclear magnetization and Larmor precession

A hydrogen nucleus carries spin and behaves like a tiny magnet. In a static field $B_0$ the individual spin orientations are nearly random, but a slight Boltzmann bias (on the order of a few per million) produces a net magnetization $M_0$ along $B_0$. This macroscopic magnetization is the source of every MRI signal.

Tip the magnetization away from $B_0$ and it precesses around the field like a spinning top, at an angular frequency proportional to the field strength:

$$\omega_0 = \gamma B_0$$

The constant $\gamma$ is the gyromagnetic ratio; for protons $\gamma/2\pi = 42.58$ MHz/T. At 1.5 T this is about 64 MHz, at 3 T about 128 MHz — the MRI "signal" lives in roughly the FM-radio band.

RF excitation and the rotating frame

At thermal equilibrium the magnetization points along $B_0$ (the $z$ axis) and cannot be detected. We therefore apply a magnetic field $B_1$ oscillating at the Larmor frequency (an RF pulse) from a transverse direction. On resonance, viewed from a frame rotating at $\omega_0$, the precession appears to stop and the magnetization simply tips slowly around $B_1$. The tip angle (flip angle) is set by when the pulse is switched off: a 90° pulse puts the magnetization fully into the transverse plane, a 180° pulse inverts it.

The transverse magnetization $M_{xy}$ rotates at the Larmor frequency in the laboratory frame. A rotating magnet near a coil induces a voltage (Faraday's law) — and that voltage is the measured MRI signal.

The Bloch equations and relaxation

Tipped magnetization returns to equilibrium through two independent processes, described phenomenologically by the Bloch equations. In the rotating frame they separate into two lines:

$$\frac{dM_z}{dt} = \frac{M_0 - M_z}{T_1}, \qquad \frac{dM_{xy}}{dt} = -\frac{M_{xy}}{T_2}$$

T1 (longitudinal relaxation) is the time constant with which spins hand energy to the lattice (their molecular surroundings) and $M_z$ recovers. T2 (transverse relaxation) is the time constant with which spin–spin interactions randomize the phases, destroying the vector sum of the transverse magnetization. No energy is lost — only alignment — so $T_2 \le T_1$ always holds.

The large tissue-to-tissue differences in these two time constants are the source of MRI contrast. At 1.5 T, cerebrospinal fluid (CSF) has T1 ≈ 4000 ms and T2 ≈ 2000 ms, white matter T1 ≈ 590 ms and T2 ≈ 80 ms, fat T1 ≈ 250 ms and T2 ≈ 60 ms. Recall that in CT the $\mu$ of water and soft tissue differed by only a few percent — these differences are enormous by comparison. Which difference ends up in the image is chosen by the acquisition parameters (Chapter 17).

Simulation: the magnetization vector and relaxation

Press "90° pulse" to tip the magnetization into the transverse plane and watch $M_z$ recover with T1 while $|M_{xy}|$ decays with T2. Try setting T1 and T2 to tissue values — CSF and fat live on completely different time scales. A 180° pulse inverts the magnetization, and $M_z$ recovers from $-M_0$ (inversion recovery). You can also see what repeated small flip angles do.

The FID and T2*

The signal induced in the coil right after a 90° pulse is called the free induction decay (FID). Ideally it would decay with T2, but in practice it vanishes faster: the static field $B_0$ is never perfectly uniform, so the Larmor frequency varies slightly from place to place and the phases fan out. The time constant of this apparent decay is written T2*:

$$\frac{1}{T_2^*} = \frac{1}{T_2} + \frac{1}{T_2'}$$

where $T_2'$ is the contribution of the field inhomogeneity. The key point is that the $T_2'$ dephasing is static, and therefore reversible: invert the phases with a 180° pulse and the spins that were running ahead now lag behind and catch up, and the signal returns as an echo (the spin echo, Chapter 17).

Simulation: FID and T2*

Forty-eight isochromats with slightly different frequencies fan out after a 90° pulse. Confirm that the net signal (the FID) drops faster than the reference curve of true T2 decay, and that increasing the B0 inhomogeneity shortens T2*. Press "180° pulse" mid-decay and the fan folds back: the signal returns as an echo at twice the time of the pulse.

Key points

The MRI signal comes from the macroscopic magnetization of protons in a static field. It precesses at the Larmor frequency $\omega_0 = \gamma B_0$ and can be tipped to any angle by a resonant RF pulse. Tipped magnetization returns to equilibrium with T1 (longitudinal) and T2 (transverse), and the tissue differences in these constants are the source of contrast. B0 inhomogeneity produces the faster apparent decay T2*, whose static part can be rewound by a 180° pulse. But so far the signal carries no information about where it came from. Encoding position is the job of gradient fields and k-space — the next chapter.

References

• Bloch F. Nuclear Induction. Physical Review 70, 460–474 (1946).
• Hahn EL. Spin Echoes. Physical Review 80, 580–594 (1950).
• Nishimura DG. Principles of Magnetic Resonance Imaging. Stanford University (2010).
• McRobbie DW et al. MRI from Picture to Proton, 3rd ed. Cambridge University Press (2017).